Wireless Expanders
WWireless Expanders
Shirel Attali ∗ Merav Parter † David Peleg ‡ Shay Solomon § Abstract
This paper introduces an extended notion of expansion suitable for radio networks. A graph G = ( V, E ) is said to be an ( α w , β w )- wireless expander if for every subset S ⊆ V s.t. | S | ≤ α w · | V | ,there exists a subset S (cid:48) ⊆ S s.t. there are at least β w · | S | vertices in V \ S that are adjacent in G to exactly one vertex in S (cid:48) . The main question we ask is the following: to what extent are ordinaryexpanders also good wireless expanders? We answer this question in a nearly tight manner. Onthe positive side, we show that any ( α, β )-expander with maximum degree ∆ and β ≥ / ∆ is also a( α w , β w ) wireless expander for β w = Ω( β/ log(2 · min { ∆ /β, ∆ · β } )). Thus the wireless expansion can besmaller than the ordinary expansion by at most a factor that is logarithmic in min { ∆ /β, ∆ · β } , which,in turn, depends on the average degree rather than the maximum degree of the graph. In particular, forlow arboricity graphs (such as planar graphs), the wireless expansion matches the ordinary expansionup to a constant factor. We complement this positive result by presenting an explicit construction ofa “bad” ( α, β )-expander for which the wireless expansion is β w = O ( β/ log(2 · min { ∆ /β, ∆ · β } ).We also analyze the theoretical properties of wireless expanders and their connection to uniqueneighbor expanders, and then demonstrate their applicability: Our results (both the positive and thenegative) yield improved bounds for the spokesmen election problem that was introduced in the seminalpaper of Chlamtac and Weinstein [7] to devise efficient broadcasting for multihop radio networks. Ournegative result yields a significantly simpler proof than that from the seminal paper of Kushilevitz andMansour [11] for a lower bound on the broadcast time in radio networks. ∗ The Weizmann Institute of Science. Email: [email protected] . † The Weizmann Institute of Science. Email: [email protected] . ‡ The Weizmann Institute of Science. Email: [email protected] . § IBM Research. Email: [email protected] . a r X i v : . [ c s . D S ] F e b Introduction
An expander is a sparse graph that has strong connectivity properties [10]. There are several definitionsfor expanders, with natural connections between them. We focus on the following combinatorial definition.
Expanders:
Let G = ( V, E ) be an undirected graph. For a set S ⊂ V , let Γ( S ) denote the set ofneighbors of the verices of S , and define Γ − ( S ) = Γ( S ) \ S . We say that G is an ( α, β ) expander , forpositive parameters α and β , if | Γ − ( S ) | ≥ β · | S | for every S ⊆ V s.t. | S | ≤ α · | V | .One of the main advantages of expanders is that they enable fast and effective dissemination ofinformation from a small group of vertices to the outside world. This property becomes less immediatewhen we consider using the expansion property in the context of wireless communication networks. Suchnetworks can be represented by a specific kind of graphs, called radio networks [8]. A radio network is anundirected (multihop) network of processors that communicate in synchronous rounds in the followingmanner. In each step, a processor can either transmit or keep silent. A processor receives a message ina given step if and only if it keeps silent and precisely one of its neighbors transmits in this step. If noneof its neighbors transmits, it hears nothing. If more than one neighbor (including itself) transmits in agiven step, then none of the messages is received. In this case we say that a collision occurred. It isassumed that the effect at processor u of more than one of its neighbors transmitting is the same as ofno neighbor transmitting, i.e., a node cannot distinguish a collision from silence.The usual definition of expanders is not enough to ensure fast message propagation in radio networks.Consider, for example, a radio network C + consisting of a complete graph C with one more vertex s ,the source, connected to two vertices x and y from C . Obviously this is a good expander, but in thiscase, after the first step of broadcast, if all the vertices that received the message (i.e., the three vertices s , x and y ) transmit it simultaneously to all their neighbors, then no one will hear it. This motivatesconsidering another definition of expanders, namely, unique neighbor expanders (or unique expanders, inshort) [2]. Unique neighbor expanders:
Let G = ( V, E ) be an undirected graph. We say that G is an ( α u , β u )- unique neighbor expander if for every S ⊆ V s.t. | S | ≤ α u · | V | , there are at least β u · | S | vertices in V \ S that are adjacent in G to exactly one vertex in S .Clearly, if G is a unique expander with good parameters, then broadcasting on it can be fast (again,by requiring all the vertices that received the message to send it to all their neighbors). Unfortunately,it seems that unique neighbor expansion might be hard to come by. For example, while the graph C + described above is a good (ordinary) expander, it is clearly not a good unique expander, as can be realizedby considering the set S = { x, y, s } . (In general, ordinary expanders might have rather small uniqueneighbor expansion, as will be shown soon.) In addition, explicit constructions of unique expanders arerather scarce and known only for a limited set of parameters [2, 6].The key observation triggering the current paper is that the property required from unique expandersmight be stronger than necessary. This is because there is no reason to require all the vertices that receivedthe message to send it. Rather, it may be enough to pick a subset X of this set, that has a large set ofunique neighbors, and require only the vertices of X to transmit. This may be an attractive alternativesince such a property may be easier to guarantee than unique neighbor expansion, and therefore may beachievable with better parameters α and β . (Note, e.g., that this property holds for our example graph C + .) This observation thus motivates our definition for a new variant of expanders. Wireless expanders:
Let G = ( V, E ) be an undirected graph. We say that G is an ( α w , β w )- wirelessexpander if for every S ⊆ V s.t. | S | ≤ α w · | V | , there exists a subset S (cid:48) ⊆ S s.t. there are at least β w · | S | vertices in V \ S that are adjacent to exactly one vertex in S (cid:48) .In this paper we are interested in investigating the properties of wireless expanders and the relation-ships between these graphs and the classes of ordinary expanders and unique neighbor expanders. We1sk the following questions: by how much does the relaxed definition of wireless expanders (compared tounique neighbor expanders) help us in providing expanders with better parameters that are suitable forradio network communication? More specifically, given an ( α, β )-expander, can we prove that it is alsoan ( α w , β w )-wireless expander with α w = f ( α, β ) and β w = g ( α, β ), for some functions f and g ? We present several results relating the parameters of the different notions of expanders. We begin byinvestigating the relationships between ordinary expanders and the more strict notion of unique neighborexpanders. • Let G = ( V, E ) be a d -regular graph that is an ( α u , β u )-unique neighbor expander, and let λ = λ denote the second largest eigenvalue of its adjacency matrix, given by a uv = 1 if ( u, v ) ∈ E and a uv =0 otherwise. Then G is an ( α, β )-expander with α = α u and β ≥ (1 − /d ) · β u + ( d − λ ) /d · (1 − α u ). • Suppose G = ( V, E ) is an ( α, β )-expander with maximum degree ∆. Then it is also an ( α u , β u )-unique expander with α u = α , and β u ≥ β − ∆. On the other hand, we show that there is an( α, β ) bipartite expander whose unique expansion is β u ≤ β − ∆.We then turn to consider our new relaxed notion of wireless expander. Our key contribution is in providingnearly tight characterization for the relation between ordinary expanders and wireless expanders. On thepositive side, using the probabilistic method, we show: Theorem 1.1 (Positive Result)
For every ∆ ≥ , β ≥ / ∆ , every ( α, β ) -expander G with maxi-mum degree ∆ is a also an ( α w , β w )-wireless expander with α w ≥ α and β w = Ω( β/ log(2 · min { ∆ /β, ∆ · β } )) . Our probabilistic argument has some similarity to the known decay method [5], which is a standardtechnique for coping with collisions in radio networks. Roughly speaking, in the decay protocol of [5],time is divided into phases of log n rounds and in the i th round of each phase, each node that holds amessage transmits it with probability 2 − i . Hence, each node that has a neighbor that holds a message,receives it within O (log n ) phases. We use the idea of the decay method to show the existence of a subset S (cid:48) ⊆ S with a large unique neighborhood in Γ( S ).An important feature of our argument is that it bounds the deviation of the wireless expansion from theordinary expansion as a function of the average-degree rather than the maximum degree. As β gets closerto ∆ or to 1 / ∆, this finer dependence leads to significantly better results than what could be achieved usingthe standard decay argument; our argument is also arguably simpler than the standard decay argument.As a technical note, we use the probabilistic method to prove a lower bound of Ω( β/ log(2 · ∆ /β )) on β w , and then we push it up to the bound of Theorem 1.1 via a separate deterministic argument. Asa corollary, for the important family of low arboricity graphs , which includes planar graphs and moregenerally graphs excluding a fixed minor, the wireless expansion matches the ordinary expansion up toa constant factor. (Indeed, the arboricity is at least min { ∆ /β, ∆ · β } ; see Section 2.1 for the definitionof arboricity.) In particular, this shows that radio broadcast in low arboricity graphs can be done muchmore efficiently than what was previously known!Beyond the probabilistic argument, we also provide explicit deterministic arguments that obtain betterparameters (by a constant factor); these are deferred to the appendix.We also show that asymptotically, no tighter connection can be established: Theorem 1.2 (Negative Result)
There exists an ( α, β ) -expander with maximum degree ∆ , whosewireless expansion is β w = O ( β/ log(2 · min { ∆ /β, ∆ · β } ) . Spokesman Electionproblem introduced in the seminal paper of [7], where given a bipartite graph G = ( S, N, E ), the goalis to compute a subset S (cid:48) ⊆ S with the maximum number of unique neighbors Γ ( S (cid:48) ) in N . Morespecifically, we provide tight bounds for this problem, which apply to any expansion and average degreeparameters, whereas the previous result of [7] applies only to one specific (very large) expansion parameterand only with respect to the maximum degree (rather than the average degree, which is a finer measure).In Section 4.2.1, we provide a detailed comparison to the bounds obtained by [7].Finally, another application of our negative result, and of our explicit core graph in particular, isin the context of broadcast lower bounds in radio networks. In their seminal paper, Kushilevitz andMansour [11] proved that there exist networks in which the expected time to broadcast a message isΩ( D log( n/D )), where D is the network diameter and n is the number of vertices, and this lower boundis tight for any D = Ω(log n ) due to a highly nontrivial upper bound by Czumaj and Rytter [9]. Sincethe upper bound of [9] holds with high probability, it implies that the lower bound Ω( D log( n/D )) of [11]also holds with high probability. Newport [12] presented an interesting alternative proof to the one byKushilevitz and Mansour. Although short and elegant, Newport’s proof relies on two fundamental resultsin this area, due to Alon et al. [1] and Alon et al. [3] – Lemma 3.1 in [12] – whose proof is intricate.Also, as with Kushilevitz and Mansour’s proof, Newport only proves an expected lower bound on thebroadcast time, with the understanding that a high probability bound follows from [9]. By unwindingthe ingredients of Newport’s proof, the resulting proof (especially for a high probability bound on thebroadcast time) is long and intricate. Using the properties of our explicit core graph construction, wederive a simple and self-contained proof for the same lower bound, arguably much simpler than thatof [11, 12]. An important advantage of our proof over [11, 12] is that it gives a high probability bound onthe broadcast time directly , i.e., without having to take a detour through the upper bound of [9].Summarizing, besides the mathematical appeal of wireless expanders and their connections to well-studied types of expanders, we demonstrate that they find natural applications in the well-studied areaof radio networks. We anticipate that a further study of wireless expanders will reveal additional appli-cations, also outside the scope of radio networks, and we thus believe it is of fundamental importance. In Section 2 we introduce the notation and definitions used throughout. We investigate the relationsbetween ordinary expanders and unique neighbor expanders in Section 3. Section 4 is devoted to ournew notion of wireless expanders, where we present nearly tight characterization for the relation betweenordinary expanders and wireless expanders. We start (Section 4.1) with describing our basic framework;the positive and negative results are presented in Section 4.2 and Section 4.3, respectively. (As mentioned,some positive results are deferred to the appendix. These improve on the parameters provided in Section4.2 by constant factors, using explicit deterministic arguments.) Our results for the Spokesman Electionproblem [7] are given in Section 4.2.1. Finally, Section 5 is devoted to our alternative lower bound proofof Ω( D log( n/D )) on the broadcast time in radio networks.3 Preliminaries
For an undirected graph G = ( V, E ), vertex v ∈ V and a subset S ⊆ V , denote the set of v ’s neighborsin G by Γ( v ) = { u | ( u, v ) ∈ E } , and let Γ( S ) = (cid:83) v ∈ S Γ( v ) be the neighborhood of a vertex set S in G (including neighbors that belong to S itself), and Γ − ( S ) = Γ( S ) \ S be the set of neighbors externalto S . Also define Γ( v, S ) = Γ( v ) ∩ S as the neighbors of v in the subset S . The expansion of S is theratio | Γ − ( S ) | / | S | . The unique-neighborhood of S , denoted by Γ ( S ), is the set of vertices outside S thathave a unique neighbor from S . The unique-neighbor expansion of S is the ratio | Γ ( S ) | / | S | . Let S (cid:48) be an arbitrary subset of S . The S -excluding neighborhood of S (cid:48) , denoted by Γ S ( S (cid:48) ), is the set of allvertices outside S that have at least one neighbor from S (cid:48) . Similarly, the S -excluding unique-neighborhood of S (cid:48) , denoted by Γ S ( S (cid:48) ), is the set of all vertices outside S that have a unique neighbor from S (cid:48) . Inparticular, Γ ( S ) = Γ S ( S ). The wireless expansion of S is the maximum ratio | Γ S ( S (cid:48) ) | / | S | over allsubsets S (cid:48) of S . For two sets S, T ⊂ V , let e ( S, T ) be the set of edges connecting S and T . For vertex v ∈ V , let deg( v ) = deg G ( v ) denote the degree of v in G , i.e., the number of v (cid:48) s neighbors, and let∆( G ) = max { deg( v ) | v ∈ V } be the maximum degree over all the vertices in G . For set S ⊂ V andvertex v ∈ V , let deg( v, S ) = deg( v ) ∩ S be the number of v (cid:48) s neighbors that are in S . For two vertices v, u ∈ V , let d ( u, v ) be the distance between u and v (i.e., the length of the shortest path connectingthem), and let D = D ( G ) = max { d ( u, v ) | u, v ∈ V } be the diameter of the graph, i.e. the maximumdistance between any two vertices.We use the combinatorial definition for (vertex) expansion, which requires that every (not too large)set of vertices of the graph has a relatively large set of neighbors. Specifically, an n -vertex graph G iscalled an ( α, β ) vertex expander for positive parameters α and β , if every subset S ⊆ V s.t. | S | ≤ αn hasmany external neighbors, namely, | Γ − ( S ) | ≥ β · | S | . The (ordinary) expansion β ( G ) of G is defined as theminimum expansion over all vertex sets S ⊆ V of size | S | ≤ αn , namely, β ( G ) = min {| Γ − ( S ) | / | S | | S ⊆ V, | S | ≤ αn } . A similar definition appears in the literature for bipartite graph, namely, a bipartite graph G = ( L, R, E ) with sides L and R , such that every edge from E ⊂ L × R connects one vertex of L andone vertex of R is called an ( α, β ) bipartite vertex expander if every subset S ⊂ L s.t. | S | ≤ α | L | hasat least β | S | neighbors in R . It is usually assumed that the two sides L and R of the bipartition are of(roughly) the same size.A graph G = ( V, E ) has arboricity η = η ( G ) if η = max U ⊆ V (cid:24) | E ( U ) || U | − (cid:25) , where E ( U ) = { ( u, v ) ∈ E | u, v ∈ U } . Thus the arboricity is the same (up to a factor of 2) as themaximum average degree over all induced subgraphs of G . It is easy to see that for any ( α, β )-expanderwith maximum degree ∆, the arboricity is at least min { ∆ /β, ∆ · β } . Let us now define formally the notions of unique and wireless expanders. Let G = ( V, E ) be an n -vertexundirected graph. We say that G is an ( α u , β u )- unique expander [2] if for every S ⊆ V s.t. | S | ≤ α u n , thereare at least β u · | S | vertices in V \ S that are adjacent to exactly one vertex in S , namely, | Γ ( S ) | ≥ β u · | S | .The unique-neighbor expansion β u ( G ) of G is defined as the minimum unique-neighbor expansion overall vertex sets S ⊆ V with | S | ≤ α u n , namely, β u ( G ) = min {| Γ ( S ) | / | S | | S ⊆ V, | S | ≤ α u n } . We say that G is an ( α w , β w )- wireless expander if for every S ⊆ V s.t. | S | ≤ α w n , there exists asubset S (cid:48) ⊆ S s.t. there are at least β w · | S | vertices in V \ S that are adjacent in G to exactly one vertex4n S (cid:48) , i.e., | Γ S ( S (cid:48) ) | ≥ β w · | S | . The wireless expansion β w ( G ) of G is defined as the minimum wirelessexpansion over all sets S ⊆ V with | S | ≤ α w n , namely, β w ( G ) = min { max {| Γ S ( S (cid:48) ) | / | S | | S (cid:48) ⊆ S } | S ⊆ V, | S | ≤ α w n } . In our arguments, we usually fix α and study the relations between the β -values for different notions ofexpanders. The following connection is easy to verify. Observation 2.1 If α = α u = α w , then β ( G ) ≥ β w ( G ) ≥ β u ( G ) . β and β u Let G = ( V, E ) be a d -regular undirected graph and let A = A G = ( a uv ) u,v ∈ V be its adjacency matrixgiven by a uv = 1 if ( u, v ) ∈ E and a uv = 0 otherwise. Since G is d -regular, the largest eigenvalue of A is d , corresponding to the all-1 eigenvector (as 1 /d · A is a stochastic matrix). Let λ = λ denote the secondlargest eigenvalue of G . Lemma 3.1
If a d -regular graph G = ( V, E ) is an ( α u , β u ) -unique expander, then it also an ( α, β ) -expander with α = α u and β ≥ (1 − /d ) · β u + ( d − λ ) · (1 − α u ) /d . Proof:
Alon and Spencer [4] prove that every partition of the set of vertices V into two disjoint subsets A and B satisfies | e ( A, B ) | ≥ ( d − λ ) · | A | · | B | / | V | . In our case (i.e. A = S, B = V \ S = ¯ S , and | S | ≤ α u · | V | ) we get that | e ( S, ¯ S ) | ≥ ( d − λ ) · | S | · | ¯ S || V |≥ ( d − λ ) · | S | · ( | V | − α u · | V | ) | V | = ( d − λ ) · | S | · (1 − α u ) . Moreover, by the expansion properties, there exists a set U of at least β u · | S | vertices in Γ − ( S ) thathave a unique neighbor in S . From uniqueness, we have e ( S, U ) = | U | ≥ β u | S | . Thus, there are at least( d − λ ) · | S | · (1 − α u ) − | U | edges in e ( S, ¯ S ) that are not connect to the vertices in U (i.e. in e ( S, ¯ S \ U )).Now, because G is d -regular, we get that there exist at least | U | + (( d − λ ) · | S | · (1 − α u ) − | U | ) /d verticesin Γ − ( S ). Hence, we get | Γ − ( S ) | ≥ | U | + (( d − λ ) · | S | · (1 − α u ) − | U | ) d = (cid:18) − d (cid:19) · | U | + ( d − λ ) · (1 − α u ) d · | S |≥ (cid:18) − d (cid:19) · β u | S | + ( d − λ ) · (1 − α u ) d · | S | = (cid:18)(cid:18) − d (cid:19) · β u + ( d − λ ) · (1 − α u ) d (cid:19) · | S | thus, G is a ( α, β )-expander with α ≥ α u and β ≥ (1 − /d ) · β u + (( d − λ ) · (1 − α u ) /d .It is known (and easy to verify) that ordinary expanders whose expansion is close to the (maximum)degree in the graph are also good unique expanders, or formally: Lemma 3.2
Suppose G = ( V, E ) is an ( α, β ) -expander with maximum degree ∆ . Then it is also a unique ( α u , β u ) -expander, with α u = α and β u ≥ β − ∆ . emark. Substituting β = (1 − ε )∆ (for ε ≤ / β u ≥ (1 − ε )∆.The lower bound 2 β − ∆ on the unique-neighbor expansion β u provided by Lemma 3.2 is meaningfulonly when β is larger than ∆ /
2. The following example shows that this lower bound 2 β − ∆ is tight. Lemma 3.3
For any ∆ and β such that ∆ / ≤ β ≤ ∆ , there is an ( α, β ) bipartite expander G bad =( S, N, E ) with maximum degree ∆ whose unique expansion is β u ≤ β − ∆ . Proof:
Construct the graph G bad as follows. Let S = { v , . . . , v s } , with s = | S | , and suppose thateach vertex v i ∈ S has exactly ∆ neighbors, all of which are in N . (For technical convenience, we define v = v s , v = v s +1 ; that is, the vertices v and v s are not different than the other vertices (they shouldnot be viewed as “endpoints”, but rather part of an implicit “cycle”). Moreover, for each i = 1 , . . . , s ,the vertices v i and v i +1 have exactly ∆ − β common neighbors; that is, | Γ( v i ) ∩ Γ( v i +1 ) | = ∆ − β . Moreconcretely, writing Γ( v i ) = { v i , . . . , v ∆ i } , we have thatΓ( v i ) ∩ Γ( v i +1 ) = { v β +1 i , . . . , v ∆ i } = { v i +1 , . . . , v ∆ − βi +1 } . In other words, the “last” ∆ − β neighbors v β +1 i , . . . , v ∆ i of v i are the “first” ∆ − β neighbors v i +1 , . . . , v ∆ − βi +1 of v i +1 , respectively. (See Figure 1 for an illustration.)Figure 1: An illustration of a worst-case scenario for the unique-neighbor expansion.
This means that for each i = 1 , . . . , s , the first (resp., last) ∆ − β neighbors of v i are also neighbors of v i − (resp, v i +1 ). The remaining ∆ − − β ) = 2 β − ∆ neighbors of v i , however, are uniquely covered by v i . It follows that the number of vertices in the neighborhood of S that are uniquely covered by verticesfrom S is equal to s (2 β − ∆). Consequently, the unique neighbor expansion β u is 2 β − ∆, as claimed.Noting that the ordinary expansion is β completes the proof of the lemma. Remarks. (1) The meaning of Lemma 3.3 is that a graph with high (ordinary) expansion may haveunique neighbor expansion of zero. For example, in the graph G bad described in the proof of Lemma 3.3,the unique-neighbor expansion is 2 β − ∆, but the wireless expansion is at least max { β − ∆ , ∆ / } . Tosee that, let S (cid:48) be a subset of S and suppose S (cid:48) = S ∪ S ∪ . . . ∪ S k such that each S i is a sequenceof consecutive vertices, i.e., using the previous notations, for S i of size l , S i = { v j , .., v j + l } for someindex 1 ≤ j ≤ s . Suppose also that between every two sets S i and S j there is at least one vertexthat is not in S (cid:48) (in other words, we can’t expand S i to be a longer secuence in S (cid:48) ). Therefore, tocompute β w , it is enough to compute the expansion parameter for each S i = { v j , . . . , v j + l } . Considertwo options for choosing the set S (cid:48)(cid:48) ⊂ S i . The first choice is to take S (cid:48)(cid:48) = S i . Then we get an expansion6f f ( l ) = ( l ∆ − l − − β )) /l = ((2 − l )∆ + 2( l − β ) /l . The second choice is to take into S (cid:48)(cid:48) every second vertex in the sequence of S i . Then we get an expansion of g ( l ) = l ∆ / (2 l ) if l iseven, and g ( l ) = ( l + 1)∆ / (2 l ) if l is odd. (In the case where S (cid:48) = S we get in the first choice anexpansion of f ( l ) = l (2 β − ∆) /l = 2 β − ∆ and in the second an expansion of g ( l ) = ( l − / (2 l )).Thus, β w ≥ min { max { g ( l ) , f ( l ) } | l > } . As f ( l ) and g ( l ) are both decreasing functions, we getthat β w ≥ max { lim l →∞ g ( l ) , lim l →∞ f ( l ) } = max (cid:8) β − ∆ , ∆2 (cid:9) . This calculation also shows that if β = ∆ /
2, then the unique-neighbor expansion becomes 0, but the wireless expansion becomes ∆ / S and N differ bya factor of β . Also, it does not provide an ordinary non-bipartite expander, because the expansion isachieved only on one side, from S towards N . Nevertheless, one can plug this “bad” bipartite graph ontop of an ordinary ( α, β )-expander with a possibly good unique-neighbor expansion, so that the graphresulting from this tweak is an ordinary ( α, β )-expander with a unique-neighbor expansion bounded by2 β − ∆. Notice, however, that the maximum degree in the resulting graph, denoted by ∆ (cid:48) , may be aslarge as the sum of the maximum degrees of the “bad” bipartite graph and the ( α, β )-expander that westarted from. For example, if ∆ (cid:48) = 2∆, then the unique-neighbor expansion of the resulting graph isbounded by 2 β − ∆ = 2 β − ∆ (cid:48) /
2. Since we apply a similar tweak in Section 4.3 (in the context of wirelessexpansion rather than unique expansion), we omit the exact details of this rather simple tweak from theextended abstract.
Consider an arbitrary (ordinary) ( α, β )-expander G . As shown in Section 3, the unique-neighbor ex-pansion β u provided by G may be zero even if the ordinary expansion β is high. In what follows wedemonstrate that the wireless expansion β w ( G ) of G cannot be much lower than its ordinary expansion β ( G ). Moreover, we prove asymptotically tight bounds on the ratio β ( G ) /β w ( G ). This yields a strongseparation between the unique-neighbor expansion and the wireless expansion, which provides a natu-ral motivation for studying wireless expanders, particularly in applications where we are given a fixed expander network (that cannot be changed).First let us observe that by Obs. 2.1, Lemma 3.2 yields the following bound on β w . Lemma 4.1
Suppose G = ( V, E ) is an ( α, β ) -expander with maximum degree ∆ . Then it is also awireless ( α w , β w ) -expander with α w = α and β w ≥ β − ∆ . Throughout what follows, we simplify the discussion by focusing attention to an arbitrary bipartitegraph G S = ( S, N, E S ) with sides S and N , such that | N | ≥ β · | S | . We assume that no vertex of G S isisolated, i.e., all vertex degrees are at least 1.Note that this bipartition can be thought of as representing all edges in the original graph G thatconnect an arbitrary vertex set S with its neighborhood N = Γ − ( S ). While in G there might be edgesinternal to S and/or N , ignoring these edges has no effect whatsoever on the expansion bounds.Our goal is to show the existence of a subset S (cid:48) of S in the graph G S , whose S -excluding unique-neighborhood Γ S ( S (cid:48) ) is not much smaller than the entire neighborhood N of S . Of course, this wouldimply that the wireless expansion of an arbitrary set S in G (of any size) is close to its ordinary expansion,yielding the required result. Let δ S (resp., δ N ) be the average degree of the set S (resp., N ) in the graph G S . That is, δ S = (cid:80) u ∈ S deg( u, N ) / | S | and δ N = (cid:80) u (cid:48) ∈ N deg( u (cid:48) , S ) / | N | . Clearly, δ N , δ S ≥
1. In this section, we show that β w can be bounded from below as a function of min { δ S , δ N } .7e begin by considering an ( α, β )-expander G for β ≥
1. We now show:
Lemma 4.2
For every β, ∆ ≥ , there exists a subset S ∗ ⊆ S , satisfying that | Γ S ( S ∗ ) | = Ω( | N | / log 2 δ N ) = Ω( β/ log 2 δ N ) · | S | . Hence, β w = Ω( β/ log 2(∆ /β )) . Proof:
Since β ≥
1, we have | S | ≤ | N | , 1 ≤ δ N ≤ δ S and δ N ≤ ∆ /β . The proof relies on theprobabilistic method. First, consider the set N (cid:48) of all vertices from N with degree at most 2 δ N . Notethat | N (cid:48) | ≥ | N | / N (cid:48) have positive degree. We now divide the subset N (cid:48) into k = (cid:98) log 2 δ N (cid:99) subsets depending on their degree in S , where the i th subset N i consists of all vertices u ∈ N (cid:48) with deg( u, S ) ∈ [2 i , i +1 ). Let N j be the largest subset among these k subsets. We have that | N j | ≥ | N | /k = Ω( | N | / log 2 δ N ) = Ω( | N | / log 2(∆ /β )). We next show that there exists a subset S ∗ ⊆ S such that Γ S ( S ∗ ) contains a constant fraction of the vertices of N j .Consider a random subset S (cid:48) ⊆ S obtained by sampling each vertex u ∈ S independently withprobability 1 / j . For every vertex u ∈ N j , let X ( u ) ∈ { , } be the indicator random variable that takesvalue 1 if u has exactly one neighbor in S (cid:48) . As deg( u, S ) ∈ [2 j , j +1 ), we have thatIE (() X ( u )) = IP( X ( u ) = 1) = deg( u, S ) / j · (1 − / j ) deg( u,S ) − ≥ (1 − / j ) j +1 − ≥ e − . Hence, (cid:80) u ∈ N j IE (() X ( u )) = Ω( | N j | ) = Ω( β | S | / log 2(∆ /β )). We get that the expected number of verticesin N that are uniquely covered by a random subset S (cid:48) is Ω( β | S | / log 2(∆ /β )). Hence, there exists a subset S ∗ ⊆ S with | Γ S ( S ∗ ) | = Ω( β/ log 2(∆ /β )) · | S | . The lemma follows.In Appendix A, we provide a sequence of deterministic arguments that obtain better bounds for β w (byconstant factors) compared to the probabilistic argument shown above.We now turn to consider the case β <
1. In this case the bound on the wireless expansion dependson δ S , namely, on the average degree in the larger set S . We show: Lemma 4.3
For every ∆ ≥ and β ∈ [1 / ∆ , , there exists a subset S ∗ ⊆ S , satisfying that | Γ S ( S ∗ ) | =Ω( β/ log δ S ) · | S | . Since δ S ≤ β · ∆ , we have β w = Ω( β/ log 2(∆ · β )) . Proof:
Let S (cid:48) ⊆ S be the set of all vertices u ∈ S with deg( u, N ) ≤ δ S , and note that | S (cid:48) | ≥ | S | /
2. Let N (cid:48) = Γ − ( S (cid:48) ) be the set of neighbors of S (cid:48) in N . By the expansion of G , we have | N (cid:48) | ≥ β · | S (cid:48) | ≥ β | S | / S (cid:48)(cid:48) ⊆ S (cid:48) satisfying Γ − ( S (cid:48)(cid:48) ) = N (cid:48) and | S (cid:48)(cid:48) | ≤ | N (cid:48) | . To see this,initially set S (cid:48)(cid:48) to be empty. Iterate over the vertices of S (cid:48) and add a vertex u ∈ S (cid:48) to S (cid:48)(cid:48) only if itcovers a new vertex of N (cid:48) (i.e., it has a new neighbor in N (cid:48) that has not been covered before). Then | S (cid:48)(cid:48) | ≤ | N (cid:48) | and hence in the induced bipartite graph G (cid:48) with sides S (cid:48)(cid:48) and N (cid:48) , the expansion measure β ’, with β (cid:48) = | N (cid:48) | / | S (cid:48)(cid:48) | , is at least 1. The average degree of a vertex u ∈ N (cid:48) in the graph G (cid:48) is boundedby | E ( G (cid:48) ) | / | N (cid:48) | ≤ δ S · | S (cid:48)(cid:48) | / | N (cid:48) | ≤ δ S . Employing the argument of Lemma 4.2 on the bipartite graph G (cid:48) , we get that there exists a subset S ∗ ⊆ S (cid:48)(cid:48) satisfying | Γ S (cid:48)(cid:48) ( S ∗ ) | = Ω( | N (cid:48) | / log 4 δ S ) = Ω( β/ log 2 δ S ) | S | .Since δ S ≤ ∆ · β , it follows that β w = Ω( β/ log 2(∆ · β )).Theorem 1.1 follows from Lemmas 4.2 and 4.3. Motivated by broadcasting in multihop radio networks, Chalmtac and Weinstien [7] defined the spokesmenelection problem . In this problem, given a bipartite graph G = ( S, N, E ), the goal is to compute a subset S (cid:48) ⊆ S with the maximum number of unique neighbors Γ ( S (cid:48) ) in N . This problem was shown in [8]to be NP-hard. In [7], an approximation scheme is presented that computes a subset S (cid:48) ⊆ S with | Γ ( S (cid:48) ) | ≥ | N | / log | S | , and this approximation scheme was then used to devise efficient broadcastingalgorithms for multihop radio networks. 8he bounds provided in Lemmas 4.2 and 4.3 refine and strengthen upon the bound of [7]. Our boundsshow that | Γ ( S (cid:48) ) | cannot be smaller than | N | by more than a factor that is logarithmic in 2 min { δ N , δ S } ,which depends on the average degree in G , whereas the bound of 4.3 did not preclude the possibility of | Γ ( S (cid:48) ) | being smaller than | N | by a factor of log | S | . Note that min { δ N , δ S } is always upper bounded by | S | , but can be much smaller than it. In particular, min { δ N , δ S } is always low in low arboricity graphs(even if the maximum degree is huge), regardless of | S | .We remark that our randomized approach of choosing the subset S (cid:48) ⊆ S is extremely simple, and inparticular, it yields a much simpler solution to the Spokesman Election problem than that of [7]. Sincethe solution to this problem was used in [7] to devise efficient broadcasting algorithms for multihop radionetworks, our solution can be used to obtain simpler broadcasting algorithms for multihop radio networksthan those of [7].In the next section (Section 4.3), we show that our positive results for ( α, β )-expanders are essentiallythe best that one can hope for, by providing a “bad” expander example. A bad graph expander examplefor the related Spokesman Election problem was given in [7], but our graph example is stronger than thatof [7] in several ways, and is based on completely different ideas. The graph example of [7] is tailoredfor the somewhat degenerate case where | N | = Ω( | S | !), whence N is exponentially larger than S , thusthe expansion of the bad graph (and the degree) is huge. In addition, in their example, one cannotuniquely cover more than | N | / log( | S | ) = | N | / log log | N | vertices of N , leaving a big gap between theirpositive and negative results. Our bad graph example, in contrast, works for any expansion parameter β . Moreover, similarly to our positive result, the bounds implied by our negative result depend on theaverage degree of the graph rather than the maximum degree or the size of S . In particular, by taking β to be constant and ∆ to be sufficiently large, our graph example shows that one cannot cover more than | N | / log | N | vertices of N , which not only matches our positive result, but also closes the gap left by [7]. In this section we present a “bad graph ” expander construction. The description of our construction isgiven in three stages. First, in Section 4.3.1 we construct a bipartite graph G S = ( S, N, E S ) with sides S and N that satisfies two somewhat contradictory requirements: On the one hand, for every subset S (cid:48) of S , | Γ( S (cid:48) ) | ≥ log 2 | S | · | S (cid:48) | . Hence the ordinary expansion of G S , denoted by β , is at least log 2 | S | .On the other hand, for every subset S (cid:48) of S , | Γ S ( S (cid:48) ) | ≤ (2 / log 2 | S | ) · | N | . Hence the wireless expansionof G S , denoted β w , satisfies β w ≤ β (2 / log 2 | S | ). Although this graph is an ordinary bipartite expander(according to the definition given in Section 2.1), note that the size of N is greater than that of S by afactor of log 2 | S | . Also, it does not provide an ordinary non-bipartite expander, because the expansionis achieved only on one side, from S towards N . Nevertheless, it provides the core of our worst-caseexpander, and is henceforth referred to as the core graph . Next, in Section 4.3.2 we describe a generalizedcore graph G ∗ S = ( S ∗ , N ∗ , E ∗ S ) with an arbitrary expansion β ∗ , while preserving the same upper boundon the wireless expansion. Finally, in Section 4.3.3 we plug the generalized core graph on top of anordinary expander G ( V, E ) with a possibly good wireless expansion, such that N ∗ ⊆ V and S ∗ ∩ V = ∅ ,and demonstrate that the resulting graph ˜ G = ( V ∪ S ∗ , E ∪ E ∗ S ) is an ordinary expander with a similarexpansion but a poor wireless expansion. While the generalized core graph is bipartite, the ordinaryexpander G that we started from does not have to be bipartite. If the original expander G is bipartite,we can ensure that the expander resulting from our modification will also be bipartite. For any integer s ≥ , there is a bipartite graph G S = ( S, N, E S ) such that:1. s := | S | and | N | = s log 2 s .2. Each vertex in S has degree s − . . The maximum degree ∆ N of a vertex in N is s , and the average degree δ N of a vertex in N is atmost s/ log 2 s .4. For every subset S (cid:48) of S , | Γ( S (cid:48) ) | ≥ log 2 s · | S (cid:48) | . (Hence the ordinary expansion, denoted β , is atleast log 2 s .)5. For every subset S (cid:48) of S , | Γ S ( S (cid:48) ) | ≤ s = (2 / log 2 s ) · | N | . (Hence the wireless expansion, denoted β w , satisfies β w ≤ β (2 / log 2 s ) .) Proof:
We assume for simplicity that s is an integer power of 2, which may effect the bounds in thestatements of the lemma by at most a small constant. To describe the edge set E S of G S , consider aperfect binary tree T S with s leaves (and s − z of T S with aunique vertex of S . Each vertex v of T S is associated with a set N v of vertices from N ; all these vertex setsare pairwise disjoint, and we have N = (cid:83) v ∈ T S N v . For a vertex v at level i of the tree, i = 0 , , . . . , log s ,the set N v contains s/ i vertices. Thus the sizes of these vertex sets decrease geometrically with thelevel, starting with the set N rt at the root rt that consists of s vertices, and ending with singletons atthe leaves. Denote by N i the union of the sets N v over all i -level vertices in T S . For all i = 0 , , . . . , log s ,we have | N i | = s , hence | N | = s log 2 s . For a leaf z in T S , let A ( z ) denote the set of its ancestors in T S (including z itself), and let ˆ N z = (cid:83) w ∈ A ( z ) N w . Define E ( z ) = { ( z, v ) | v ∈ ˆ N z } . Then E S = (cid:83) z ∈ S E ( z ).(See Fig. 2 for an illustration.) Figure 2 Observation 4.5
There is an edge between vertex z ∈ S and vertex v ∈ N iff the unique vertex w in T S such that v ∈ N w is an ancestor of z in T S . Note that the degree of each vertex z ∈ S , namely | E ( z ) | , is equal to (cid:80) log si =0 i = 2 s −
1. On the otherhand, the degrees of vertices in N are not uniform. For a vertex v in T S , each vertex in N v is incident onthe descendant leaves of v . This means that if v is at level i of T S , then all vertices in N v have degree2 log s − i = s/ i . Hence, the maximum degree ∆ N of a vertex in N is s and the average degree δ N of avertex in N is given by δ N = 1 | N | ( log s (cid:88) i =0 | N i | ( s/ i ))10 1 | N | ( log s (cid:88) i =0 s i ) ≤ s s log 2 s = 2 s log 2 s . Next, we lower bound the expansion β of the graph G S . Fix an arbitrary set S (cid:48) ⊆ S of size k , for any1 ≤ k ≤ s , and consider the set of k leaves in T S identified with S (cid:48) , denoted by s , . . . , s k . Recall thatthe level of the root rt is 0, the level of its children is 1, etc., the level of the leaves of T S is log s ; inwhat follows we say that a vertex has inverse-level j if its level in T S is log s − j . For each vertex v atinverse-level j in T S , the associated vertex set N v has size 2 j . Next, we distinguish between inverse-levelsat most (cid:98) log k (cid:99) and higher inverse-levels. For any inverse-level 0 ≤ j ≤ (cid:98) log k (cid:99) , the number of ancestorsof the k leaves s , . . . , s k in the tree T S is at least k/ j , hence the union of the corresponding vertex sets isof size at least k . (The lower bound is realized when the k leaves are consecutive to each other in T S .) Foreach inverse-level higher than (cid:98) log k (cid:99) , the number of ancestors of the k leaves s , . . . , s k may be as smallas 1, but the vertex set associated with such an ancestor is of size at least k . It follows that the union ofthe corresponding vertex sets at each level is lower bounded by k , and so the union of the vertex sets of allancestors of the k leaves s , . . . , s k over all levels is at least (log s +1) · k . By Observation 4.5, all the verticesin this union are neighbors of the vertices in S (cid:48) , thereby yielding | Γ( S (cid:48) ) | ≥ (log s + 1) · k = log 2 s · | S (cid:48) | . Itfollows that β ≥ log 2 s .It remains to upper bound the wireless expansion β w of the graph G S . Fix an arbitrary set S (cid:48) ⊆ S ,and recall that Γ S ( S (cid:48) ) denotes the set of all vertices outside S that have a single neighbor from S (cid:48) .For a vertex v in T S , let D ( v ) denote the set of its descendants in T S (including v itself), and letˇ N v = (cid:83) w ∈ D ( v ) N w . We argue that for any vertex v at inverse-level j , for j = 0 , , . . . , log s , it holds that | Γ S ( S (cid:48) ) ∩ ˇ N v | ≤ j +1 −
1. The proof is by induction on j . Basis j = 0 . In this case v is a leaf, henceˇ N v = N v = { v } , and so | Γ S ( S (cid:48) ) ∩ ˇ N v | ≤ j +1 − Induction step: Assume the correctness of thestatement for all smaller values of j , and prove it for j . Consider an arbitrary vertex v at level j , anddenote its left and right children by v L and v R , respectively. Suppose first that S (cid:48) contains at least oneleaf z L from the subtree of v L and at least one leaf z R from the subtree of v R . By Observation 4.5, everyvertex in N v is incident to both z L and z R , hence no vertex of N v belongs to Γ S ( S (cid:48) ). It follows thatΓ S ( S (cid:48) ) ∩ ˇ N v = (Γ S ( S (cid:48) ) ∩ ˇ N v L ) ∪ (Γ S ( S (cid:48) ) ∩ ˇ N v R ) . By the induction hypothesis, we conclude that | Γ S ( S (cid:48) ) ∩ ˇ N v | = | Γ S ( S (cid:48) ) ∩ ˇ N v L | + | Γ S ( S (cid:48) ) ∩ ˇ N v R |≤ · (2 j − ≤ j +1 − . We henceforth assume that no leaf in the subtree of either v L or v R , without loss of generality v L ,belongs to S (cid:48) . Hence, by Observation 4.5 again, no vertex of ˇ N v L belongs to Γ( S (cid:48) ) ⊇ Γ S ( S (cid:48) ), which givesΓ S ( S (cid:48) ) ∩ ˇ N v = (Γ S ( S (cid:48) ) ∩ N v ) ∪ (Γ S ( S (cid:48) ) ∩ ˇ N v R ) . Obviously | (Γ S ( S (cid:48) ) ∩ N v ) | ≤ | N v | = 2 j . By the inductionhypothesis, we obtain | Γ S ( S (cid:48) ) ∩ ˇ N v | = | Γ S ( S (cid:48) ) ∩ N v | + | Γ S ( S (cid:48) ) ∩ ˇ N v R | ≤ j + 2 j − j +1 − . Thiscompletes the proof of the induction.Since ˇ N rt = N , applying the induction statement for the root rt of T S yields | Γ S ( S (cid:48) ) | = | Γ S ( S (cid:48) ) ∩ ˇ N rt | ≤ log s +1 − ≤ s = (2 / log 2 s ) · | N | . It follows that β w ≤ β (2 / log 2 s ), which completes the proof of the lemma. Notice that the expansion of the graph provided by Lemma 4.4 is logarithmic in the size of its vertex setand also in the maximum and average degree (both in S and in N ). In what follows we show how toconstruct a generalized core graph that has an arbitrary expansion. Lemma 4.6
For any integer ∆ ∗ ≥ and any β ∗ satisfying (2 e ) / ∆ ∗ ≤ β ∗ ≤ ∆ ∗ / (2 e ) (where e is thebase of the natural logarithm), there exists a bipartite graph G ∗ S = ( S ∗ , N ∗ , E ∗ S ) with sides S ∗ and N ∗ ofmaximum degree ∆ ∗ , such that . | S ∗ | ≤ ∆ ∗ / , | N ∗ | = β ∗ · | S ∗ | .2. For every subset S (cid:48) of S ∗ , | Γ( S (cid:48) ) | ≥ β ∗ · | S (cid:48) | . (Thus, ordinary expansion is at least β ∗ .)3. For every subset S (cid:48) of S ∗ , | Γ S ∗ ( S (cid:48) ) | ≤ (4 / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } )) ·| N ∗ | . (Hence the wireless expansion, denoted β w , satisfies β w ≤ β ∗ (4 / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } )) .) To prove Lemma 4.6, we first present the following two lemmas which generalize Lemma 4.4 to get anarbitrary expansion.
Lemma 4.7
For any integer s ≥ and any β > log 2 s , there exists a bipartite graph ˆ G S = ( S, ˆ N , ˆ E S ) such that1. s := | S | and | ˆ N | = s · β .2. Each vertex in S has degree (2 s − · ( β/ log 2 s ) .3. The maximum degree ∆ ˆ N of a vertex in ˆ N is s , and the average degree δ ˆ N of a vertex in ˆ N is atmost s/ log 2 s .4. For every subset S (cid:48) of S , | Γ( S (cid:48) ) | ≥ β · | S (cid:48) | . (Hence the ordinary expansion is at least β .)5. For every subset S (cid:48) of S , | Γ S ( S (cid:48) ) | ≤ s · ( β/ log 2 s ) = (2 / log 2 s ) ·| ˆ N | . (Hence the wireless expansion,denoted β w , satisfies β w ≤ β (2 / log 2 s ) .) Proof:
We assume for simplicity that k = β/ log 2 s is an integer, and modify the construction usedto prove Lemma 4.4 by creating k copies v , . . . , v k for each vertex v in N . Thus each vertex set N v is “expanded” by a factor of k ; denote the expanded vertex set by ˆ N v . The vertex set ˆ N of ˆ G S is theunion of all copies of all vertices in N , or in other words, it is the union of all the expanded vertex sets,i.e., ˆ N = (cid:83) v ∈ T S ˆ N v . The edge set ˆ E S of ˆ G S is obtained by translating each edge ( v, u ) in the originalgraph G S , where v ∈ N , into the k edges ( v , u ) , . . . , ( v k , u ) in ˆ G S . Other than this modification, theconstruction remains intact. Note that S remains unchanged, and the degree of vertices in ˆ N is the sameas the degree of vertices in N in the original graph G S (both the maximum and average degree). On theother hand, we now have | ˆ N | = ( s log 2 s ) · ( β/ log 2 s ) = s · β . Moreover, the expansion increases from atleast log 2 s to at least β , and the degree of vertices in S increases from 2 s − s − · ( β/ log 2 s ).Finally, note that for every subset S (cid:48) of S , | Γ S ( S (cid:48) ) | increases by a factor of β/ log 2 s , hence | Γ S ( S (cid:48) ) | is atmost 2 s · ( β/ log 2 s ) = (2 / log 2 s ) · | ˆ N | , thus the wireless expansion β w satisfies β w ≤ β (2 / log 2 s ). Lemma 4.8
For any integer s ≥ and any β ≤ log 2 s , there exists a bipartite graph ˇ G S = ( ˇ S, N, ˇ E S ) with sides ˇ S and N , such that1. | ˇ S | = s · (log 2 s/β ) and | N | = s log 2 s .2. Each vertex in ˇ S has degree s − .3. The maximum degree ∆ N of a vertex in N is s · (log 2 s/β ) , and the average degree δ N of a vertexin N is at most s/β .4. For every subset S (cid:48) of ˇ S , | Γ( S (cid:48) ) | ≥ β · | S (cid:48) | . (Hence the ordinary expansion is at least β .)5. For every subset S (cid:48) of ˇ S , | Γ S ( S (cid:48) ) | ≤ s = (2 / log 2 s ) · | N | . (Hence the wireless expansion, denoted β w , satisfies β w ≤ β (2 / log 2 s ) .) roof: We assume for simplicity that k = log 2 s/β is an integer, and modify the construction used toprove Lemma 4.4 by creating k copies v , . . . , v k for each vertex v in S . The vertex set ˇ S of ˇ G S is the unionof all copies of all vertices in S , and the edge set ˇ E S is obtained by translating each edge ( v, u ) in theoriginal graph G S , where v ∈ S , into the k edges ( v , u ) , . . . , ( v k , u ) in ˇ G S . Other than this modification,the construction remains intact. Note that N remains unchanged, and the degree of vertices in ˇ S is thesame as the degree of vertices in S in the original graph G S (both the maximum and average degree).On the other hand, we now have | ˇ S | = s · (log 2 s/β ). Moreover, the expansion decreases from at leastlog 2 s to at least β , and the degree of vertices in N increases by a factor of log 2 s/β . Finally, note thatfor every subset S (cid:48) of ˇ S , | Γ S ( S (cid:48) ) | remains at most 2 s = (2 / log 2 s ) · | N | , thus the wireless expansion β w remains unchanged, satisfying β w ≤ β (2 / log 2 s ).We are now ready to complete to proof of Lemma 4.6. Proof: [Lemma 4.6] Since β ∗ ≤ ∆ ∗ / (2 e ), we may write ∆ ∗ = 2 s · ( β ∗ / log 2 s ), for s ≥ e . Suppose firstthat β ∗ > log 2 s . In this case we take G ∗ S to be the graph provided by Lemma 4.7 for (cid:100) s (cid:101) and β ∗ = β ;we assume for simplicity that s is an integer, but this assumption has a negligible effect. The maximumdegree in the graph is (2 s − · ( β ∗ / log 2 s ), which is bounded by ∆ ∗ := 2 s · ( β ∗ / log 2 s ). This in particularyields ∆ ∗ ≥ s , and so | S ∗ | = s ≤ ∆ ∗ /
2. We also have | N ∗ | = β ∗ · | S ∗ | . The second assertion followsimmediately from Lemma 4.7(4). It remains to prove the third assertion. Lemma 4.7(5) implies that forevery subset S (cid:48) of S ∗ , | Γ S ∗ ( S (cid:48) ) | ≤ s · ( β ∗ / log 2 s ) = (2 / log 2 s ) · | N ∗ | . Observe thatmin { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } = ∆ ∗ /β ∗ = 2 s/ log 2 s ≤ s. Hence 2 / log 2 s ≤ / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } ), which implies that | Γ S ∗ ( S (cid:48) ) | ≤ (2 / log 2 s ) · | N ∗ |≤ (2 / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } )) · | N ∗ | . We henceforth assume that β ∗ ≤ log 2 s . Since β ∗ ≥ (2 e ) / ∆ ∗ , we may write ∆ ∗ = 2 s (cid:48) · (log 2 s (cid:48) /β ∗ ), for s (cid:48) ≥ e/
2. Next, we argue that β ∗ ≤ log 2 s (cid:48) . Since β ∗ ≤ log 2 s and as ∆ ∗ is equal to both 2 s · ( β ∗ / log 2 s )and 2 s (cid:48) · (log 2 s (cid:48) /β ), it follows that((2 s (cid:48) ) / (2 s )) log(2 s (cid:48) ) log(2 s ) = ( β ∗ ) ≤ log (2 s ) . Thus (2 s (cid:48) ) · log(2 s (cid:48) ) ≤ (2 s ) · log(2 s ), and so s (cid:48) ≤ s . Next, we prove that (2 s (cid:48) ) / log(2 s (cid:48) ) ≤ (2 s ) / log(2 s )by taking logarithms for both hand sides and noting that the function f ( x ) = x − log x is monotoneincreasing for x > log e and that s ≥ s (cid:48) ≥ e/
2. Rearranging, we get( β ∗ ) = ((2 s (cid:48) ) / (2 s )) log(2 s (cid:48) ) log(2 s ) ≤ log (2 s (cid:48) ), thus β ∗ ≤ log 2 s (cid:48) .In this case we take G ∗ S to be the graph provided by Lemma 4.8 for (cid:100) s (cid:48) (cid:101) and β ∗ = β ; we again assumefor simplicity that s is an integer, but this assumption has a negligible effect. The maximum degreein the graph is max { s (cid:48) − , s (cid:48) · (log 2 s (cid:48) /β ) } , which is bounded by ∆ ∗ := 2 s (cid:48) · (log 2 s (cid:48) /β ∗ ). Note that | S ∗ | = s (cid:48) · (log 2 s (cid:48) /β ∗ ) = ∆ ∗ / | N ∗ | = s (cid:48) log 2 s (cid:48) = β ∗ · | S ∗ | . The second assertion follows immediatelyfrom Lemma 4.8(4). It remains to prove the third assertion. Lemma 4.8(5) implies that for every subset S (cid:48) of S ∗ , | Γ S ∗ ( S (cid:48) ) | ≤ s (cid:48) = (2 / log 2 s (cid:48) ) · | N ∗ | . Observe thatmin { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } ≤ ∆ ∗ · β ∗ = 2 s (cid:48) · log 2 s (cid:48) . Hence 2 / log 2 s (cid:48) = 4 / log((2 s (cid:48) ) ) ≤ / log(2 s (cid:48) · log 2 s (cid:48) ) ≤ / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } ) , which implies that | Γ S ∗ ( S (cid:48) ) | ≤ (2 / log 2 s (cid:48) ) · | N ∗ |≤ (4 / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } )) · | N ∗ | . .3.3 Worst-Case Expanders Let G be an arbitrary ( α, β )-expander on n vertices with maximum degree ∆, and let 0 < (cid:15) < / (cid:15) will determine the extent by which the parameters of interest blowup due to the modification that we perform on the original graph G to obtain poor wireless expansion.There is a tradeoff between the wireless expansion and the other parameters: The stronger our upperbound on the wireless expansion is, the larger the blow-up in the other parameters becomes.For technical reasons, we require that ∆ · β ≥ / (1 − (cid:15) ). We start by constructing the generalizedcore graph G ∗ S = ( S ∗ , N ∗ , E ∗ S ) provided by Lemma 4.6 for ∆ ∗ = (cid:15) · ∆ and expansion β ∗ = β/(cid:15) , thusyielding | S ∗ | ≤ ∆ ∗ / (cid:15) (∆ /
2) and | N ∗ | = β ∗ · | S ∗ | = ( β/(cid:15) ) · | S ∗ | . Our worst-case expander ˜ G is obtainedby plugging G ∗ S on top of G . The vertices of S ∗ are not part of the original vertex set of G , but are rathernew vertices added to it. The vertices of N ∗ are chosen arbitrarily from V ( G ). Remark. If G is a bipartite expander, expanding from the left side L to the right side R , and if we want˜ G to remain bipartite and to expand from ˜ L to ˜ R , then ˜ L will be defined as the union of L and S ∗ , and˜ R will be defined as the union of R and a dummy vertex set of the same size as S ∗ , to guarantee that | ˜ L | = | ˜ R | .In what follows we analyze the properties of ˜ G . Denoting the number of vertices in ˜ G by ˜ n , we have n ≤ ˜ n ≤ n + 2 | S ∗ | ≤ n + 2 (cid:15) (∆ / ≤ (1 + (cid:15) ) · n. Write ˜∆ = (1 + (cid:15) ) · ∆, and note that the maximumdegree in ˜ G is bounded by ∆ + ∆ ∗ ≤ ∆ + (cid:15) · ∆ = ˜∆ . Claim 4.9 ˜ G is an ordinary ( ˜ α, ˜ β ) -expander, where ˜ β = (1 − (cid:15) ) · β, ˜ α = (1 − (cid:15) ) · α . Proof:
Since ˜ n < (1 + (cid:15) ) · n and as ˜ α = (1 − (cid:15) ) · α , it follows that ˜ α · ˜ n ≤ (1 − (cid:15) ) α · (1 + (cid:15) ) · n =(1 − (cid:15) ) α · n < α · n. Consider an arbitrary set X of at most ˜ α · ˜ n ≤ α · n vertices from ˜ G . By Lemma 4.6(2),the expansion in G ∗ S is at least β ∗ = β/(cid:15) , hence | Γ − ( X ∩ S ∗ ) | ≥ ( β/(cid:15) ) · | X ∩ S ∗ | . If | X ∩ S ∗ | ≥ (cid:15) · | X | ,then we have | Γ − ( X ) | ≥ | Γ − ( X ∩ S ∗ ) | ≥ ( β/(cid:15) ) · | X ∩ S ∗ | ≥ ( β/(cid:15) ) · ( (cid:15) · | X | ) = β · | X | > ˜ β · | X | . Otherwise, | X \ S ∗ | ≥ (1 − (cid:15) ) · | X | , and as the expansion in G is at least β , we have | Γ − ( X ) | ≥ | Γ − ( X \ S ∗ ) | ≥ β · | X \ S ∗ | ≥ β · (1 − (cid:15) ) · | X | = ˜ β · | X | . Recall that ∆ · β ≥ / (1 − (cid:15) ), and note that ˜∆ · ˜ β = (1 + (cid:15) )∆ · (1 − (cid:15) ) β ≥
1. We also havethat ˜∆ / ˜ β > ∆ /β ≥
1. Hence the term log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ) is non-negative, and the upper bound O ( ˜ β/ ( (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) in the following claim is well-defined. Claim 4.10
The wireless expansion ˜ β w of ˜ G satisfies ˜ β w = O ( ˜ β/ ( (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) . Proof:
Note that ˜ β w is trivially upper bounded by β , thus the claim holds vacuously whenever (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ) <
2. We may henceforth assume that (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ) ≥ , which impliesthat both ˜∆ / ˜ β and ˜∆ · ˜ β are at least 2 /(cid:15) . Since (cid:15) < /
2, it follows that∆ ∗ · β ∗ = ∆ · β ≥ ( ˜∆ / (1 + (cid:15) )) · ( ˜ β/ (1 − (cid:15) )) ≥ ˜∆ · ˜ β ≥ /(cid:15) ≥ e and ∆ ∗ /β ∗ = (cid:15) (∆ /β ) ≥ (cid:15) ( ˜∆ / (1 + (cid:15) )) / ( ˜ β/ (1 − (cid:15) ))= (cid:15) ((1 − (cid:15) ) / (1 + (cid:15) )) · ( ˜∆ / ˜ β ) ≥ (cid:15) ((1 − (cid:15) ) / (1 + (cid:15) )) · /(cid:15) ≥ e. In particular, we have (2 e ) / ∆ ∗ ≤ β ∗ ≤ ∆ ∗ / (2 e ), as required in Lemma 4.6. Since all edges adjacent tothe vertices of S ∗ belong to the core graph G ∗ S with parameters ∆ ∗ and β ∗ , Lemma 4.6(3) implies thatfor every subset S (cid:48) of S ∗ , | Γ S ∗ ( S (cid:48) ) | ≤ (4 / log(min { ∆ ∗ /β ∗ , ∆ ∗ · β ∗ } )) · | N ∗ | (4(1 + (cid:15) ) / ( (cid:15) (1 − (cid:15) ) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) · | N ∗ | . ≤ (12 / ( (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) · | N ∗ | . = (12 / ( (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) · β · | S ∗ | . ≤ (24 / ( (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) · ˜ β · | S ∗ | . (It is easily verified that the third and last inequalities hold for (cid:15) < / (cid:15) . Corollary 4.11
For any n, ∆ , β and < (cid:15) < / such that ∆ · β ≥ / (1 − (cid:15) ) , if there exists an ordinary ( α, β ) -expander G on n vertices with maximum degree ∆ , then there exists an ( ˜ α, ˜ β ) -expander ˜ G on ˜ n vertices with maximum degree ˜∆ and wireless expansion ˜ β w , where: (1) ∆ ≤ ˜∆ ≤ (1 + (cid:15) ) · ∆ ; (2) n ≤ ˜ n ≤ (1 + (cid:15) ) · n ; (3) ˜ β = (1 − (cid:15) ) · β ; (4) ˜ α = (1 − (cid:15) ) · α ; and (5) ˜ β w = O ( ˜ β/ ( (cid:15) · log(min { ˜∆ / ˜ β, ˜∆ · ˜ β } ))) . One may use Corollary 4.11 in conjunction with known constructions of explicit expanders (such asRamanujan graphs), which achieve near-optimal expansion for any degree parameter. Taking (cid:15) to be asufficiently small constant thus completes the proof of Theorem 1.2.
In this section we provide a simple proof for obtaining a tight lower bound of Ω( D log( n/D )) on thebroadcast time in radio networks, which holds both in expectation and with high probability.Consider our core bipartite graph G S = ( S, N, E S ) from Lemma 4.4, with sides S and N , where s = | S | and | N | = s log 2 s . Suppose that we connect an additional vertex rt to all vertices of S andinitiate a (radio) broadcast at rt in the resulting graph. By Lemma 4.4(5), one cannot uniquely covermore than 2 s vertices (i.e., a (2 / (log 2 s ))-fraction) of N using any subset S (cid:48) ⊆ S . It follows that at anyround after the first, the broadcast may reach at most 2 s new vertices of N , which yields the followingcorollary. Corollary 5.1
The number of rounds needed for the broadcast to reach a (2 i/ (log 2 s )) -fraction of N isat least i , for any ≤ i ≤ ((log 2 s ) / . Next, we construct a graph G of diameter Θ( D ), for an arbitrary parameter D = Ω(log n ), in whichthe number of rounds needed to complete a broadcast is Ω( D log( n/D )).The core graph G S has | S | + | N | = s (1 + log 2 s ) = s (log 4 s ) vertices. We take D/ G S , G S , . . . , G D/ S , each containing roughly n/D vertices. Thus we take s so that n/D ≈ s (log 4 s ), and so log s = Θ(log( n/D )). Denote the sides of G iS by S i and N i . We connect the root rt = rt to all vertices of S , and for each 1 ≤ i ≤ D/
2, we randomly sample a vertex from N i , denotedby rt i , and connect it (unless i = D/
2) to all vertices of S i +1 . This completes the construction of thegraph G . It is easy to verify that the diameter of G is Θ( D ), and to be more accurate, the diameter is D + 2. In what follows we assume that none of the processors associated with the vertices of the graphinitially have any topological information on the graph (except for its size and diameter). This ratherstandard assumption was also required in the proof of Kushilevitz and Mansour [11].Consider a broadcast initiated at rt . We make the following immediate observation. Observation 5.2
The message must reach rt i − before reaching rt i , for ≤ i ≤ D/ . Denote by R i the random variable for the number of rounds needed for the message to be sent from rt i − to rt i , for each i , and let R be the random variable for the number of rounds needed to send the messagefrom rt to rt D/ . We thus have R = R + R + . . . + R D/ .15y Corollary 5.1, the number of rounds needed for the broadcast message to reach half of the verticesof N (from rt = rt ) is at least ((log 2 s ) /
4) + 1 = Θ(log( n/D )). Since rt was sampled randomlyfrom all vertices N and as none of the processors have any topological information on the graph, rt received this message within this many rounds with probability at most 1 /
2, hence R = Ω(log( n/D ))with constant probability. By Observation 5.2, the only way for the message to reach any vertex of S , and later rt , is via rt , hence we can repeat this argument, and carry it out inductively. Sincethe D/ R , R , . . . , R D/ are independently and identically distributed, and as D = Ω(log n )(where the constant hiding in the Ω-notation is sufficiently large), a Chernoff bound implies that IP( R =Ω( D log( n/D ))) ≥ − n − c , where c is a constant as big as needed. For the expectation bound, notethat IE (() R i ) > (log 2 s ) / n/D )) by Corollary 5.1, for each i , and by linearity of expectationwe obtain IE (() R ) = IE (() R ) + IE (() R ) + . . . + IE (() R D/ ) = Ω( D log( n/D )). (The assumption that D = Ω(log n ) is used for deriving the high probability bound but not the expectation bound.) Acknowledgments.
We are grateful to Mohsen Ghaffari for the useful discussions on the probabilistic arguments of Section4.2.
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Distributed Computing - 28th InternationalSymposium, DISC 2014, Austin, TX, USA, October 12-15, 2014. Proceedings , pages 258–272, 2014. ppendix A Deterministic and Constructive Analysis with Improved Bounds
A.1 Bounds depending on the maximum degree
A.1.1 A naive approach
In this section we provide a simple argument showing that when the maximum degree is small, the wirelessexpansion β w is not much smaller than the ordinary expansion β . Recall that we consider an arbitrarybipartite graph G S = ( S, N, E S ) with sides S and N , such that | N | = β · | S | . We assume that no vertexof G S is isolated, i.e., all vertex degrees are at least 1. In what follows we define s = | S | , γ = | N | . Lemma A.1 In G S = ( S, N, E S ) , if the maximum degree is ∆ , then there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ/ ∆ . Proof:
We describe a procedure for computing vertex sets S uni ⊆ S and N uni ⊆ N , such that | N uni | ≥ γ/ ∆and every vertex of N uni has a unique neighbor in S uni .Initialize N uni = S uni = ∅ , N tmp = N, S tmp = S . At each step of the procedure, the sets N uni and S uni (respectively, N tmp and S tmp ) grow (resp., shrink). The procedure maintains the following invariantthroughout. Invariant: (I1) S tmp ∪ S uni ⊆ S and S tmp ∩ S uni = ∅ .(I2) N tmp ∪ N uni ⊆ N and N tmp ∩ N uni = ∅ .(I3) Every vertex of N uni has a unique neighbor in S uni .(I4) Every vertex of N tmp has at least one neighbor in S tmp , but has no neighbor in S uni .For a vertex x ∈ N tmp , recall that Γ( x, S tmp ) is the set of neighbors of x in S tmp . At each step wepick a vertex v ∈ N tmp minimizing | Γ( v, S tmp ) | , i.e., a vertex with a minimum number of neighbors in S tmp . (By invariant ( I | Γ( v, S tmp ) | ≥ Q v be the set of all vertices in N tmp thatare incident on at least one vertex of Γ( v, S tmp ). By the choice of v , for any vertex u in Q v satisfyingΓ( u, S tmp ) ⊆ Γ( v, S tmp ), we must have Γ( u, S tmp ) = Γ( v, S tmp ). We partition Q v into two subsets Q (cid:48) v and Q (cid:48)(cid:48) v , where Q (cid:48) v contains all vertices u for which Γ( u, S tmp ) = Γ( v, S tmp ) and Q (cid:48)(cid:48) v contains the remainingvertices of Q v (all of which must have a neighbor in S tmp \ Γ( v, S tmp )). Obviously we have Q (cid:48) v ⊇ { v } , so | Q (cid:48) v | ≥ w of Γ( v, S tmp ) from S tmp to S uni ; note that w is incidenton all vertices of Q (cid:48) v . Then we remove all other vertices of Γ( v, S tmp ) from S tmp , which prevents thesevertices from entering S uni later on, thus guaranteeing that all vertices in Q (cid:48) v will have w as their uniqueneighbor in S uni . Subsequently, all vertices of Q (cid:48) v are moved from N tmp to N uni . (See Figure 3 for anillustration.)In addition, to prevent violating invariant ( I
4) now and invariant ( I
3) in the future, all neighbors of w that belong to Q (cid:48)(cid:48) v are removed from N tmp (they are incident to w which has just moved to S uni , and theymight have neighbors in S tmp that will be moved to S uni later on). It is clear that the first three invariants( I − ( I
3) continue to hold following this step. As for invariant ( I u of N tmp at the beginning of this step. We know that u had no neighbors in S uni at the beginning of thestep. If u is not a neighbor of w , then u had no neighbors in S uni also at the end of the step. Otherwise,If u is a neighbor of w , then its only new neighbor in S uni at the end of the step is w and u was removedfrom N tmp (it either moves to N uni if it belongs to Q (cid:48) v , or it is removed altogether if it belongs to Q (cid:48)(cid:48) v ).This shows that every vertex u of N tmp has no neighbor in S uni at the end of the step. Next, if u has aneighbor outside Γ( v, S tmp ), then this neighbor remains in S tmp following the step (since only the verticesiigure 3: An illustration of a single step of the procedure. The dashed lines represent edges that connectvertices in Q v with vertices in S tmp , where v is a vertex in N tmp minimizing | Γ( v, S tmp ) | . The vertices in Q (cid:48) v are colored black, and they move from N tmp to N uni ; the vertices in Q (cid:48)(cid:48) v are colored green, and they areremoved from N tmp ; the vertices in Γ( v, S tmp ) are colored red, and they are removed from S tmp , except for w which moves to S uni . of Γ( v, S tmp ) are removed from S tmp during the step). Otherwise, we have Γ( u, S tmp ) ⊆ Γ( v, S tmp ), whichby the choice of v implies that Γ( u, S tmp ) = Γ( v, S tmp ). By definition, u ∈ Q (cid:48) v , and is thus removed from N tmp during the step. This shows that at the end of this step, every vertex of N tmp has at least oneneighbor in S tmp , so ( I
4) holds.This procedure terminates once N tmp = ∅ . By invariant ( I N uni has a uniqueneighbor in S uni . At each step of the procedure, we move | Q (cid:48) v | ≥ N tmp to N uni , all of whichare neighbors of some vertex w ∈ Γ( v, S tmp ), and remove some of the other (at most ∆ −
1) neighbors of w from N tmp . Consequently, at least one vertex among every ∆ vertices removed from N tmp must moveto N uni . Since initially we have N tmp = N , it follows that | N uni | ≥ γ/ ∆.Note that the proof of this lemma takes into account the maximum degree ∆ S of a vertex in S , ratherthan the maximum degree ∆ in the entire graph. Corollary A.2
Suppose G is an ( α, β ) -expander with maximum degree ∆ . Then it is also an ( α w , β w )-wireless expander, with α w = α and β w ≥ β/ ∆ . A.1.2 Procedure Partition
Our next goal is to strengthen Corollary A.2. In this section we describe a procedure, hereafter namedProcedure
Partition , which lies at the core of our lower bounds on the wireless expansion. This procedureis then employed in various scenarios to conclude that the wireless expansion is close to the ordinaryexpansion. The procedure partitions N into N uni , N many , N tmp and S into S uni and S tmp , such that thefollowing conditions hold. (In what follows we refer to these conditions as the “partition conditions”.)(P1) Every vertex of N uni has a unique neighbor in S uni .(P2) Every vertex of N tmp has at least one neighbor in S tmp , but has no neighbor in S uni .(P3) | N uni | ≥ | N many | .(P4) Either N tmp = ∅ , or | E tmp | ≤ | E uni | holds, where E uni (resp., E tmp ) denotes the set of edgesconnecting all vertices in S tmp with vertices in N uni (resp., N tmp ).iit the outset, we initialize N uni = N many = S uni = ∅ , N tmp = N, S tmp = S . At each step of theprocedure, the sets N uni and S uni grow and the set N tmp and S tmp shrink. The set N many also grows, butnot necessarily at each step; it contains “junk” vertices that once belonged to N uni , but were removedfrom N uni due to new vertices added to S uni .The first three aforementioned conditions are maintained throughout the execution of the procedure.(Notice that initially all three of them hold trivially.) On the other hand, condition ( P
4) is required tohold only when the procedure terminates.For a vertex x ∈ S tmp , denote by N tmp ( x ) (resp., N uni ( x )) the set of neighbors of x in N tmp (resp., N uni ).At each step we pick a vertex v ∈ S tmp maximizing gain ( v ) := | N tmp ( v ) | − | N uni ( v ) | . Assuming gain ( v ) >
0, we move v from S tmp to S uni ; to preserve condition ( P N uni ( v )from N uni to N many . Next, we move all vertices of N tmp ( v ) from N tmp to N uni . Since gain ( v ) > P
3) holds. The reason condition ( P
2) holds is because once a vertex of S tmp moves to S uni ,all its neighbors in N tmp are moved to N uni . Obviously the sets N uni , N many , N tmp (resp., S uni , S tmp ) forma partition of N (resp., S ).Procedure Partition terminates once S tmp becomes empty or once gain ( v ) ≤ v ∈ S tmp . Inthe former case, condition ( P
2) implies that N tmp = ∅ , and we are done. In the latter case, we have | N tmp ( v ) | ≤ | N uni ( v ) | for any v ∈ S tmp , yielding | E tmp | = (cid:88) v ∈ S tmp | N tmp ( v ) | ≤ (cid:88) v ∈ S tmp | N uni ( v ) | = 2 | E uni | . (1)(See Figure 4 for an illustration.)Figure 4: An illustration of the edge sets E uni and E tmp which connect S tmp to N uni and N tmp , respectively.The vertices in N tmp ( v ) and N uni ( v ) , as well as the edges connecting them to v , are colored red; here we have gain ( v ) = | N tmp ( v ) | − | N uni ( v ) | = − . A.1.3 Constructive lower bound for β w in terms of the average degree Let N = Γ − ( S ) and γ = | N | , and denote by δ the average degree of a vertex in N , i.e., δ = (1 /γ ) (cid:80) v ∈ N deg( v, S ).(As all vertex degrees are at least 1, we have δ ≥ δ rather than the maximumdegree ∆. Lemma A.3
In the graph G S there exists a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ/ (8 δ ) . iii roof: Denote by N δ the set of vertices of N = Γ − ( S ) with degree at most 2 δ . Observe that at leasthalf the vertices of N have degree at most twice the average, implying that | N δ | ≥ γ/
2. We applyProcedure
Partition , but consider the vertex set N δ rather than N . Thus we obtain a partition of N δ rather than N into N δ uni , N δ many , N δ tmp and a partition of S into S uni and S tmp satisfying the partitionconditions ( P − ( P | N δ uni | ≥ γ/ (8 δ ).Suppose first that the procedure terminates because N δ tmp = ∅ . By partition condition ( P | N δ uni | ≥| N δ uni | + | N δ many | = | N δ | . It follows that | N δ uni | ≥ | N δ | ≥ γ ≥ γ δ . (2)We henceforth assume that | E tmp | ≤ | E uni | . By definition, each vertex in N δ has at most 2 δ neighborsin S . Condition ( P
1) implies that each vertex in N δ uni has a single neighbor in S uni , so it has at most2 δ − S tmp , yielding | E uni | ≤ (2 δ − | N δ uni | . By condition ( P N δ tmp isincident on at least one edge of E tmp , and so | E tmp | ≥ | N δ tmp | . It follows that | N δ tmp | ≤ | E tmp | ≤ | E uni | ≤ (4 δ − | N δ uni | . Hence, 4 δ · | N δ uni | = (2 + (4 δ − | N δ uni |≥ | N δ uni | + | N δ many | + | N δ tmp | = | N δ | ≥ γ , which yields | N δ uni | ≥ γ/ (8 δ ) . For every S ⊂ V denote by δ S the average degree of a vertex in N = Γ − ( S ), i.e., δ S = (1 / | N | ) (cid:80) v ∈ N deg( v, S )and denote ¯ δ = max { δ S | S ⊂ V, | S | ≤ αn } . Corollary A.4
Let G = ( V, E ) be an ( α, β ) -expander. Then(1) G is an ( α w , β w ) -wireless expander with α w = α and β w ≥ β/ (8¯ δ ) ≥ β/ (8∆) , where ∆ is the maximumdegree in the graph.(2) In the regime β ≥ , we have δ S ≤ ∆ /β , for every S such that | S | ≤ αn , thus ¯ δ ≤ ∆ /β and we get β w ≥ β / (8∆) . A.1.4 “Convenient” degree constraints
The following lemmas show that if many vertices in Γ − ( S ) have roughly the same degree, then theordinary expansion β and the wireless expansion β w of G S are roughly the same. Lemma A.5 In G S , for any c > and any i ∈ { , , . . . , log c | S |} , there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ | N ( i ) | / c ) , where N ( i ) denotes the set of vertices in N with degree in [ c i − , c i ) for i < log c | S | and for i = log c | S | is the set of vertices in N with degree in [ c i − , c i ] = [ | S | /c, | S | ] . Corollary A.6 In G S , for any c > there is a subset S (cid:48) of S such that | Γ S ( S (cid:48) ) | ≥ log c c ) log ∆ · γ . Proof:
The previous lemma implies also that for every c > i ∈ { , , . . . , log c ∆ } (rather thenlog c | S | ), there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ | N ( i ) | / c ), where N ( i ) denotes the set of verticesin N with degree in [ c i − , c i ) for i < log c ∆ and for i = log c ∆ is the set of vertices in N with degree iniv c i − , c i ] = [∆ /c, ∆]. Observe that there exists an index j ∈ { , , . . . , log c ∆ } s.t. | N ( j ) | ≥ γ/ log c ∆, forthis index j , we get | Γ S ( S (cid:48) ) | ≥ | N ( j ) | / c ) ≥ γ c ) log c ∆ .The maximum of f ( c ) = log c/ (2(1 + c )) is attained at c ≈ . ≈ . Corollary A.7
Let G = ( V, E ) be an ( α, β ) -expander with maximum degree ∆ . Then it is also an ( α w , β w ) -wireless expander with α w ≥ α and β w ≥ . ∆ · β . A.2 Bounds depending on the average degree
Recall that δ denotes the average degree of a vertex in N = Γ − ( S ). In case δ is known, we can state astronger bound than that of Corollary A.7, using δ in place of ∆. Corollary A.8 In G S , for any c > and t > there is a subset S (cid:48) of S such that | Γ S ( S (cid:48) ) | ≥ (cid:18) − t (cid:19) c ) log c ( tδ ) · γ. Corollary A.9 In G S , for every (cid:15) > , and for sufficiently large δ , there is a subset S (cid:48) of S such that | Γ S ( S (cid:48) ) | ≥ . (cid:15) ) log ( δ ) · γ. Corollary A.10
Let G = ( V, E ) be an ( α, β ) -expander with maximum degree ∆ and let (cid:15) > . Supposethat for every S , δ S is large enough. Then G is also an ( α w , β w ) -wireless expander with α w ≥ α and β w ≥ . (cid:15) ) log (¯ δ ) · β . Proof:
Given S ⊂ V with | S | ≤ α | V | , write γ = | Γ − ( S ) | and let G S = ( S, Γ − ( S ) , e ( S, Γ − ( S )). Notethat as G is an ( α, β )-expander, γ ≥ β | S | and by Corollary A.9, there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ . (cid:15) ) log ( δ S ) · γ ≥ . (cid:15) ) log ( δ S ) · β | S | ≥ . (cid:15) ) log (¯ δ ) · β | S | . Hence β w ≥ . (cid:15) ) log (¯ δ ) · β. Lemma A.11
Suppose there exists c > and t > such that for every subset N (cid:48) of N in G S ofsufficiently large size (say, of size at least ( γ/ − /t ) ), the average degree δ (cid:48) of a vertex in N (cid:48) is atleast tδ/c . Then there is a subset S (cid:48) of S such that | Γ S ( S (cid:48) ) | ≥ γ c ) (cid:18) − t (cid:19) . Proof:
We apply Procedure
Partition , but consider the vertex set N tδ of vertices in N = Γ − ( S )with degree at most tδ . Thus we obtain a partition of N tδ rather than N into N tδ uni , N tδ many , N tδ tmp and apartition of S into S uni and S tmp satisfying the partition conditions ( P − ( P | N tδ uni | ≥ | N tδ | / c ). This complete the proof as | N tδ | ≥ γ (1 − /t ).If | N tδ tmp | < ( γ/ − /t ), as | N tδ | ≥ γ (1 − /t ) and by using partition condition ( P
3) we get2 | N tδ uni | ≥ | N tδ uni | + | N tδ many | ≥ ( γ/ − /t ), hence | N tδ uni | ≥ ( γ/ − /t ) ≥ ( γ/ (2(1 + c )))(1 − /t ). δ that that satisfies (cid:15) ln( δ ) − ln(ln δ ) − ln(1 + (cid:15) ) − ≥ i.e., satisfies (cid:15) ln( δ S ) − ln(ln δ S ) − ln(1 + (cid:15) ) − ≥ vtherwise, if | N tδ tmp | ≥ ( γ/ − /t ), in particular nonempty and it must hold that | E tmp | ≤ | E uni | . Bydefinition, each vertex in N tδ has at most tδ neighbors in S . Condition ( P
1) implies that each vertex in N tδ uni has a single neighbor in S uni , so it has at most tδ − S tmp , yielding | E uni | ≤ ( tδ − | N tδ uni | . By condition ( P N tδ tmp is incident only on edges of E tmp . Since | N tδ tmp | ≥ ( γ/ − /t ),the average degree in this set is at least tδ/c . Therefore, | E tmp | ≥ ( tδ/c ) | N tδ tmp | . It follows that tδc | N δ tmp | ≤ | E tmp | ≤ | E uni | ≤ tδ − | N tδ uni | . Hence 2 (cid:18) tδc + tδ (cid:19) | N tδ uni | ≥ (cid:18) · tδc + 2( tδ − (cid:19) | N tδ uni |≥ tδc · ( | N tδ uni | + | N tδ many | + | N tδ tmp | )= tδc | N tδ | , which yields | N tδ uni | ≥ | N tδ | c ) . Corollary A.12
Let G = ( V, E ) be an ( α, β ) -expander and suppose there exists c > and t > suchthat for every subset S of V of size | S | ≤ αn and for every subset M of Γ − ( S ) of sufficiently large size(say, of size at least ( | Γ − ( S ) | / − /t ) ), the average degree δ (cid:48) of a vertex in M is at least ( tδ S ) /c . Then G is also an ( α w , β w ) -wireless expander with α w ≥ α and β w ≥ β c ) (cid:18) − t (cid:19) . Proof:
The proof follows similar lines as those in the proof of Corollary A.10.
A.2.1 Near-optimal boundsLemma A.13 In G S there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ/ (9 log(2 δ )) . Proof:
We prove the existence of vertex sets S uni ⊆ S and N uni ⊆ N = Γ − ( S ), such that | N uni | ≥ γ/ (9 log(2 δ )) and every vertex of N uni has a unique neighbor in S uni . The proof is by induction on γ , forall values of δ ≥
1. (Since δ ≥
1, we have log(2 δ ) ≥ Basis: γ ≤ . Let v be an arbitrary vertex of S with at least one neighbor in N , let S uni = { v } , and let N uni be the (non-empty) neighborhood of v . We thus have | N uni | ≥ ≥ γ/ (9 log(2 δ )). Induction step: Assume the correctness of the statement for all smaller values of γ , and prove it for γ . We apply Procedure
Partition (with the bipartite graph induced by the sets S and N ). If the procedureterminates because N tmp = ∅ , then we have | N uni | ≥ | N | / N tmp (cid:54) = ∅ , i.e., γ (cid:48) = | N tmp | ≥
1. In particular, it must hold that | E tmp | ≤ | E uni | . Denote by δ (cid:48) the average degree of a vertex in N tmp , counting only neighbors that belong to S tmp . By partition condition ( P N tmp is contained in S tmp ; confusing as itmight be, we do not make use of this property here. We do use, however, another property guaranteed bypartition condition ( P N tmp has at least one neighbor in S tmp , which implies that δ (cid:48) ≥ δ (cid:48) ) ≥
1. Since N tmp is non-empty, it must hold that | E tmp | ≥
1. Hence | E uni | ≥ | E tmp | / ≥ / | E uni | ≥
1. Consequently, we have | N uni | ≥
1, which in turn yields 1 ≤ γ (cid:48) ≤ γ − γ (cid:48) / log(2 δ (cid:48) ) ≥ γ/ log(2 δ ). By the induction hypothesis for γ (cid:48) (restricting ourselvesto the subgraph of G S induced by the vertex sets S tmp and N tmp ), we conclude that there is a subset ˜ S of S tmp with | Γ S tmp ( ˜ S ) ∩ N tmp | ≥ γ (cid:48) / (9 log(2 δ (cid:48) )), yielding | Γ S ( ˜ S ) | ≥ | Γ S tmp ( ˜ S ) ∩ N tmp | ≥ γ (cid:48) δ (cid:48) ) ≥ γ δ ) . We may henceforth assume that γ (cid:48) log(2 δ (cid:48) ) < γ log(2 δ ) . (3)Observe that | E uni | + | E tmp | ≤ | E S | = δ · γ . By definition, | E tmp | = δ (cid:48) · γ (cid:48) . It follows that3 δ (cid:48) · γ (cid:48) = 3 | E tmp | ≤ | E uni | + | E tmp | ) ≤ δ · γ, yielding log(2 δ (cid:48) ) ≤ log(2 δ ) + log (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19) . (4)Plugging Equation (4) into Equation (3), we obtain γ (cid:48) < γ log(2 δ ) (cid:18) log(2 δ ) + log (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19)(cid:19) . (5)We may assume that | N uni | < γ/
9, as otherwise | N uni | ≥ γ/ ≥ γ/ (9 log(2 δ )) and we are done. Bypartition condition ( P | N uni | ≥ | N many | . Hence γ = | N uni | + | N many | + γ (cid:48) ≤ | N uni | + γ (cid:48) , yielding( γ − γ (cid:48) ) / ≤ | N uni | < γ/
9. Hence 2 / γ/γ (cid:48) ) ≤ /
7, which giveslog (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19) ≤ log (cid:18) (cid:19) ≤ − . It follows that γ log(2 δ ) (cid:18) log(2 δ ) + log (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19)(cid:19) ≤ γ log(2 δ ) (cid:18) log(2 δ ) − (cid:19) . (6)Plugging Equation (6) into Equation (5) gives γ (cid:48) ≤ γ log(2 δ ) (cid:18) log(2 δ ) − (cid:19) = γ − γ δ ) ≤ | N uni | + γ (cid:48) − γ δ ) , implying that | N uni | ≥ γ/ (9 log(2 δ )). Corollary A.14
Let G = ( V, E ) be an ( α, β ) -expander. Then,(1) G is an ( α w , β w ) -wireless expander with α w = α and β w ≥ β/ (9 log(2¯ δ )) ≥ β/ (9 log(2∆)) , where ∆ is the maximum degree in the graph.(2) In the regime β ≥ , we have δ S ≤ ∆ /β , thus ¯ δ ≤ ∆ /β , and hence β w ≥ β/ (9 log(2∆ /β )) . Corollary A.15 In G S there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ min (cid:26) γ δ , γ (cid:27) . vii roof: We prove that if δ < S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ/
20 and if δ ≥ S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ/ (9 log δ ). The proof is by induction on γ . Basis: γ ≤ . If δ <
2, by Lemma A.13 there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ/ (9 log(2 δ )) > γ/ ≥ γ/
20. For δ ≥
2, let v be an arbitrary vertex of S with at least one neighbor in N , let S (cid:48) = { v } , then | Γ( v ) | = | Γ S ( S (cid:48) ) | ≥ ≥ γ/ (9 log δ ). Induction step: Assume the correctness of the statement for all smaller values of γ , and prove it for γ . If δ <
2, then the same proof holds as in the basis case. Let assume δ ≥ δ ≥
1. Weapply Procedure
Partition (with the bipartite graph induced by the sets S and N ). If the procedureterminates because N tmp = ∅ , then we have | N uni | ≥ | N | / N tmp (cid:54) = ∅ , i.e., γ (cid:48) = | N tmp | ≥
1. In particular, it must hold that | E tmp | ≤ | E uni | . Denote by δ (cid:48) the average degree of a vertex in N tmp , counting only neighbors that belongto S tmp . By partition condition ( P
2) each vertex of N tmp has at least one neighbor in S tmp , whichimplies that δ (cid:48) ≥
1, thus log(2 δ (cid:48) ) ≥
1. Since N tmp is non-empty, it must hold that | E tmp | ≥
1. Hence | E uni | ≥ | E tmp | / ≥ /
2, yielding | E uni | ≥
1. Consequently, we have | N uni | ≥
1, which in turn yields1 ≤ γ (cid:48) ≤ γ − δ (cid:48) <
2. By Lemma A.13, there is a subset S (cid:48) of S s.t. | Γ S ( S (cid:48) ) | ≥ | Γ S tmp ( S (cid:48) ) ∩ N tmp | ≥ γ (cid:48) δ (cid:48) ) ≥ γ (cid:48) . (7)If γ (cid:48) < γ (9 / γ = | N uni | + | N many | + γ (cid:48) ≤ | N uni | + γ (cid:48) , we get | N uni | ≥ γ/
20. So we can assume γ (cid:48) ≥ γ (9 / | Γ S ( S (cid:48) ) | ≥ γ (cid:48) ≥ γ . The second case is when δ (cid:48) ≥ δ ≥ γ (cid:48) / log δ (cid:48) ≥ γ/ log δ . By the induction hypothesis for γ (cid:48) (restricting ourselves to thesubgraph of G S induced by the vertex sets S tmp and N tmp ), we conclude that there is a subset ˜ S of S tmp with | Γ S tmp ( ˜ S ) ∩ N tmp | ≥ γ (cid:48) / (9 log δ (cid:48) ), yielding | Γ S ( ˜ S ) | ≥ | Γ S tmp ( ˜ S ) ∩ N tmp | ≥ γ (cid:48) δ (cid:48) ≥ γ δ . We may henceforth assume that γ (cid:48) log δ (cid:48) < γ log δ . (8)Observe that | E uni | + | E tmp | ≤ | E S | = δ · γ . By definition, | E tmp | = δ (cid:48) · γ (cid:48) . It follows that3 δ (cid:48) · γ (cid:48) = 3 | E tmp | ≤ | E uni | + | E tmp | ) ≤ δ · γ, yielding log δ (cid:48) ≤ log δ + log (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19) . (9)Plugging Equation (9) into Equation (8), we obtain γ (cid:48) < γ log δ (cid:18) log δ + log (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19)(cid:19) . (10)We may assume that | N uni | < γ/
9, as otherwise | N uni | ≥ γ/ ≥ γ/ (9 log δ ) and we are done. Bypartition condition ( P | N uni | ≥ | N many | . Hence γ = | N uni | + | N many | + γ (cid:48) ≤ | N uni | + γ (cid:48) , yielding( γ − γ (cid:48) ) / ≤ | N uni | < γ/
9. Hence 2 / γ/γ (cid:48) ) ≤ /
7, which giveslog (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19) ≤ log (cid:18) (cid:19) ≤ − . viiit follows that γ log δ (cid:18) log δ + log (cid:16) γ (cid:17) − log (cid:18) γ (cid:48) (cid:19)(cid:19) ≤ γ log δ (cid:18) log δ − (cid:19) . (11)Plugging Equation (11) into Equation (10) gives γ (cid:48) ≤ γ log δ (cid:18) log δ − (cid:19) = γ − γ δ ≤ | N uni | + γ (cid:48) − γ δ , implying that | N uni | ≥ γ/ (9 log δ ).By corollaries A.13, A.8 and A.15 we get the following result. Denote M G ( x ) = max min { / (9 log x ) , / } , / (9 log(2 x )) , max { (1 − /t )(2 . / log( tx )) | t > } . Corollary A.16 In G S , there is a subset S (cid:48) of S with | Γ S ( S (cid:48) ) | ≥ γ · M G ( δ ) . Observation A.17 max { min { γ/ (9 log δ ) , γ/ } , γ/ (9 log(2 δ )) } is given by γ/ (9 log(2 δ )) if δ ≤ / γ/ if / ≤ δ ≤ / γ/ (9 log δ ) otherwise . Moreover, for every (cid:15) > , if δ satisfies (cid:15) ln( δ ) − ln(ln δ ) − ln(1 + (cid:15) ) − ≥ , then max { γ (1 − /t )(2 . / log( tδ )) | t > } = γ . (cid:15) ) log( δ ) . In that case, max { γ/ (9 log δ ) , max { γ (1 − /t )(1 / (2(1 + c ) log c ( tδ )) | t > }} ≥ γ . (cid:15) ) log( δ ) if and only if (cid:15) < . ,i.e., to understand which expression is the maximum, we need to take (cid:15) (cid:48) = min { (cid:15) | (cid:15) ln( δ ) − ln(ln δ ) − ln(1 + (cid:15) ) − ≥ } and then check if (cid:15) (cid:48) < . or not. Let G = ( V, E ) be an ( α, β )-expander, and for every S in V , denote γ S = | Γ − ( S ) | . As G is an( α, β )-expander, γ S ≥ β | S | . Then, Corollary A.16 yields the following bound on β w . Lemma A.18
Let G = ( V, E ) be an ( α, β ) -expander. Then,(1) G is an ( α w , β w ) -wireless expander with α w = α and β w ≥ β · M G (¯ δ ) . (2) In the regime β ≥ , we have δ S ≤ ∆ /β , thus ¯ δ ≤ ∆ /β , and hence β w ≥ β · M G (∆ /β ) . Proof:
Let S in V s.t. | S | ≤ αn , and let G S = ( S, Γ − ( S ) , E S ) be the corresponding graph. Then, by Corollary A.16, | Γ S ( S (cid:48) ) | ≥ γ S · M G ( δ S ) ≥ β | S | · M G ( δ S ) . Now, M G ( x ) is a decreasing function, and as δ S ≥ ¯ δ , we get that M G ( δ S ) ≥ M G (¯ δ ) andthus | Γ S ( S (cid:48) ) | ≥ β | S | · M G (¯ δ ) . Moreover, in the regime β ≥
1, we have δ S ≤ ∆ /β , thus ¯ δ ≤ ∆ /β , andhence | Γ S ( S (cid:48) ) | ≥ β | S | · M G (∆ /β ).The bounds presented in Section A.2 on β w are functions of ¯ δ (like the inequality β w ≥ β/ (9 log(2¯ δ ))that we proved in Corollary A.14). Theses bounds are usually hard to use, since in most cases we cannotgive an evaluation of ¯ δ . But there are cases in which we can evaluate ¯ δ , and get a better lower bound for β w than β/β/