Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time
aa r X i v : . [ c s . D M ] F e b LOCAL MULTICOLORING ALGORITHMS:
COMPUTING A NEARLY-OPTIMAL TDMA SCHEDULE IN CONSTANT TIME
FABIAN KUHN MIT, Computer Science and Artificial Intelligence Lab32 Vassar St, Cambridge, MA 02139, USA
E-mail address : [email protected] A BSTRACT . We are given a set V of autonomous agents (e.g. the computers of a distributed system)that are connected to each other by a graph G = ( V, E ) (e.g. by a communication network connectingthe agents). Assume that all agents have a unique ID between and N for a parameter N ≥ | V | and that each agent knows its ID as well as the IDs of its neighbors in G . Based on this limitedinformation, every agent v must autonomously compute a set of colors S v ⊆ C such that the colorsets S u and S v of adjacent agents u and v are disjoint. We prove that there is a deterministic algorithmthat uses a total of | C | = O (∆ log( N ) /ε ) colors such that for every node v of G (i.e., for everyagent), we have | S v | ≥ | C |· (1 − ε ) / ( δ v +1) , where δ v is the degree of v and where ∆ is the maximumdegree of G . For N = Ω(∆ log ∆) , Ω(∆ + log log N ) colors are necessary even to assign at leastone color to every node (i.e., to compute a standard vertex coloring). Using randomization, it ispossible to assign an (1 − ε ) / ( δ + 1) -fraction of all colors to every node of degree δ using only O (∆ log | V | /ε ) colors w.h.p. We show that this is asymptotically almost optimal. For graphs withmaximum degree ∆ = Ω(log | V | ) , Ω(∆ log | V | / log log | V | ) colors are needed in expectation, evento compute a valid coloring.The described multicoloring problem has direct applications in the context of wireless ad hoc andsensor networks. In order to coordinate the access to the shared wireless medium, the nodes of sucha network need to employ some medium access control (MAC) protocol. Typical MAC protocolscontrol the access to the shared channel by time (TDMA), frequency (FDMA), or code divisionmultiple access (CDMA) schemes. Many channel access schemes assign a fixed set of time slots,frequencies, or (orthogonal) codes to the nodes of a network such that nodes that interfere with eachother receive disjoint sets of time slots, frequencies, or code sets. Finding a valid assignment of timeslots, frequencies, or codes hence directly corresponds to computing a multicoloring of a graph G .The scarcity of bandwidth, energy, and computing resources in ad hoc and sensor networks, as wellas the often highly dynamic nature of these networks require that the multicoloring can be computedbased on as little and as local information as possible.
1. Introduction
In this paper, we look at a variant of the standard vertex coloring problem that we name graph multicoloring . Given an n -node graph G = ( V, E ) , the goal is to assign a set S v of colors to eachnode v ∈ V such that the color sets S u and S v of two adjacent nodes u ∈ V and v ∈ V are disjointwhile at the same time, the fraction of colors assigned to each node is as large as possible and the Key words and phrases: distributed algorithms, graph coloring, local algorithms, medium access control, multicolor-ing, TDMA, wireless networks.For space reasons, most proofs are omitted from this extended abstract. A full version can be received from theauthor’s web site at http://people.csail.mit.edu/fkuhn/publications/multicoloring.pdf . c (cid:13) F. Kuhn CC (cid:13) Creative Commons Attribution-NoDerivs License
14 F. KUHN total number of colors used is as small as possible. In particular, we look at the following distributed variant of this multicoloring problem. Each node has a unique identifier (ID) between and N foran integer parameter N ≥ n . The nodes are autonomous agents and we assume that every agenthas only very limited, local information about G . Specifically, we assume that every node v ∈ V merely knows its own ID as well as the IDs of all its neighbors. Based on this local information,every node v needs to compute a color set S v such that the color sets computed by adjacent nodesare disjoint. Since our locality condition implies that every node is allowed to communicate witheach neighbor only once, we call such a a distributed algorithm a one-shot algorithm .We prove nearly tight upper and lower bounds for deterministic and randomized algorithmssolving the above distributed multicoloring problem. Let ∆ be the largest degree of G . We show thatfor every ε ∈ (0 , , there is a deterministic multicoloring algorithm that uses O (∆ log( N ) /ε ) colors and assigns a (1 − ε ) / ( δ + 1) -fraction of all colors to each node of degree δ . Note thatbecause a node v of degree δ does not know anything about the topology of G (except that itself has δ neighbors), no one-shot multicoloring algorithm can assign more than a / ( δ + 1) -fraction of thecolors to all nodes of degree δ (the nodes could be in a clique of size δ + 1 ). The upper bound proofis based on the probabilistic method and thus only establishes the existence of an algorithm. Wedescribe an algebraic construction yielding an explicit algorithm that achieves the same bounds upto polylogarithmic factors. Using O (∆ log N ) colors, for a value ε > , the algorithm assigns a ε/ O ( δ ε log N ) -fraction of all colors to nodes of degree δ . At the cost of using O (∆ log ∗ N log N ) colors, it is even possible to improve the fraction of colors assigned to each node by a factor of log N . The deterministic upper bound results are complemented by a lower bound showing thatif N = Ω(∆ log ∆) , even for the standard vertex coloring problem, every deterministic one-shotalgorithm needs to use at least Ω(∆ + log log N ) colors.If we allow the nodes to use randomization (and only require that the claimed bounds areobtained with high probability), we can do significantly better. In a randomized one-shot algorithm,we assume that every node can compute a sequence of random bits at the beginning of an algorithmand that nodes also know their own random bits as well as the random bits of the neighbors whencomputing the color set. We show that for ε ∈ (0 , , with high probability, O (∆ log( n ) /ε ) colorssuffice to assign a (1 − ε ) / ( δ + 1) -fraction of all colors to every node of degree δ . If log n ≤ ∆ ≤ n − ε for a constant ε > , we show that every randomized one-shot algorithm needs atleast Ω(∆ log n/ log log n ) colors. Again, the lower bound even holds for standard vertex coloringalgorithms where every node only needs to choose a single color.Synchronizing the access to a common resource is a typical application of coloring in networks.If we have a c -coloring of the network graph, we can partition the resource (and/or time) into c partsand assign a part to each node v depending on v ’s color. In such a setting, it seems natural to use amulticoloring instead of a standard vertex coloring and assign more than one part of the resource toevery node. This allows to use the resource more often and thus more efficiently.The most prominent specific example of this basic approach occurs in the context of mediaaccess control (MAC) protocols for wireless ad hoc and sensor networks. These networks consistof autonomous wireless devices that communicate with each other by the use of radio signals. Iftwo or more close-by nodes transmit radio signals at the same time, a receiving node only hearsthe superposition of all transmitted signals. Hence, simultaneous transmissions of close-by nodesinterfere with each other and we thus have to control the access to the wireless channel. A stan-dard way to avoid interference between close-by transmissions is to use a time (TDMA), frequency(FDMA), or code division multiple access (CDMA) scheme to divide the channel among the nodes.A TDMA protocol divides the time into time slots and assigns different time slots to conflictingnodes. When using FDMA, nodes that can interfere with each other are assigned different frequen-cies, whereas a CDMA scheme uses different (orthogonal) codes for interfering nodes. Classically, OCAL MULTICOLORING ALGORITHMS 615
TDMA, FDMA, and CDMA protocols are implemented by a standard vertex coloring of the graphinduced by the interference relations. In all three cases, it would be natural to use the more generalmulticoloring problem in order to achieve a more effective use of the wireless medium. EfficientTDMA schedules, FDMA frequency assignments, or CDMA code assignments are all directly ob-tained from a multicoloring of the interference graph where the fraction of colors assigned to eachnodes is as large as possible. It is also natural to require that the total number of colors is small. Thiskeeps the length of a TDMA schedule or the total number of frequencies or codes small and thushelps to improve the efficiency and reduce unnecessary overhead of the resulting MAC protocols.In contrast to many wired networks, wireless ad hoc and sensor networks typically consist ofsmall devices that have limited computing and storage capabilities. Because these devices operateon batteries, wireless nodes also have to keep the amount of computation and especially commu-nication to a minimum in order to save energy and thus increase their lifetime. As the nodes ofan ad hoc or sensor network need to operate without central control, everything that is computed,has to be computed by a distributed algorithm by the nodes themselves. Coordination between thenodes is achieved by exchanging messages. Because of the resource constraints, these distributedalgorithms need to be as simple and efficient as possible. The messages transmitted and receivedby each node should be as few and as short as possible. Note that because of interference, thebandwidth of each local region is extremely limited. Typically, for a node v , the time needed toeven receive a single message from all neighbors is proportional to the degree of v (see e.g. [19]).As long as the information provided to each node is symmetric, it is clear that every node needs toknow the IDs of all adjacent nodes in G in order to compute a reasonably good multicoloring of G .Hence, the one-shot multicoloring algorithms considered in this paper base their computations onthe minimum information needed to compute a non-trivial solution to the problem. Based on theabove observations, even learning the IDs of all neighbors requires quite a bit of time and resources.Hence, acquiring significantly more information might already render an algorithm inapplicable inpractice. As a result of the scarcity of resources, the size and simplicity of the wireless devices used insensor networks, and the dependency of the characteristic of radio transmissions on environmentalconditions, ad hoc and sensor networks are much less stable than usual wired networks. As a con-sequence, the topology of these networks (and of their interference graph) can be highly dynamic.This is especially true for ad hoc networks, where it is often even assumed that the nodes are mobileand thus can move in space. In order to adapt to such dynamic conditions, a multicoloring needsto be recomputed periodically. This makes the resource and time efficiency of the used algorithmseven more important. This is particularly true for the locality of the algorithms. If the computationof every node only depends on the topology of a close-by neighborhood, dynamic changes also onlyaffect near-by nodes.The remainder of the paper is organized as follows. In Section 2, we discuss related work.The problem is formally defined in Section 3. We present the deterministic and randomized upperbounds in Section 4 and the lower bounds in Section 5.
2. Related Work
There is a rich literature on distributed algorithms to compute classical vertex colorings (seee.g. [1, 4, 11, 15, 16, 21]). The paper most related to the present one is [15]. In [15], deterministicalgorithms for the standard coloring problem in the same distributed setting are studied (i.e., every It seems that in order to achieve a significant improvement on the multicolorings computed by the algorithms pre-sented in this paper, every node would need much more information. Even if every node knows its complete O (log ∆) -neighborhood, the best deterministic coloring algorithm that we are aware of needs Θ(∆ ) colors.
16 F. KUHN node has to compute its color based on its ID and the IDs of its neighbors). The main result is a
Ω(∆ / log ∆) lower bound on the number of colors. The first paper to study distributed coloringis a seminal paper by Linial [16]. The main result of [16] is an Ω(log ∗ n ) -time lower bound forcoloring a ring with a constant number of colors. As a corollary of this lower bound, one obtains an Ω(log log N ) lower bound on the number of colors for deterministic one-shot coloring algorithms asstudied in this paper. Linial also looks at distributed coloring algorithms for general graph and showsthat one can compute an O (∆ ) -coloring in time O (log ∗ n ) . In order to color a general graph withless colors, the best known distributed algorithms are significantly slower. Using randomization,an O (∆) -coloring can be obtained in time O ( √ log n ) [14]. Further, the fastest algorithm to obtain a (∆ + 1) -coloring is based on an algorithm to compute a maximal independent set by Luby [17] andon a reduction described in [16] and has time complexity O (log n ) . The best known deterministicalgorithms to compute a (∆ + 1) -coloring have time complexities O ( √ log n ) and O (∆ log ∆ +log ∗ n ) and are described in [21] and [15], respectively. For special graph classes, there are moreefficient deterministic algorithms. It has long been known that in rings [4] and bounded degreegraphs [11, 16], a (∆ + 1) -coloring can be computed in time O (log ∗ n ) . Very recently, it hasbeen shown that this also holds for the much larger class of graphs with bounded local independentsets [26]. In particular, this graph class contains all graph classes that are typically used to modelwireless ad hoc and sensor networks. Another recent result shows that graphs of bounded arboricitycan be colored with a constant number of colors in time O (log n ) [3].Closely related to vertex coloring algorithms are distributed algorithms to compute edge col-orings [5, 12, 22]. In a seminal paper, Naor and Stockmeyer were the first to look at distributedalgorithms where all nodes have to base their decisions on constant neighborhoods [20]. It is shownthat a weak coloring with f (∆) colors (every node needs to have a neighbor with a different color)can be computed in time if every vertex has an odd degree. Another interesting approach is takenin [9] where the complexity of distributed coloring is studied in case there is an oracle that givessome nodes a few bits of extra information.There are many papers that propose to use some graph coloring variant in order to computeTDMA schedules and FDMA frequency or CDMA code assignments (see e.g. [2, 10, 13, 18, 24,25, 27]). Many of these papers compute a vertex coloring of the network graph such that nodesat distance at most have different colors. This guarantees that no two neighbors of a node usethe same time slot, frequency, or code. Some of the papers also propose to construct a TDMAschedule by computing an edge coloring and using different time slots for different edges. Clearly,it is straight-forward to use our algorithms for edge colorings, i.e., to compute a multicoloring ofthe line graph. With the exception of [13] all these papers compute a coloring and assign only onetime slot, frequency, or code to every node or edge. In [13], first, a standard coloring is computed.Based on this coloring, an improved slot assignment is constructed such that in the end, the numberof slots assigned to a node is inversely proportional to the number of colors in its neighborhood.
3. Formal Problem Description
Throughout the paper, we use log( · ) to denote logarithms to base and ln( · ) to denote nat-ural logarithms, respectively. By log ( i ) x and by ln ( i ) x , we denote the i -fold applications of thelogarithm functions log and ln to x , respectively . The log star function is defined as log ∗ n := In [6], it is claimed that an O (∆) coloring can be computed in time O (log ∗ ( n/ ∆)) . However, the argumentation in[6] has a fundamental flaw that cannot be fixed [23]. We have log (0) x = ln (0) x = x , log ( i +1) x = log(log ( i ) x ) , and ln ( i +1) x = ln(ln ( i ) x ) . Note that we also use log i x = (log x ) i and ln i x = (ln x ) i OCAL MULTICOLORING ALGORITHMS 617 min i { log ( i ) n ≤ } . We also use the following standard notations. For an integer n ≥ , [ n ] = { , . . . , n } . For a finite set Ω and an integer k ∈ { , . . . , | Ω |} , (cid:0) Ω k (cid:1) = { S ∈ Ω : | S | = k } . Theterm with high probability (w.h.p.) means with probability at least − /n c for a constant c ≥ . The multicoloring problem that was introduced in Section 1 can be formally defined as follows.
Definition 3.1 (Multicoloring) . An ( ρ ( δ ) , k ) -multicoloring γ of a graph G = ( V, E ) is a mapping γ : V → [ k ] that assigns a set γ ( v ) ⊂ [ k ] of colors to each node v of G such that ∀{ u, v } ∈ E : γ ( u ) ∩ γ ( v ) = ∅ and such that for every node v ∈ V of degree δ , | γ ( v ) | /k ≥ ρ ( δ ) / ( δ + 1) .We call ρ ( δ ) the approximation ratio of a ( ρ ( δ ) , k ) -multicoloring. Because in a one-shot al-gorithm (cf. the next section for a formal definition), a node of degree δ cannot distinguish G from K δ +1 , the approximation ratio of every one-shot algorithm needs to be at most .The multicoloring problem is related to the fractional coloring problem in the following way.Assume that every node is assigned the same number c of colors and that the total number of colorsis k . Taking every color with fraction /c then leads to a fractional ( k/c ) -coloring of G . Hence, inthis case, k/c is lower bounded by the fractional chromatic number χ f ( G ) of G . As outlined in the introduction, we are interested in local algorithms to compute multicoloringsof an n -node graph G = ( V, E ) . For a parameter N ≥ n , we assume that every node v has a uniqueID x v ∈ [ N ] . In deterministic algorithms, every node has to compute a color set based on its ownID as well as the IDs of its neighbors. For randomized algorithms, we assume that nodes also knowthe random bits of their neighbors. Formally, a one-shot algorithm can be defined as follows. Definition 3.2 (One-Shot Algorithm) . We call a distributed algorithm a one-shot algorithm if everynode v performs (a subset of) the following three steps:1. Generate sequence R v of random bits (deterministic algorithms: R v = ∅ )2. Send x v , R v to all neighbors3. Compute solution based on x v , R v , and the received informationAssume that G is a network graph such that two nodes u and v can directly communicate witheach other iff they are connected by an edge in G . In the standard synchronous message passing model, time is divided into rounds and in every round, every node of G can send a message to eachof its neighbors. One-shot algorithms then exactly correspond to computations that can be carriedout in a single communication round.For deterministic one-shot algorithms, the output of every node v is a function of v ’s ID x v andthe IDs of v ’s neighbors. We call this information on which v bases its decisions, the one-hop view of v . Definition 3.3 (One-Hop View) . Consider a node v with ID x v and let Γ v be the set of IDs of theneighbors of v . We call the pair ( x v , Γ v ) the one-hop view of v .Let ( x u , Γ u ) and ( x v , Γ v ) be the one-hop views of two adjacent nodes. Because u and v areneighbors, we have x u ∈ Γ v and that x v ∈ Γ u . It is also not hard to see that ∀ x u , x v ∈ [ N ] and ∀ Γ u , Γ v ∈ [ N ] such that x u = x v , x u ∈ Γ v \ Γ u , x v ∈ Γ u \ Γ v , (3.1)there is a labeled graph that has two adjacent nodes u and v with one-hop views ( x u , Γ u ) and ( x v , Γ v ) , respectively. Assume that we are given a graph with maximum degree ∆ (i.e., for all
18 F. KUHN one-hop views ( x v , Γ v ) , we have | Γ v | ≤ ∆ ). A one-shot vertex coloring algorithm maps everypossible one-hop view to a color. A correct coloring algorithm must assign different colors to twoone-hop views ( x u , Γ u ) and ( x v , Γ v ) iff they satisfy Condition (3.1). This leads to the definitionof the neighborhood graph N ( N, ∆) [15] (the general notion of neighborhood graphs has beenintroduced in [16]). The nodes of N ( N, ∆) are all one-hop views ( x v , Γ v ) with | Γ v | ≤ ∆ . Thereis an edge between ( x u , Γ u ) and ( x v , Γ v ) iff the one-hop views satisfy Condition (3.1). Hence, aone-shot coloring algorithm must assign different colors to two one-hop views iff they are neighborsin N ( N, ∆) . The number of colors that are needed to properly color graphs with maximum degree ∆ by a one-shot algorithm therefore exactly equals the chromatic number χ (cid:0) N ( N, ∆) (cid:1) of theneighborhood graph (see [15, 16] for more details). Similarly, a one-shot ( ρ ( δ ) , k ) -multicoloringalgorithm corresponds to a ( ρ ( δ ) , k ) -multicoloring of the neighborhood graph.
4. Upper Bounds
In this section, we prove all the upper bounds claimed in Section 1. We first prove that anefficient deterministic one-shot multicoloring algorithm exists in Section 4.1. Based on similarideas, we derive an almost optimal randomized algorithm in Section 4.2. Finally, in Section 4.3, weintroduce constructive methods to obtain one-shot multicoloring algorithms. For all algorithms, weassume that the nodes know the size of the ID space N as well as ∆ , an upper bound on the largestdegree in the network. It certainly makes sense that nodes are aware of the used ID space. Note thatit is straight-forward to see that there cannot be a non-trivial solution to the one-shot multicoloringproblem if the nodes do not have an upper bound on the maximum degree in the network. The existence of an efficient, deterministic one-shot multicoloring algorithm is established bythe following theorem.
Theorem 4.1.
Assume that we are given a graph with maximum degree ∆ and node IDs in [ N ] .Then, for all < ε ≤ , there is a deterministic, one-shot (cid:0) − ε, O (∆ log( N ) /ε ) (cid:1) -multicoloringalgorithm.Proof. We use permutations to construct colors as described in [15]. For i = 1 , . . . , k , let ≺ i bea global order on the ID set [ N ] . A node v with -hop view ( x v , Γ v ) includes color i in its colorset iff ∀ y ∈ Γ v : x v ≺ i y . It is clear that with this approach the color sets of adjacent nodesare disjoint. In order to show that nodes of degree δ obtain a ρ/ ( δ + 1) -fraction of all colors, weneed to show that for all δ ∈ [∆] , all x ∈ [ N ] , and all Γ ∈ (cid:0) [ N ] \{ x } δ (cid:1) , for all y ∈ Γ , x ≺ i y forat least kρ/ ( δ + 1) global orders ≺ i . We use the probabilistic method to show that a set of size k = 2(∆ + 1) ln( N ) /ε of global orders ≺ i exists such that every node of degree δ ∈ [∆] gets atleast an (1 − ε ) / ( δ + 1) -fraction of the k colors. Such a set implies that there exists an algorithmthat satisfies the claimed bounds for all graphs with maximum degree ∆ and IDs in [ N ] .Let ≺ , . . . , ≺ k be k global orders chosen independently and uniformly at random. The prob-ability that a node v with degree δ and -hop view ( x v , Γ v ) gets color i is / ( δ + 1) (note that | Γ v | = δ ). Let X v be the number of colors that v gets. We have E [ X v ] = k/ ( δ + 1) ≥ k/ (∆ + 1) .Using a Chernoff bound, we then obtain P (cid:20) X v < (1 − ε ) · kδ + 1 (cid:21) = P [ X v < (1 − ε ) · E [ X v ]] < e − ε E [ X v ] / ≤ N ∆+1 . (4.1) OCAL MULTICOLORING ALGORITHMS 619
Algorithm 1
Explicit Deterministic Multicoloring Algorithm: Basic Construction
Input: one-hop view ( x, Γ) , parameter ℓ ≥ Output: set S of colors, initially S = ∅ for all ( α , α , . . . , α ℓ ) ∈ F q × F q × · · · × F q ℓ do β ,x := ϕ ,x ( α ) ; ∀ y ∈ Γ : β ,y := ϕ ,y ( α ) for i := 1 to ℓ do β i,x := ϕ i,β i − ,x ( α i ) ; ∀ y ∈ Γ : β i,y := ϕ i,β i − ,y ( α i ) if ∀ y ∈ Γ : β ℓ,x = β ℓ,y then S := S ∪ ( α , α , . . . , α ℓ , β ℓ,x ) The total number of different possible one-hop views can be bounded as |N ( N, ∆) | = N · P ∆ δ =1 (cid:0) N − δ (cid:1) < N ∆+1 . By a union bound argument, we therefore get that with positive probability,for all δ ∈ [∆] , all possible one-hop views ( x v , Γ v ) with | Γ v | = δ get at least (1 − ε ) · k/ ( δ + 1) colors. Hence, there exists a set of k global orders on the ID set [ N ] such that all one-hop viewsobtain at least the required number of colors. Remark:
Note that if we increase the number of permutations (i.e., the number of colors) by aconstant factor, all possible one-hop views ( x, Γ) with | Γ | = δ get a (1 − ε ) / ( δ + 1) -fraction of allcolors w.h.p. We will now show that with the use of randomization, the upper bound of Section 4.1 can besignificantly improved if the algorithm only needs to be correct w.h.p. We will again use randompermutations. The problem of the deterministic algorithm is that the algorithm needs to assign alarge set of colors to all roughly N ∆ possible one-hop views. With the use of randomization, weessentially only have to assign colors to n randomly chosen one-hop views.For simplicity, we assume that every node knows the number of nodes n (knowing an upperbound on n is sufficient). For an integer parameter k > , every v ∈ V chooses k independentrandom numbers x v, , . . . , x v,k ∈ [ kn ] and sends these random numbers to all neighbors. Weuse these random numbers to induce k random permutations on the nodes. Let Γ( v ) be the set ofneighbors of a node v . A node v selects all colors i for which x v,i < x u,i for all u ∈ Γ( v ) . Theorem 4.2.
Choosing k = 6(∆ + 1) ln( n ) /ε leads to a randomized one-shot algorithm thatcomputes a (1 − ε, k ) -multicoloring w.h.p. Remark:
In the above algorithm, every node has to generate O (∆ log ( n ) /ε ) random bits andsend these bits to the neighbors. Using a (non-trivial) probabilistic argument, it is possible to showthat the same result can be achieved using only O (log n ) random bits per node. We have shown in Section 4.1 that there is a deterministic one-shot algorithm that almostmatches the lower bound (cf. Theorem 5.2). Unfortunately, the techniques of Section 4.1 do notyield an explicit algorithm. In this section, we will present constructive methods to obtain a one-shot multicoloring algorithm.We develop the algorithm in two steps. First, we construct a multicoloring where in the worstcase, every node v obtains the same fraction of colors independent of v ’s degree. We then showhow to increase the fraction of colors assigned to low-degree nodes. For an integer parameter ℓ ≥ ,
20 F. KUHN let q , . . . , q ℓ be prime powers and let d , . . . , d ℓ be positive integers such that q d +10 ≥ N and q d i +1 i ≥ q i − for i ≥ . For a prime power q and a positive integer d , let P ( q, d ) be the set of all q d +1 polynomials of degree at most d in F q [ z ] , where F q is the finite field of order q . We assumethat that we are given an injection ϕ from the ID set [ N ] to the polynomials in P ( q , d ) andinjections ϕ i from F q i − to P ( q i , d i ) for i ≥ . For a value x in the respective domain, let ϕ i,x bethe polynomial assigned to x by injection ϕ i . The first part of the algorithm is an adaptation of atechnique used in a coloring algorithm described in [16] that is based on an algebraic constructionof [7]. There, a node v with one-hop view ( x, Γ) selects a color (cid:0) α, ϕ ,x ( α ) (cid:1) , where α ∈ F q is avalue for which ϕ ,x ( α ) = ϕ ,y ( α ) for all y ∈ Γ (we have to set q and d such that this is alwayspossible). We make two modifications to this basic algorithm. Instead of only selecting one value α ∈ F q such that ∀ y ∈ Γ : ϕ ,x ( α ) = ϕ ,y ( α ) , we select all values α for which this is true. Wethen use these values recursively (as if ϕ i,x ( α i ) was the ID of v ) ℓ times to reduce the dependenceof the approximation ratio of the coloring on N . The details of the first step of the algorithm aregiven by Algorithm 1. Lemma 4.3.
Assume that for ≤ i ≤ ℓ , q i ≥ f i ∆ d i where f i > . Then, Algorithm 1 constructsa multicoloring with q ℓ · Q ℓi =0 q i colors where every node at least receives a λ/q ℓ -fraction of allcolors where λ = Q ℓi =0 (1 − /f i ) .Proof. All colors that are added to the color set in line 6 are from F q × F q × · · · × F q ℓ × F q ℓ . It istherefore clear that the number of different colors is q ℓ · Q ℓi =0 q i as claimed. From the condition inline 5, it also follows that the color sets of adjacent nodes are disjoint.To determine the approximation ratio, we count the number of colors, a node v with one-hopview ( x, Γ) gets. First note that the condition in line 5 of the algorithm implies that (and is thereforeequivalent to demand that) β i,x = β i,y for all y ∈ Γ and for all i ∈ { , . . . , ℓ } because β i,x = β i,y implies β j,x = β j,y for all j ≥ i . We therefore need to count the number of ( α , . . . , α ℓ ) ∈ F q × · · · × F q ℓ for which β i,x = β i,y for all i ∈ { , . . . , ℓ } and all y ∈ Γ . We prove by inductionon i that for i < ℓ , there are at least Q ij =0 q j · (1 − /f j ) tuples ( α , . . . , α i ) ∈ F q × · · · F q i with β j,x = β j,y for all j ≤ i . Let us first prove the statement for i = 0 . Because the IDs of adjacentnodes are different, we know that ϕ ,x = ϕ ,y for all y ∈ Γ . Two different degree d polynomialscan be equal at at most d values. Hence, for every y ∈ Γ , ϕ ,x ( α ) = ϕ ,y ( α ) for at most d values α . Thus, since | Γ | ≤ ∆ , there are at least q − ∆ d ≥ q · (1 − /f ) values α for which ϕ ,x = ϕ ,y for all y ∈ Γ . This establishes the statement for i = 0 . For i > , the argument is analogous. Let ( α , . . . , α i − ) ∈ F q × · · · × F q i − be such that β j,x = β j,y for all y ∈ Γ and all j < i . Because β i − ,x = β i − ,y , we have ϕ i,x = ϕ i,y . Thus, with the same argument as for i = 0 , there are at least q i · (1 − /f i ) values α i such that β i,x = β i,y for all y ∈ Γ . Therefore, the number of colors in thecolor set of every node is at least Q ℓi =0 q i · (cid:0) − /f i (cid:1) = λ · Q ℓi =0 q i . This is a ( λ/q ℓ ) -fraction of allcolors.The next lemma specifies how the values of q i , d i , and f i can be chosen to obtain an efficientalgorithm. Lemma 4.4.
Let ℓ be such that ln ( ℓ ) N > max { e, ∆ } . For ≤ i ≤ ℓ , we can then choose q i , d i , and f i such that Algorithm 1 computes a multicoloring with O ( ℓ ∆) ℓ +2 · log ∆ N · log ∆ ln ( ℓ ) N colors and such that every node gets at least a / (cid:0) e / ∆ (cid:6) log ∆ ln ( ℓ ) N (cid:7)(cid:1) -fraction of all colors. The number of colors that Algorithm 1 assigns to nodes with degree almost ∆ is close tooptimal even for small values of ℓ . If we choose ℓ = Θ(log ∗ N − log ∗ ∆) , nodes of degree Θ(∆) even receive at least a ( d/ ∆) -fraction of all colors for some constant d . Because the number ofcolors assigned to a node v is independent of v ’s degree, however, the coloring of Algorithm 1 is far OCAL MULTICOLORING ALGORITHMS 621
Algorithm 2
Explicit Deterministic Multicoloring Algorithm: Small Number of Colors
Input: one-hop view ( x, Γ) , instances A i ,N for i ∈ (cid:2) ⌈ log ∆ ⌉ (cid:3) of Algorithm 1, parameter ε ∈ [0 , Output: set S of colors, initially S = ∅ for all i ∈ (cid:2) ⌈ log ∆ ⌉ (cid:3) do ω i := l(cid:0) ∆ / i − (cid:1) ε · (cid:12)(cid:12) C ⌈ log ∆ ⌉ ,N (cid:12)(cid:12) / (cid:12)(cid:12) C i ,N (cid:12)(cid:12)m for all i ∈ (cid:8) ⌈ log | Γ |⌉ , . . . , ⌈ log ∆ ⌉ (cid:9) do for all c ∈ C i ,N [ x, Γ] do for all j ∈ [ ω i ] do S := S ∪ ( c, i, j ) from optimal for low-degree nodes. In the following, we show how to improve the algorithm in thisrespect.Let A ∆ ,N be an instance of Algorithm 1 for nodes with degree at most ∆ and let C ∆ ,N be thecolor set of A ∆ ,N . Further, for a one-hop view ( x, Γ) , let C ∆ ,N [ x, Γ] be the colors assigned to ( x, Γ) by Algorithm A ∆ ,N . We run instances A i ,N for all i ∈ (cid:2) ⌈ log ∆ ⌉ (cid:3) . A node v with degree δ choosesthe colors of all instances for which i ≥ δ . In order to achieve the desired trade-offs, we introducean integer weight ω for each color c , i.e., instead of adding color c , we add colors (1 , c ) , . . . , ( ω, c ) .The details are given by Algorithm 2. The properties of Algorithm 2 are summarized by the nexttheorem. The straight-forward proof is omitted. Theorem 4.5.
Assume that in the instances of Algorithm 1, the parameter ℓ is chosen such that forall ∆ , A ∆ ,N assigns at least a f ( N ) / ∆ -fraction of the colors to every node. Then, for a parameter ε ∈ [0 , , Algorithm 2 computes a (cid:0) Ω( f ( N ) ε/δ ε ) , O ( |C ,N | · ∆ ε /ε ) (cid:1) -multicoloring. Corollary 4.6.
Let ε ∈ [0 , and ℓ ≥ be a fixed constant in all used instances of Algorithm1. Then, Algorithm 2 computes an (cid:0) ε/ O ( δ ε log ∆ ln ( ℓ ) N ) , O (∆ ℓ +2 · log ∆ N · log ∆ ln ( ℓ ) N ) (cid:1) -multicoloring. In particular, choosing ℓ = 0 leads to an (cid:0) ε/ O ( δ ε log ∆ N ) , O (∆ log N ) (cid:1) -multicoloring. Taking the maximum possible value for ℓ in all used instances of Algorithm 1 yieldsan (cid:0) ε/ O ( δ ε ) , ∆ O (log ∗ N − log ∗ ∆) · log ∆ N (cid:1) -multicoloring.
5. Lower Bounds
In this section, we give lower bounds on the number of colors required for one-shot multicol-oring algorithms. In fact, we even derive the lower bounds for algorithms that need to assign onlyone color to every node, i.e., the results even hold for standard coloring algorithms.It has been shown in [15] that every deterministic one-shot c -coloring algorithm A can beinterpreted as a set of c antisymmetric relations on the ID set [ N ] . Assume that A assigns a colorfrom a set C with | C | = c to every one-hop view ( x, Γ) . For every color α ∈ C , there is a relation ⊳ α such that for all x, y ∈ [ N ] x ⊳ α y ∨ y ⊳ α x . Algorithm A can assign color α ∈ C to a one-hopview ( x, Γ) iff ∀ y ∈ Γ : x ⊳ α y .For α ∈ C , let Bad α ( x ) := { y ∈ [ N ] : x ⊳ α y } be the set of IDs that must not be adjacent toan α -colored node with ID x . To show that there is no deterministic, one-shot c -coloring algorithm,we need to show that for every c antisymmetric relations ⊳ α , . . . , ⊳ α c on [ N ] , there is a one-hopview ( x, Γ) such that ∀ i ∈ [ c ] : Γ ∩ Bad α i ( x ) = ∅ . The following lemma is a generalization ofLemma 4.5 in [15] and key for the deterministic and the randomized lower bounds. As the proof isalong the same lines as the proof of Lemma 4.5 in [15], it is omitted here.
22 F. KUHN
Lemma 5.1.
Let X ⊆ [ N ] be a set of IDs and let t , . . . , t ℓ and k , . . . , k ℓ be positive integers suchthat t i · (cid:0) λ ( | X | − c ) t i − c (cid:1) > c ( k i − for ≤ i ≤ ℓ and a parameter λ ∈ [0 , . Then there exists an ID set X ′ ⊆ X with | X ′ | > (1 − ℓ · λ ) · ( | X | − c ) such that for all i ∈ [ ℓ ] , ∀ x ∈ X ′ , ∀ α , . . . , α t i ∈ C : t i X j =1 (cid:12)(cid:12) Bad α j ( x ) ∩ X (cid:12)(cid:12) ≥ k i , ∀ x ∈ X ′ , ∀ α ∈ C : Bad α ( x ) ∩ X = ∅ . Based on several applications of Lemma 5.1 (and based on an
Ω(log log N ) lower bound in[16]), it is possible to derive an almost tight lower bound for deterministic one-shot coloring algo-rithms. Due to lack of space, we only state the result here. Theorem 5.2. If N = Ω(∆ log ∆) , every deterministic one-shot coloring algorithm needs at least Ω(∆ + log log N ) colors. To obtain a lower bound for randomized multicoloring algorithms, we can again use the toolsderived for the deterministic lower bound by applying Yao’s principle. On a worst-case input, thebest randomized algorithm cannot perform better than the best deterministic algorithm for a givenrandom input distribution. Choosing the node labeling at random allows to again only considerdeterministic algorithms.We assume that the n nodes are assigned a random permutation of the labels , . . . , n (i.e.,every label occurs exactly once). Note that because we want to prove a lower bound, assuming themost restricted possible ID space makes the bound stronger. For an ID x ∈ [ n ] , we sort all colors α ∈ C by increasing values of | Bad α ( x ) | and let α x,i be the i th color in this sorted order. Further,for x ∈ [ n ] , we define b x,i := (cid:12)(cid:12) Bad α x,i ( x ) (cid:12)(cid:12) . In the following, we assume that c = κ · ∆ ⌊ ln n ⌋⌈ ln ln n ⌉ + 2 and n ≥ and n ≥ ∆ · ln n (5.1)for a constant < κ ≤ that will be determined later. By applying Lemma 5.1 in different ways,the next lemma gives lower bounds on the values of b x,i for n/ IDs x ∈ [ n ] . Lemma 5.3.
Assume that c and n are as given by Equation (5.1) and let < ρ < / be apositive constant. Further, let ˜ t = (cid:6) ρ ln n/ ln ln n (cid:7) and t i = 2 i − · ⌊ ln n ⌋ for ≤ i ≤ ℓ where ℓ = ⌈ ln ln n ⌉ + 2 . Then, for at least n/ of all IDs x ∈ [ n ] , we have b x, ≥ ln ln n κ · ln n · n ∆ − , b x, ˜ t ≥ ρ κ · n ∆ − , b x,t i ≥ i − · (cid:18) κ · n ∆ − (cid:19) for ≤ i ≤ ℓ. In order to prove the lower bound, we want to show that for a randomly chosen one-hop view ( x, Γ) with | Γ | = ∆ , the probability that there is a color α ∈ C for which Γ ∩ Bad α ( x ) = ∅ issufficiently small. Instead of directly looking at random one-hop views ( x, Γ) with | Γ | = ∆ , wefirst look at one-hop views with | Γ | ≈ ∆ /e that are constructed as follows. Let X ⊆ [ n ] be the setof IDs x of size | X | ≥ n/ for which the bounds of Lemma 5.3 hold. We choose x R uniformlyat random from X . The remaining n − IDs are independently added to a set Γ R with probability p = ∆ en . For a color α ∈ C , let E α be the event that Γ R ∩ Bad α ( x R ) = ∅ , i.e., E α is the event thatcolor α cannot be assigend to the randomly chosen one-hop view ( x R , Γ R ) . OCAL MULTICOLORING ALGORITHMS 623
Lemma 5.4.
The probability that the randomly chosen one-hop view cannot be assigned one of the c colors in C is bounded by P " \ α ∈ C E α ≥ Y α ∈ C P (cid:2) E α (cid:3) ≥ Y α ∈ C (cid:16) − e − ∆ en ·| Bad α ( x R ) | (cid:17) = c Y i =1 (cid:18) − e − ∆ · bxR,ien (cid:19) . Proof.
Note first that for α ∈ C , we have P (cid:2) E α (cid:3) = P (cid:2) Γ R ∩ Bad α ( x R ) = ∅ (cid:3) = (1 − p ) | Bad α ( x R ) | ≤ e − p | Bad α ( x R ) | = e − ∆ en ·| Bad α ( x R ) | . It therefore remains to prove that the probability that all events E α occur can be lower bounded bythe probability that would result for independent events. Let us denote the colors in C by α , . . . , α c .We then have P " \ α ∈ C E α = c Y i =1 P E α i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i − \ j =1 E α j ≥ c Y i =1 P (cid:2) E α i (cid:3) . (5.2)The inequality holds because the events E α are positively correlated. Knowing that an element froma set Bad α ( x R ) is in Γ R cannot decrease the probability that an element from a set Bad α ′ ( x R ) isin Γ R . Note that this is only true because the IDs are independently added to Γ R . More formally,Inequality (5.2) can also directly be followed from the FKG inequality [8].For space reasons, the following two lemmas are given without proof. Lemma 5.5.
Assume that c and n are given as in (5.1) where the constant κ is chosen sufficientlysmall and let ρ > be a constant as in Lemma 5.3. There is a constant n > such that for n ≥ n , P (cid:2)T α ∈ C E α (cid:3) > n ρ . Lemma 5.6.
Let ( x, Γ) be a one-hop view chosen uniformly at random from all one-hop views with | Γ | = ∆ . If ∆ ≥ e (ln n + 2) and n , c , and ρ are as before, the probability that none of the c colorscan be assigned to ( x, Γ) is at least / (8 n ρ ) . In the following, we call a node u together with ∆ neighbors v , . . . , v ∆ , a ∆ -star. Theorem 5.7.
Let G be a graph with n nodes and n ε disjoint ∆ -stars for a constant ε > . On G , every randomized one-shot coloring algorithm needs at least Ω(∆ log n/ log log n ) colors inexpectation and with high probability.Proof. W.l.o.g., we can certainly assume that n ≥ n for a sufficiently large constant n . We choose ρ ≤ ε/ and consider n ε of the n ε disjoint ∆ -stars. Let us call these n ε ∆ -stars S , . . . , S n ε .Assume that the ID assignment of the n nodes of G is chosen uniformly at random from all IDassignments with IDs , . . . , n . The IDs of the star S are perfectly random. We can thereforedirectly apply Lemma 5.6 and obtain that the probability that the center node of S gets no coloris at least / (8 n ρ ) . Consider star S . The IDs of the nodes of S are chosen at random amongthe n − ∆ − IDs that are not assigned to the nodes of S . Applying Lemma 5.6 we get that theprobability that S does not get a color is at least / (8( n − ∆ − ρ ) ≥ / (8 n ρ ) independentlyof whether S does get a color. The probability that the starts S , . . . , S n ε all get a color thereforeis at most n ε − Y i =0 (cid:18) − n − i (∆ + 1)) ρ (cid:19) ≤ (cid:18) − n ρ (cid:19) n ε ≤ e − nε n ρ ≤ e − n ρ / . Hence, there is a constant η > such that η ∆ ln n/ ln ln n colors do not suffice with probability atleast − e − n ρ / for a positive constant ρ . The lemma thus follows.
24 F. KUHN
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