Bounding Interference in Wireless Ad Hoc Networks with Nodes in Random Position
BBounding Interference in Wireless Ad Hoc Networks with Nodes inRandom Position ∗ Majid Khabbazian † Stephane Durocher ‡ Alireza Haghnegahdar § October 28, 2011
Abstract
The interference at a wireless node s can be modelled by the number of wireless nodes whose trans-mission ranges cover s . Given a set of positions for wireless nodes, the interference minimization problemis to assign a transmission radius (equivalently, a power level) to each node such that the resulting com-munication graph is connected, while minimizing the maximum interference. We consider the modelintroduced by von Rickenback et al. (2005), in which each transmission range is represented by a balland edges in the communication graph are symmetric. The problem is NP-complete in two dimensions(Buchin 2008) and no polynomial-time approximation algorithm is known. Furthermore, even in onedimension (the highway model), the problem’s complexity is unknown and the maximum interference ofa set of n wireless nodes can be as high as Θ( √ n ) (von Rickenback et al. 2005). In this paper we showhow to solve the problem efficiently in settings typical for wireless ad hoc networks. In particular, weshow that if node positions are represented by a set P of n points selected uniformly and independentlyat random over a d -dimensional rectangular region, for any fixed d , then the topology given by the closureof the Euclidean minimum spanning tree of P has maximum interference O (log n ) with high probabil-ity. We extend this bound to a general class of communication graphs over a broad set of probabilitydistributions. Next we present a local algorithm that constructs a graph from this class; this is the firstlocal algorithm to provide an upper bound on the expected maximum interference. Finally, we discussan empirical evaluation of our algorithm with a suite of simulation results. keywords: interference, topology control, minimum spanning tree, random distribution, expectation Establishing connectivity in a wireless network can be a complex task for which various (sometimes conflict-ing) objectives may need to be optimized. To permit a packet to be routed from any origin node to anydestination node in the network, the corresponding communication graph must be connected (or stronglyconnected if unidirectional communication links are permitted). In addition to requiring connectivity, vari-ous properties can be imposed on the network, including low power consumption [21, 27], bounded averagetraffic load [9, 14], small average hop distance between sender-receiver pairs [1], low dilation ( t -spanner)[1,3,6,7,16,22,25], and minimal interference; this latter objective, minimizing interference (and, consequently,minimizing the required bandwidth), is the focus of much recent research [1, 2, 5, 12, 18–20, 23, 24, 27–31] andof this paper.We adopt the interference model introduced by von Rickenbach et al. [30] (see Section 1.2). We modeltransmission in a wireless network by assigning to each wireless node p a radius of transmission r ( p ), such that ∗ This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). † University of Winnipeg, Winnipeg, Canada, [email protected] ‡ University of Manitoba, Winnipeg, Canada, [email protected] § University of British Columbia, Vancouver, Canada, [email protected] a r X i v : . [ c s . C G ] N ov very node within distance r ( p ) of p can receive a transmission from p , whereas no node a greater distancefrom p can. Consequently, the interference at node p is the number of nodes that have p within their respectiveradii of transmission. Given a set of wireless nodes whose positions are represented by a set of points P , weconsider the problem of identifying a connected network on P that minimizes the maximum interference.The problem of constructing the network is equivalent to that of assigning a transmission radius to eachnode. That is, once the transmission radius of each node is fixed, the corresponding communication graphand its associated maximum interference are also fixed. Conversely, once a graph is fixed, the transmissionradius of each node is determined by the distance to its furthest neighbour.Given a set of points P in the plane, finding a connected graph on P that minimizes the maximuminterference is NP-complete [5]. A polynomial-time algorithm exists that returns a solution with maximuminterference O ( √ n ), where n = | P | [12]. Even in one dimension, for every n there exists a set of n points P such thta any graph on P has maximum interference Ω( √ n ) [30]. All such known examples involve specificconstructions (i.e., exponential chains). We are interested in investigating a more realistic class of wireless adhoc networks: those whose node positions observe common random distributions that better model actualwireless ad hoc networks.When nodes are positioned on a line (often called the highway model ), a simple heuristic is to assign toeach node a radius of transmission that corresponds to the maximum of the distances to its respective nearestneighbours to the left and right. In the worst case, such a strategy can result in Θ( n ) maximum interferencewhen an optimal solution has only Θ( √ n ) maximum interference [30]. Recently, Kranakis et al. [20] showedthat if n nodes are positioned uniformly at random on an interval, then the maximum interference providedby this heuristic is Θ( √ log n ) with high probability.In this paper, we examine the corresponding problem in two and higher dimensions. We generalizethe nearest-neighbour path used in the highway model to the Euclidean minimum spanning tree (MST),and show that with high probability, the maximum interference of the MST of a set of n points selecteduniformly at random over a d -dimensional region [0 , d is O (log n ), for any fixed d ≥
1. Our techniquesdiffer significantly from those used by Kranakis et al. to achieve their results in one dimension. As we showin Section 3, our results also apply to a broad class of random distributions, denoted D , that includes boththe uniform random distribution and realistic distributions for modelling random motion in mobile wirelessnetworks, as well as to a large class of connected spanning graphs that includes the MST.In Section 3.4 we present a local algorithm that constructs a topology whose maximum interference is O (log n ) with high probability when node positions are selected according to a distribution in D . Previouslocal algorithms for topology control (e.g., the cone-based local algorithm (CBTC) [21]) attempt to reducetransmission radii (i.e., power consumption), but not necessarily the maximum interference. Although re-ducing transmission radii at many nodes is often necessary to reduce the maximum interference, the twoobjectives differ; specifically, some nodes may require large transmission radii to minimize the maximuminterference. Ours is the first local algorithm to provide a non-trivial upper bound on maximum interfer-ence. Our algorithm can be applied to any existing topology to refine it and further reduce its maximuminterference. Consequently, our solution can be used either independently, or paired with another topologycontrol strategy. Finally, we discuss an empirical evaluation of our algorithm with a suite of simulationresults in Section 4. We represent the position of a wireless node as a point in Euclidean space, R d , for some fixed d ≥
1. Forsimplicity, we refer to each node by its corresponding point. Similarly, we represent a wireless network byits communication graph, a geometric graph whose vertices are a set of points P ⊆ R d . Given a (simple andundirected) graph G , we employ standard graph-theoretic notation, where V ( G ) denotes the vertex set of G and E ( G ) denotes its edge set. We say vertices u and v are k -hop neighbours if there is a simple path oflength k from u to v in G . When k = 1 we say u and v are neighbours. In the majority of instances, two or three dimensions suffice to model an actual wireless network. Our results are presented interms of an arbitrary d since this permits expressing a more general result without increasing the complexity of the correspondingnotation.
2e assume a uniform range of communication for each node and consider bidirectional communicationlinks, each of which is represented by an undirected graph edge connecting two nodes. Specifically, eachnode p has some radius of transmission , denoted by the function r : P → R + , such that a node q receivesa transmission from p if and only if dist( p, q ) ≤ r ( p ), where dist( p, q ) = || p − q || denotes the Euclideandistance between points p and q in R d . For simplicity, suppose each node has an infinite radius of reception,regardless of its radius of transmission. Definition 1 (Communication Graph)
A graph G is a communication graph with respect to a point set P ⊆ R d and a function r : P → R + if1. V ( G ) = P , and2. for all vertices p and q in V ( G ) , { p, q } ∈ E ( G ) ⇔ dist( p, q ) ≤ min { r ( p ) , r ( q ) } . (1)Together, set P and function r uniquely determine the corresponding communication graph G . Alter-natively, a communication graph can be defined as the closure of a given embedded graph. Specifically, ifinstead of being given P and r , we are given an arbitrary graph H embedded in R d , then the set P is triviallydetermined by V ( H ) and a transmission radius for each node p ∈ V ( H ) can be assigned to satisfy (1) by r ( p ) = max q ∈ Adj( p ) dist( p, q ) , (2)where Adj( p ) = { q | { q, p } ∈ E ( H ) } denotes the set of vertices adjacent to p in H . The communicationgraph determined by H is the unique edge-minimal supergraph of H that satisfies Definition 1. We denotethis graph by H (cid:48) and refer to it as the closure of graph H . Therefore, a communication graph G can bedefined either as a function of a set of points P and an associated mapping of transmission radii r : P → R + ,or as the closure of a given embedded graph H (where G = H (cid:48) ). Definition 2 (Interference)
Given a communication graph G the interference at node p in V ( G ) is inter G ( p ) = |{ q | q ∈ V ( G ) \ { p } and dist( q, p ) ≤ r ( q ) }| and the maximum interference of G is inter( G ) = max p ∈ V ( G ) inter G ( p ) . In other words, the interference at node p , denoted inter G ( p ), is the number of nodes q such that node p lieswithin q ’s radius of transmission. This does not imply the existence of the edge { p, q } in the correspondingcommunication graph; such an edges exists if and only if the relationship is reciprocal, i.e., q also lies with p ’s radius of transmission.Given a point set P , let G ( P ) denote the set of connected communication graphs on P . Let OPT( P )denote the optimal maximum interference attainable over graphs in G ( P ). That is,OPT( P ) = min G ∈G ( P ) inter( G ) = min G ∈G ( P ) max p ∈ V ( G ) inter G ( p ) . Thus, given a set of points P representing the positions of wireless nodes, the interference minimizationproblem is to find a connected communication graph G on P that spans P such that the maximum interferenceis minimized (i.e., its maximum interference is OPT( P )). In this paper we examine the maximum interferenceof the communication graph determined by the closure of MST( P ), where MST( P ) denotes the Euclideanminimum spanning tree of the point set P . Our results apply with high probability, which refers to probabilityat least 1 − n − c , where n = | P | denotes the number of networks nodes and c ≥ Related Work
In this paper we consider the bidirectional interference model (defined in Section 1.2). This model wasintroduced by von Rickenback et al. [30], who gave a polynomial-time approximation algorithm that findsa solution with maximum interference O ( n / · OPT( P )) for any given set of points P on a line, and aone-dimensional construction showing that OPT( P ) ∈ Ω( √ n ) in the worst case, where n = | P | . Halld´orssonand Tokuyama [12] gave a polynomial-time algorithm that returns a solution with maximum interference O ( √ n ) for any given set of n points in the plane. Buchin [5] showed that finding an optimal solution (onewhose maximum interference is exactly OPT( P )) is NP-complete in the plane. Tan et al. [29] gave an O ( n n O (OPT( P )) )-time algorithm for finding an optimal solution for any given set of points P on a line.Kranakis et al. [20] showed that for any set of points P selected uniformly at random from the unit interval,the maximum interference of the nearest-neighbour path (MST) has maximum interference Θ( √ log n ) withhigh probability. Finally, Sharma et al. [28] consider heuristic solutions to the two-dimensional problem. If communication links are not bidirectional (i.e., edges are directed) and the communication graph is requiredto be strongly connected, then the worst-case maximum interference decreases. Under this model, vonRickenback et al. [31] and Korman [18] give polynomial-time algorithms that return solutions with maximuminterference O (log n ) for any given set of points in the plane, and a one-dimensional construction showingthat in the worst case OPT( P ) ∈ Ω(log n ). In addition to results that examine the problem of minimizing the maximum interference, some work hasaddressed the problem of minimizing the average interference, e.g., Tan et al. [29] and Moscibroda andWattenhofer [24].
Before presenting our results on random sets of points, we begin with a brief discussion regarding the pos-sibility of generalizing existing algorithms that provide approximate solutions for one-dimensional instancesof the interference minimization problem (in an adversarial deterministic input setting).Since the problem of identifying a graph that achieves the optimal (minimum) interference is NP-hard intwo or more dimensions [5], it is natural to ask whether one can design a polynomial-time algorithm to re-turn a good approximate solution. Although Rickenback et al. [30] give a Θ( n / )-approximate algorithm inone dimension [30], the current best polynomial-time algorithm in two (or more) dimensions by Halld´orssonand Tokuyama [12] returns a solution whose maximum interference is O ( √ n ); as noted by Halld´orsson andTokuyama, this algorithm is not known to guarantee any approximation factor better than the immediatebound of O ( √ n ). The algorithm of Rickenback et al. uses two strategies for constructing respective com-munication graphs, and returns the graph with the lower maximum interference; an elegant argument thatdepends on Lemma 1 bounds the resulting worst-case maximum interference by Θ( n / · OPT( P )). The twostrategies correspond roughly to a) MST( P ) (cid:48) and b) classifying every √ n th node as a hub, joining each hubto its left and right neighbouring hubs to form a network backbone, and connecting each remaining nodeto its closest hub. The algorithm of Halld´orsson and Tokuyama applies (cid:15) -nets, resulting in a strategy thatis loosely analogous to a generalization of the hub strategy of Rickenback et al. to higher dimensions. Onemight wonder whether the hybrid approach of Rickenback et al. might be applicable in higher dimensions.Specifically, can a good approximation factor be guaranteed by returning the better of the respective graphs4eturned by the (cid:15) -net algorithm of Halld´orsson and Tokuyama and the communication graph determined byMST( P ) (cid:48) ? To apply this idea directly in two or more dimensions would require generalizing the followingproperty established by von Rickenback et al.: Lemma 1 (von Rickenback et al. [30] (2005))
For any set of points P ⊆ R , OPT( P ) ∈ Ω (cid:16)(cid:112) inter(MST( P ) (cid:48) ) (cid:17) . However, von Rickenback et al. also show that for any n , there exists a set of n points P ⊆ R such thatOPT( P ) ∈ O (1) and inter(MST( P ) (cid:48) ) ∈ Θ( n ), which implies that Lemma 1 does not hold in higher dimen-sions. Consequently, techniques such as those used by von Rickenback et al. to bound the approximationfactor of their algorithm in one dimension do not immediately generalize to higher dimensions. Although using the hybrid approach of von Rickenback et al. [30] directly may not be possible, Kranakiset al. [20] recently showed that if a set P of n points is selected uniformly at random from an interval,then the maximum interference of the communication graph determined by MST( P ) (cid:48) is Θ( √ log n ) with highprobability. Throughout this section, we assume general position of points; specifically, we assume that thedistance between each pair of nodes is unique. This can be expressed formally as ∀{ p , p , q , q } ⊆ P ,dist( p , q ) = dist( p , q ) ⇔ { p , q } = { p , q } . We begin by introducing the following definitions:
Definition 3 (Primitive Edge)
An edge { p, q } ∈ E ( G ) in a communication graph G is primitive if min { r ( p ) , r ( q ) } = dist( p, q ) . Definition 4 (Bridge)
An edge { p, q } ∈ E ( G ) in a communication graph G is bridged if there is a pathjoining p and q in G consisting of at most three edges, each of which is of length less than dist( p, q ) . Definition 5 ( T ( P ) ) Given a set of points P in R d , T ( P ) is the set of all communication graphs G with V ( G ) = P such that no primitive edge { p, q } ∈ E ( G ) is bridged. Let C ( R, r, d ) be the minimum number of d -dimensional balls of radius r required to cover a d -dimensionalball of radius R . The following property follows since R d is a doubling metric space for any constant d [13](equivalently, R d and has constant doubling dimension [10, 11]): Proposition 2 If d ∈ Θ(1) and
R/r ∈ Θ(1) , then C ( R, r, d ) ∈ Θ(1) . We now bound the maximum interference of any graph in T ( P ). Theorem 3
Let P be a set of points in R d . For any graph G ∈ T ( P ) , inter( G ) ∈ O (cid:18) log (cid:18) d max ( G ) d min ( G ) (cid:19)(cid:19) , where d max ( G ) = max { s,t }∈ E ( G ) dist( s, t ) and d min ( G ) = min { s,t }∈ E ( G ) dist( s, t ) . Proof.
We first normalize the scale of P to simplify the proof. Let Q = { p · α | p ∈ P } denote a uniformscaling of P by a factor of α = 1 /d min ( G ) and let H denote the corresponding communication graph. That is, { u, v } ∈ E ( G ) ⇔ { u · α, v · α } ∈ E ( H ). Similarly, scale transmission radii such that each node’s transmissionradius in Q is α times its corresponding node’s transmission radius in P . Thus, d min ( H ) = 1 and d max ( H ) = d max ( G ) d min ( G ) . (3)5e say an edge { q , q } ∈ E ( H ) causes interference at a node p if p is within the transmission range ofeither q or q . Let p be a node in V ( H ) that has interference inter( H ). Let E ( p ) ⊆ E ( H ) be the set ofall primitive edges that cause interference at p . Since there are inter( H ) nodes whose transmission rangescover p , we get that | E ( p ) | ≥ inter( H ) /
2. That is, there are at least inter( H ) / p . Therefore, to prove the theorem it suffices to show that | E ( p ) | ∈ O (log( d max ( H ))) . (4)Let g = (cid:100) log( d max ( H )) (cid:101) . Partition E ( p ) into g + 1 subsets, E , E , . . . , E g , such that for each 0 ≤ i ≤ g , E i is the set of all edges in E ( p ) whose length is in [2 i , i +1 ). Since d max ( H ) ≤ g , it follows that E ( p ) = (cid:91) ≤ i ≤ g E i and ∀ i (cid:54) = j, E i ∩ E j = ∅ . We now show that | E i | ∈ O (1) for every i , 0 ≤ i ≤ g , from which (4) follows immediately.For each integer i , 0 ≤ i ≤ g , let V i be the set of all nodes in V ( H ) that are incident to an edge in E i andlet V (cid:48) i ⊆ V i be the set of nodes in V i that have p in their transmission radii. By our assumption of generalposition, there is an injective function from the set of primitive edges in E i to nodes in V (cid:48) i , giving that | V i | ≥ | V (cid:48) i | ≥ | E i | . (5)By definition of E i , V i , and V (cid:48) i , every node in V (cid:48) i is contained in the ball with centre p and radius 2 i +1 .Furthermore, every node v in V i is contained in the ball with centre p and radius 2 i +2 , because either v ∈ V (cid:48) i or v is adjacent to a node w in V (cid:48) i ; thus, dist( p, v ) ≤ dist( p, w ) + dist( w, v ) ≤ · i +1 . By Proposition 2, fora constant dimension d , C (2 i +1 , i − , d ) ∈ O (1) and C (2 i +2 , i − , d ) ∈ O (1). Suppose | E i | (cid:54)∈ O (1). Hence by(5), | E i | , | V i | , and | V (cid:48) i | are each ω (1). In particular, for a sufficiently large point set, | V (cid:48) i | ≥ C (2 i +1 , i − , d ) · (cid:2) C (2 i +2 , i − , d ) + 1 (cid:3) . (6)Any ball of radius 2 i +1 can be covered with C (2 i +1 , i − , d ) balls of radius 2 i − . Therefore, by (6) andthe pigeonhole principle, there must be a ball B i of radius 2 i − that contains a set of nodes V (cid:48)(cid:48) i , such that V (cid:48)(cid:48) i ⊆ V (cid:48) i and | V (cid:48)(cid:48) i | ≥ C (2 i +2 , i − , d ) + 1. Let W i be the set of nodes in V i that are adjacent to some node in V (cid:48)(cid:48) i by some edge in E i . Since the length of every edge in E i is at least 2 i and the ball B i has radius 2 i − ,every node in W i must lie outside B i . Thus, W i ∩ V (cid:48)(cid:48) i = ∅ . (7)We consider two cases: i) there is a node q in W i that is adjacent to at least two nodes in V (cid:48)(cid:48) i by edges in E i , and ii) every node in W i is adjacent to only one node in V (cid:48)(cid:48) i by some edge in E i , i.e., | W i | ≥ | V (cid:48)(cid:48) i | . Case i.
Let p and p denote two nodes in V (cid:48)(cid:48) i such that edges { p , q } and { p , q } are in E i . Without lossof generality, assume that dist( p , q ) > dist( p , q ) (by our general position assumption). Consider the path (cid:104) p , p , q (cid:105) from p to q . This path has two edges. Also, dist( p , q ) < dist( p , q ) and dist( p , p ) < dist( p , q ),because dist( p , p ) ≤ i − (as p and p are within a ball of radius 2 i − ) and dist( p , p ) ≥ i (as the edge { p , q } is in E i ). Since { p , q } is a primitive edge in H and H ∈ T ( Q ), { p , q } cannot be bridged, derivinga contradiction. Case ii.
We have | W i | ≥ | V (cid:48)(cid:48) i | ≥ C (2 i +2 , i − , d ) + 1. Since every node in W i lies in a ball of radius 2 i +2 (as W i ⊆ V i ), and a ball of radius 2 i +2 can be covered with C (2 i +2 , i − , d ) + 1 balls of radius 2 i − , theremust be a ball of radius 2 i − that contains at least two nodes q and q from W i . By (7), W i ∩ V (cid:48)(cid:48) i = ∅ .By definition, there must be two edges in E i that connect q and q to two distinct nodes p and p in V (cid:48)(cid:48) i . Without loss of generality, assume that dist( p , q ) > dist( p , q ). The length of the edge { p , q } isgreater than those of { p , p } , { p , q } and { q , q } , because dist( p , p ) ≤ i − , dist( q , q ) ≤ i − , anddist( p , q ) ≥ i (as { p , q } ∈ E i ). Therefore, every edge of the path of length three (cid:104) p , p , q , q (cid:105) from p to q has length less than dist( p , q ). (Notice that { p , p } and, similarly, { q , q } , are in E ( H ) since both p and p are inside a ball of radius 2 i − and the transmission ranges of p and p is at least 2 i , as they are6ncident to edges in E i .) Since { p , q } is a primitive edge in H and H ∈ T ( Q ), { p , q } cannot be bridged,deriving a contradiction.A contradiction is derived in both cases. Therefore, (4) holds. The result follows by (3) and (4) since set P and graph G correspond to Q and H , respectively, upon scaling by 1 /α = d min ( G ). (cid:3) In the next lemma we show that MST( P ) (cid:48) is in T ( P ). Consequently, T ( P ) is always non-empty. Lemma 4
For any set of points P ⊆ R d , MST( P ) (cid:48) ∈ T ( P ) . Proof.
The transmission range of each node p ∈ P is determined by the length of the longest edge adjacentto p in MST( P ). Suppose there is a primitive edge { p , p } ∈ MST( P ) that is bridged. Therefore, thereis a path T from p to p in MST( P ) (cid:48) that contains at most three edges, each of which is of length lessthan dist( p , p ). Removing the edge { p , p } partitions MST( P ) into two connected components, where p and p are in different components. By definition, T contains an edge that spans the two components.The two components can be joined using this edge (of length less than dist( p , p )) to obtain a new span-ning tree whose weight is less than that of MST( P ), deriving a contradiction. Therefore, no primitive edge { p , p } ∈ MST( P ) can be bridged, implying MST( P ) (cid:48) ∈ T ( P ). (cid:3) Theorem 3 implies that the interference of any graph G in T ( P ) is bounded asymptotically by thelogarithm of the ratio of the longest and shortest edges in G . While this ratio can be arbitrarily large in theworst case, we show that the ratio is bounded for many typical distributions of points. Specifically, if theratio is O ( n c ) for some constant c , then the maximum interference is O (log n ). Definition 6 ( D ) Let D denote the class of distributions over [0 , d such that for any D ∈ D and any set P of n ≥ points selected independently at random according to D , the minimum distance between any twopoints in P is greater than n − c with high probability, for some constant c (independent of n ). Theorem 5
For any integers d ≥ and n ≥ , any distribution D ∈ D , and any set P of n points, each ofwhich is selected independently at random over [0 , d according to distribution D , with high probability, forall graphs G ∈ T ( P ) , inter( G ) ∈ O (log n ) . Proof.
Let d min ( G ) = min { s,t }∈ E ( G ) dist( s, t ) and d max ( G ) = max { s,t }∈ E ( G ) dist( s, t ). Since points arecontained in [0 , d , d max ( G ) ≤ √ d . Points in P are distributed according to a distribution D ∈ D . ByDefinition 6, with high probability, d min ( G ) ≥ n − c for some constant c . Thus, with high probability, we havelog (cid:18) d max ( G ) d min ( G ) (cid:19) ≤ log (cid:32) √ dn − c (cid:33) . (8)The result follows from (8), Theorem 3, and the fact that log( n c √ d ) ∈ O (log n ) when d and c are constant. (cid:3) Lemma 6
Let D be a distribution with domain [0 , d , for which there is a constant c (cid:48) such that for anypoint x ∈ [0 , d , we have D ( x ) ≤ c (cid:48) , where D ( x ) denotes the probability density function of D at x ∈ [0 , d .Then D ∈ D . Proof.
Let p , p , . . . , p n , be n ≥ , d with distribution D . Let c (cid:48)(cid:48) =1+ log c (cid:48) +2 d and let E i , 1 ≤ i ≤ n , denote the event that there is a point p j , j (cid:54) = i , such that dist( p i , p j ) ≤ n − c (cid:48)(cid:48) .Let the random variable d min be equal to min i (cid:54) = j dist( p i , p j ). We havePr( d min ≤ n − c (cid:48)(cid:48) ) = Pr (cid:95) ≤ i ≤ n E i ≤ (cid:88) ≤ i ≤ n Pr( E i ) , (9)7here the inequality holds by the union bound. To establish an upper bound on Pr( E i ), consider a d -dimen-sional ball B i with centre p i and radius n − c (cid:48)(cid:48) . The probability that there is point p j , j (cid:54) = i , in that ballis at most c (cid:48) times the volume of B i ∩ [0 , d . The volume of B i ∩ [0 , d is at most (2 n − c (cid:48)(cid:48) ) d . Therefore,Pr( E i ) ≤ c (cid:48) (2 n − c (cid:48)(cid:48) ) d for every 1 ≤ i ≤ n . Thus, by (9), we getPr( d min > n − c (cid:48)(cid:48) ) ≥ − (cid:88) ≤ i ≤ n Pr( E i ) ≥ − n · c (cid:48) (cid:16) n − c (cid:48)(cid:48) (cid:17) d = 1 − c (cid:48) d n d +log c (cid:48) +1 ≥ − c (cid:48) d n · d +log c (cid:48) = 1 − n . Therefore, D ∈ D . Note, here c = c (cid:48)(cid:48) in Definition 6. (cid:3) Corollary 7
The uniform distribution with domain [0 , d is in D . By Corollary 7 and Theorem 5, we can conclude that if a set P of n ≥ , d , then with high probability, any communicaiton graph in G ∈ T ( P ) will have maximum interference O (log n ). This is expressed formally in the following corollary: Corollary 8
Choose any integers d ≥ and n ≥ . Let P be a set of n points, each of which is selectedindependently and uniformly at random over [0 , d . With high probability, for all graphs G ∈ T ( P ) , inter( G ) ∈ O (log n ) . Our results apply to the setting of mobility (e.g., mobile ad hoc wireless networks). Each node in a mobilenetwork must periodically exchange information with its neighbours to update its local data storing positionsand transmission radii of nodes within its local neighbourhood. The distribution of mobile nodes dependson the mobility model, which is not necessarily uniform. For example, when the network is distributedover a disc or a box-shaped region, the probability distribution associated with the random waypoint modelachieves its maximum at the centre of the region, whereas the probability of finding a node close to theregion’s boundary approaches zero [14]. Since the maximum value of the probability distribution associatedwith the random waypoint model is constant [14], by Lemma 6 and Theorem 5, we can conclude that at anypoint in time, the maximum interference of the network is O (log n ) with high probability. In general, thisholds for any random mobility model whose corresponding probability distribution has a constant maximumvalue. As discussed in Section 1.1, existing local algorithms for topology control attempt to reduce transmissionradii, but not necessarily the maximum interference. By Lemma 4 and Theorem 5, if P is a set of n pointsselected according to a distribution in D , then with high probability inter(MST( P ) (cid:48) ) ∈ O (log n ). Unfortu-nately, a minimum spanning tree cannot be generated using only local information [17]. Thus, an interestingquestion is whether each node can assign itself a transmission radius using only local information such thatthe resulting communication graph belongs to T ( P ) while remaining connected. We answer this questionaffirmatively and present the first local algorithm ( LocalRadiusReduction ), that assigns a transmission8adius to each node such that if the initial communication graph G max is connected, then the resultingcommunication graph is a connected spanning subgraph of G max that belongs to T ( P ). Consequently, theresulting topology has maximum interference O (log n ) with high probability when nodes are selected ac-cording to any distribution in D . Our algorithm can be applied to any existing topology to refine it andfurther reduce its maximum interference. Thus, our solution can be used either independently, or pairedwith another topology control strategy. The algorithm consists of three phases, which we now describe.Let P be a set of n ≥ R d and let r max : P → R + be a function that returns the maximumtransmission radius allowable at each node. Let G max denote the communication graph determined by P and r max . Suppose G max is connected. Algorithm LocalRadiusReduction assumes that each node isinitially aware of its maximum transmission radius, its spatial coordinates, and its unique identifier.The algorithm begins with a local data acquisition phase, during which every node broadcasts its identity,maximum transmission radius, and coordinates in a node data message. Each message also specifies whetherthe data is associated with the sender or whether it is forwarded from a neighbour. Every node records thenode data it receives and retransmits those messages that were not previously forwarded. Upon completingthis phase, each node is aware of the corresponding data for all nodes within its 2-hop neighbourhood. Thealgorithm then proceeds to an asynchronous transmission radius reduction phase.Consider a node u and let f denote its furthest neighbour. If u and f are bridged in G max , then u reducesits transmission radius to correspond to that of its next-furthest neighbour f (cid:48) , where dist( u, f (cid:48) ) < dist( u, f ).This process iterates until u is not bridged with its furthest neighbour within its reduced transmission radius.We formalize the local transmission radius reduction algorithm in the pseudocode in Table 1 that computesthe new transmission radius r (cid:48) ( u ) at node u .Clearly, Algorithm LocalRadiusReduction is 2-local. Since transmission radii are decreased mono-tonically (and never increased), the while loop iterates O (∆) times, where ∆ denotes the maximum vertexdegree in G max . Consequently, since each call to the subroutine Bridged terminates in O (∆ ) time, eachnode determines its reduced transmission radius r (cid:48) ( u ) in O (∆ ) time.After completing the transmission radius reduction phase, the algorithm concludes with one final ad-justement in the transmission radius to remove asymmetric edges. In this third and final phase, each node u broadcasts its reduced transmission radius r (cid:48) ( u ). Consider the set of nodes { v , . . . , v k } ⊆ Adj( u ) such thatdist( u, v i ) = r (cid:48) ( u ) for all i (when points are in general position, k = 1, and there is a unique such node v ).If r (cid:48) ( v i ) < r (cid:48) ( u ) for all i , then u can reduce its transmission radius to that of its furthest neighbour withwhich bidirectional communication is possible. Specifically, r (cid:48) ( u ) ← max v ∈ Adj( u )dist( u,v ) ≤ min { r (cid:48) ( u ) ,r (cid:48) ( v ) } dist( u, v ) . (10)The new value of r (cid:48) ( u ) as defined in (10) is straightforward to compute in O (∆) time. Lemma 9
The communication graph constructed by Algorithm
LocalRadiusReduction is in T ( P ) andis connected if the initial communication graph G max is connected. Proof.
Let G min denote the communication graph constructed by Algorithm LocalRadiusReduction .First, we prove that G min is connected if G max is connected. Let E dif = {{ u, v } | { u, v } ∈ E ( G max ) \ E ( G min ) and u and v belong to different connected components of G min } Suppose that G max is connected and G min is not connected. Therefore, E dif (cid:54) = ∅ . Let { u (cid:48) , v (cid:48) } ← arg min { u,v }∈ E dif dist( u, v ) . (11)Since { u (cid:48) , v (cid:48) } (cid:54)∈ E ( G min ), then we have that either r (cid:48) ( u (cid:48) ) < dist( u (cid:48) , v (cid:48) ) or r (cid:48) ( v (cid:48) ) < dist( u (cid:48) , v (cid:48) ). Without lossof generality, assume r (cid:48) ( u (cid:48) ) < dist( u (cid:48) , v (cid:48) ). This implies that edge { u (cid:48) , v (cid:48) } is bridged in G max since, otherwise,9 lgorithm LocalRadiusReduction ( u )1 radiusReductionComplete ← false r (cid:48) ( u ) ← r max ( u )3 f ← u // identify u ’s furthest neighbour f for each v ∈ Adj( u )5 if dist( u, v ) > dist( u, f )6 f ← v while ¬ radiusReductionComplete radiusM odif ied ← false if Bridged ( u, f )10 radiusM odif ied ← true f ← u // identify next neighbour within distance r (cid:48) ( u )12 for each v ∈ Adj( u )13 if dist( u, v ) < r (cid:48) ( u ) and dist( u, v ) > dist( u, f )14 f ← v r (cid:48) ( u ) ← dist( u, f )16 radiusReductionComplete ← ¬ radiusM odif ied return r (cid:48) ( u ) Algorithm Bridged ( a, b )1 result ← false for each v ∈ Adj( a )3 if max { dist( a, v ) , dist( v, b ) } < dist( a, b ) and v ∈ Adj( b )4 result ← true for each w ∈ Adj( v )6 if max { dist( a, v ) , dist( v, w ) , dist( w, b ) } < dist( a, b ) and w ∈ Adj( b )7 result ← true return result Table 1: Algorithm
LocalRadiusReduction (cid:48) could not reduce its transmission radius to less than dist( u (cid:48) , v (cid:48) ). By Definition 4, there is a path T between u (cid:48) and v (cid:48) in G max that contains at most three edges, each of which is of length less than dist( u (cid:48) , v (cid:48) ).Since T spans two different connected components in G min , there is an edge { u (cid:48)(cid:48) , v (cid:48)(cid:48) } in T such that u (cid:48)(cid:48) and v (cid:48)(cid:48) belong to two different connected components. Therefore, { u (cid:48)(cid:48) , v (cid:48)(cid:48) } ∈ E dif , as { u (cid:48)(cid:48) , v (cid:48)(cid:48) } ∈ E ( G max ) and { u (cid:48)(cid:48) , v (cid:48)(cid:48) } (cid:54)∈ E ( G min ). Thus, dist( u (cid:48)(cid:48) , v (cid:48)(cid:48) ) < dist( u (cid:48) , v (cid:48) ), contradicting (11). Therefore, G min is connected ifand only if G max is connected.It remains to show that G min ∈ T ( P ). Let { u, v } be any primitive edge in E ( G min ). It suffices to showthat { u, v } is not bridged in G min . By Definition 3, we have that dist( u, v ) = min { r (cid:48) ( u ) , r (cid:48) ( v ) } . Without lossof generality, assume r (cid:48) ( u ) = dist( u, v ). The edge { u, v } is not bridged in G max , otherwise the transmissionradius of u could be further reduced, resulting in the removal of { u, v } at the end of the third phase (whereasymmetric edges are removed). Consequently, { u, v } is not bridged in G min , as G min is a subgraph of G max and any edge that is bridged in G min is also bridged in G max . (cid:3) More generally, since transmission radii are only decreased, it can be shown that G min and G max havethe same number of connected components by applying Lemma 9 on every connected component of G max . We simulated our local interference minimization algorithm to evaluate its performance in static and mobilewireless networks. In both settings, each node collects the list of its 2-hop neighbours in two rounds, appliesthe algorithm to reduce its transmission radius, and then broadcasts its computed transmission radius soneighbouring nodes can eliminate asymmetric edges and possibly further reduce their transmission radii.By the end of this stage, all asymmetric edges are removed and no new asymmetric edges are generated.Consequently, a node need not broadcast its transmission radius again after it has been further reduced.We applied two mobility models to simulate mobile networks: random walk and random waypoint [15]. Inboth models each node’s initial position is a point selected uniformly at random over the simulation region.In the random walk model, each node selects a new speed and direction uniformly at random over [ v min , v max ]and [0 , π ), respectively, at regular intervals. When a node encounters the simulation region’s boundary,its direction is reversed (a rotation of π ) to remain within the simulation region with the same speed. Inthe random waypoint model, each node moves along a straight trajectory with constant speed toward adestination point selected uniformly at random over [ v min , v max ] and the simulation region, respectively.Upon reaching its destination, the node stops for a random pause time, after which it selects a new randomdestination and speed, and the process repeats. We set the simulation region’s dimensions to 1000 metres × n from 50 to 1000 in increments of 50. We fixed the maximumtransmission radius r max for each network to 100, 200, or 300 metres. To compute the average maximuminterference for static networks, for each n and r max we generated 100,000 static networks, each with n nodes and maximum transmission radius r max , distributed uniformly at random in the simulation region. Tocompute the average maximum interference for mobile networks, for each n and r max we generated 100,000snapshots for each mobility model, each with n nodes and maximum transmission radius r max . We set thespeed interval to [0 . ,
10] metres per second, and the pause time interval to [0 ,
10] seconds (in the waypointmodel). A snapshot of the network was recorded once every second over a simulation of 100,000 seconds.
We compared the average maximum interference of the topology constructed by the algorithm
LocalRa-diusReduction against the corresponding average maximum interference achieved respectively by two localtopology control algorithms: i) the local computation of the intersection of the Gabriel graph and the unit11
100 200 300 400 500 600 700 800 900 1000050100150200250300350 Number of Nodes A v e r age I n t e r f e r en c e LocalRadiusReductionCBTCGabrielUDG 100UDG 200UDG 300
Figure 1: Comparing the maximum interference of the
LocalRadiusReduction algorithm against otherlocal topology control algorithms on a static network A v e r age I n t e r f e r en c e LocalRadiusReductionCBTCGabrielUDG 100UDG 200UDG 300
Figure 2: Data from Figure 1 displayed with a bounded y -axis to emphasize relative differences12isc graph (with unit radius r max ) [4], and ii) the cone-based local topology control (CBTC) algorithm [21].In addition, we evaluated the maximum interference achieved when each node uses a fixed radius of com-munication, i.e., the communication graph is a unit disc graph of radius r max (100, 200, or 300 metres,respectively). These results are displayed in Figures 1 and 2. A v e r age I n t e r f e r en c e The Average Interference of Mobile and Static Optimized Network LocalRadiusReduction Random WalkLocalRadiusReduction Random WaypointLocalRadiusReduction Static
Figure 3: Comparing the maximum interference of the
LocalRadiusReduction algorithm on static andmobile networks using both the random walk and random waypoint mobility models A v e r age I n t e r f e r en c e The Average Interference of Mobile and Static Optimized Network LocalRadiusReduction Random WalkLocalRadiusReduction Random WaypointLocalRadiusReduction Static
Figure 4: Data from Figure 3 displayed using a logarithmic scale on the x -axisAs shown, the average maximum inteference of the unit disc graph topologies increases linearly with n . Many of the unit disc graphs generated were disconnected when the transmission radius was set to 100metres for small n . Since we require connectivity, we only considered values of n and r max for which atleast half of the networks generated were connected. When r max = 100 metres, a higher average maximuminterference was measured at n = 300 than at n = 400. This is because many networks generated for n = 300were discarded due to being disconnected. Consequently, the density of networks simulated for n = 300 washigher than the average density of a random network with n = 300 nodes, resulting in higher maximuminterference. 13lthough both the local Gabriel and CBTC algorithms performed significantly better than the unit discgraphs, the lowest average maximum interference was achieved by the LocalRadiusReduction algorithm,which is clearly seen to be logarithmic in n in Figures 3 and 4. Note that the LocalRadiusReduction algorithm reduces the maximum interference to O (log n ) with high probability, irrespective of the initialmaximum transmission radius r max .Figures 3 and 4 display the average maximum interference achieved by LocalRadiusReduction onmobile networks, plotting simulation results for both the random walk and random waypoint models, alongwith the corresponding results on a static network. Simulation results obtained using the random walkmodel closely match those obtained on a static network because the distribution of nodes at any time duringa random walk is nearly uniform [8]. The average maximum interference increases slightly but remainslogarithmic when the random waypoint model is used. The spatial distribution of nodes moving accordingto a random waypoint model is not uniform, and is maximized at the centre of the simulation region [14].Consequently, the density of nodes is high near the centre, resulting in greater interference at these nodes.Finally, we evaluated the algorithm
LocalRadiusReduction using actual mobility trace data of Pi-orkowski et al. [26], consisting of GPS coordinates for trajectories of 537 taxi vehicles recorded between May17 and June 10, 2008, driving throughout the San Fransisco Bay area. Each taxi’s trace contains between1000 and 20,000 sample points. We selected the 500 largest traces, each of which has over 8000 samplepoints. To implement our algorithm, we selected n taxis among the 500 uniformly at random, ranging from n = 50 to n = 500 in increments of 50. As seen in Figure 5, the resulting average maximum interference issimilar to that measured in our simulation results.
50 100 150 200 250 300 350 400 450 500020406080100120 The Average Interference of Real GPS−Taxi NetworkNumber of Taxis A v e r age I n t e r f e r en c e UDG 0.01LocalRadiusReduction
Figure 5: Comparing the maximum interference of the
LocalRadiusReduction against a unit disc graphon actual mobile data
Acknowledgements
Stephane Durocher would like to thank Csaba T´oth for insightful discussions related to the interferenceminimization problem in one dimension.
References [1] M. Benkert, J. Gudmundsson, H. Haverkort, and A. Wolff. Constructing minimum-interference net-works.
Comp. Geom.: Theory & App. , 40(3):179–194, 2008.142] D. Bil`o and G. Proietti. On the complexity of minimizing interference in ad-hoc and sensor networks.
Theor. Comp. Sci. , 402(1):42–55, 2008.[3] P. Bose, J. Gudmundsson, and M. Smid. Constructing plane spanners of bounded degree and low weight.
Algorithmica , 42(3–4):249–264, 2005.[4] P. Bose, P. Morin, I. Stojmenovi´c, and J. Urrutia. Routing with guaranteed delivery in ad hoc wirelessnetworks.
Wireless Net. , 7(6):609–616, 2001.[5] K. Buchin. Minimizing the maximum interference is hard.
CoRR , abs/0802.2134, 2008.[6] M. Burkhart, P. von Rickenbach, R. Wattenhofer, and A. Zollinger. Does topology control reduceinterference? In
Proc. ACM MobiHoc , pages 9–19, 2004.[7] M. Damian, S. Pandit, and S. V. Pemmaraju. Local approximation schemes for topology control. In
Proc. ACM PODC , pages 208–218, 2006.[8] A. Das Sarma, D. Nanongkai, and G. Pandurangan. Fast distributed random walks. In
Proc. ACMPODC , pages 161–170, 2009.[9] S. Durocher, E. Kranakis, D. Krizanc, and L. Narayanan. Balancing traffic load using one-turn rectilinearrouting.
J. Interconn. Net. , 10(1–2):93–120, 2009.[10] P. Fraigniaud, E. Lebhar, and Z. Lotker. A doubling dimension threshold θ (log log n ) for augmentedgraph navigability. In Proc. ESA , volume 4168 of
LNCS , pages 376–386. Springer, 2006.[11] A. Gupta, R. Krauthgamer, and J.R. Lee. Bounded geometries, fractals, and low-distortion embeddings.In
Proc. IEEE FOCS , pages 534–543, 2003.[12] M. M. Halld´orsson and T. Tokuyama. Minimizing interference of a wireless ad-hoc network in a plane.
Theor. Comp. Sci. , 402(1):29–42, 2008.[13] J. Heinonen.
Lectures on analysis on metric spaces . Springer-Verlag, New York, 2001.[14] E. Hyyti¨a, P. Lassila, and J. Virtamo. Spatial node distribution of the random waypoint mobility modelwith applications.
IEEE Trans. Mob. Comp. , 6(5):680–694, 2006.[15] D. B. Johnson and D. A. Maltz. Dynamic source routing in ad hoc wireless networks. In T. Imielinskiand H. Korth, editors,
Mobile Computing , volume 353. Kluwer Academic Publishers, 1996.[16] I. Kanj, L. Perkovi´c, and G. Xia. Computing lightweight spanners locally. In
Proc. DISC , volume 5218of
LNCS , pages 365–378. Springer, 2008.[17] M. Khan, G. Pandurangan, and V. S. Anil Kumar. Distributed algorithms for constructing approximateminimum spanning trees in wireless sensor networks.
IEEE Trans. Parallel & Dist. Sys. , 20(1):124–139,2009.[18] M. Korman. Minimizing interference in ad-hoc networks with bounded communication radius. In
Proc.ISAAC , LNCS. Springer, 2011. To appear.[19] D. Kowalski and M. Rokicki. Connectivity problem in wireless networks. In
Proc. DISC , volume 6343of
LNCS , pages 344–358. Springer, 2010.[20] E. Kranakis, D. Krizanc, P. Morin, L. Narayanan, and L. Stacho. A tight bound on the maximuminterference of random sensors in the highway model.
CoRR , abs/1007.2120, 2010.[21] L. Li, J. Y. Halpern, P. Bahl, Y.-M. Wang, and R. Wattenhofer. A cone-based distributed topology-control algorithm for wireless multi-hop networks.
IEEE/ACM Trans. Net. , 13(1):147–159, 2005.1522] X.-Y. Li, G. Calinescu, and P.-J. Wan. Distributed construction of a planar spanner and routing for adhoc wireless networks. In
Proc. IEEE INFOCOM , pages 1268–1277, 2002.[23] T. Locher, P. von Rickenbach, and R. Wattenhofer. Sensor networks continue to puzzle: Selected openproblems. In
Proc. ICDCN , volume 4904 of
LNCS , pages 25–38. Springer, 2008.[24] T. Moscibroda and R. Wattenhofer. Minimizing interference in ad hoc and sensor networks. In
Proc.ACM DIALM-POMC , pages 24–33, 2005.[25] G. Narasimhan and M. Smid.
Geometric Spanner Networks . Cambridge University Press, 2007.[26] M. Piorkowski, N. Sarafijanovic-Djukic, and M. Grossglauser. CRAWDAD data set epfl/mobility (v.2009-02-24). http://crawdad.cs.dartmouth.edu/epfl/mobility, 2009.[27] P. Santi. Topology control in wireless ad hoc and sensor networks.
ACM Comp. Surv. , 37(2):164–194,2005.[28] A. Sharma, N. Thakral, S. Udgata, and A. Pujari. Heuristics for minimizing interference in sensornetworks. In
Proc. ICDCN , volume 5408 of
LNCS , pages 49–54. Springer, 2009.[29] H. Tan, T. Lou, F. Lau, Y. Wang, and S. Chen. Minimizing interference for the highway model in wirelessad-hoc and sensor networks. In
Proc. SOFSEM , volume 6543 of
LNCS , pages 520–532. Springer, 2011.[30] P. von Rickenbach, S. Schmid, R. Wattenhofer, and A. Zollinger. A robust interference model for wirelessad hoc networks. In
Proc. IEEE IPDPS , pages 1–8, 2005.[31] P. von Rickenbach, R. Wattenhofer, and A. Zollinger. Algorithmic models of interference in wireless adhoc and sensor networks.