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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch

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Discrete Algs for Mobile Wireless Sys2 Lecture 23 Topic: Lower Bounds for Dominating Sets and Related Problems Sources: Kuhn, Moscibroda and Wattenhofer, "What Cannot Be Computed Locally!" MIT 6.885 Fall 2008 slides

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Discrete Algs for Mobile Wireless Sys3 Locality Locality means that nodes only have to communicate (even indirectly) with nodes that are close by Desirable property of a distributed algorithm: local algorithms have (the possibility of) low time complexity why bother far away nodes?

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Discrete Algs for Mobile Wireless Sys4 Locality k communication rounds means being restricted to a locality radius k. 1 rounds2 rounds3 rounds

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Discrete Algs for Mobile Wireless Sys5 Locality Can we find local algorithms for various distributed problems? means time complexity (number of rounds) is independent of network size A few positive results, e.g.: Naor & Stockmeyer: studied a class of problems called locally checkable labelings and showed there are non- trivial LCL problems that have local algorithms, including a variant of dining philosophers What about negative results (lower bounds)? Linial: coloring on a ring takes (log*n) rounds What about for dominating set and related problems?

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Discrete Algs for Mobile Wireless Sys6 Minimum Vertex Cover Minimum Vertex Cover problem: Given a graph, find smallest subset S of vertices (nodes) such that every edge is "covered" by a node in S (at least one endpoint is in S) NP-complete consider polynomial time approximation algorithms

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Discrete Algs for Mobile Wireless Sys7 Overview of [KMW] Results Any k-round MVC algorithm has an approximation ratio that is (n c/k*k /k), where n is number of nodes and c is a constant > 1/4 To ensure that the approximation ratio is no more than poly-log, k has to be at least ( (log n / log log n)), which is not local Any k-round MVC algorithm has an approximation ratio that is ( 1/k /k), where is the maximum degree To ensure that the approximation ratio is no more than poly-log, k has to be at least (log / log log ), which is not local

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Discrete Algs for Mobile Wireless Sys8 Some Special Case Graphs Consider a ring: minimum VC consists of every other node constant-time approx algorithm is to include every node approx ratio w.r.t. n is 2 Generalize to a d-regular graph Consider a tree: minimum VC consists of every other node down each branch constant-time approx algorithm is to include every non- leaf node approx ratio w.r.t. n is 2

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Discrete Algs for Mobile Wireless Sys9 Some More Special Case Graphs Consider graphs with constant max degree : constant time approx alg is to include every node approx ratio w.r.t. is constant Consider graphs that contain nodes with high degree (say, (n)): then diameter is small (say, O(1)), so in constant time, an alg can learn the entire graph and choose exactly which nodes to include approx ratio is 1 To show non-locality property, need to consider more complicated graphs…

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Discrete Algs for Mobile Wireless Sys10 Intuition for Locality-Based Lower Bounds In k rounds of communication (time k), every node can collect information about its k-neighborhood Hence, the solution of a node v in a distributed k- round computation can only depend on the k-hop neighborhood of v If two nodes u and v have the same k-hop neighborhoods, they will make the same decision: the execution of a k-round algorithm looks the same to both nodes

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Discrete Algs for Mobile Wireless Sys11 Example for Locality-Based Lower Bound How to prove such a lower bound? Let’s look at case k=2 to get the basic intuition After 1 round, nodes know their neighbors After 2 rounds, nodes know the neighbors of their neighbors

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Discrete Algs for Mobile Wireless Sys12 Two-Round Lower Bound … m2m2 … … … m … m2m2 m3m3 m nodes … … m 2 -1 m complete same view

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Discrete Algs for Mobile Wireless Sys13 Hint of Proof Construct graph G k for each k > 0 containing a bipartite subgraph S with node set C 0 U C 1 C 0 has n 0 nodes, each with d 0 neighbors in C 1 C 1 has n 1 nodes, each with d 1 neighbors in C 0 n 1 = n 0 *d 0 /d 1 C 0, n 0 = 4, d 0 = 4 C 1, n 1 = 8, d 1 = 2

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Discrete Algs for Mobile Wireless Sys14 Hint of Proof In a globally optimal solution, all edges of S (the bipartite graph) can be covered by choosing all nodes of C 1, and none of C 0, to be in the VC But in a local algorithm, decision can only be made based on k-neighborhood Construct G k so that two adjacent nodes (one in C 0 and one in C 1 ) have the same k-neighborhood and thus do the same thing (both join the VC) Since symmetry cannot be broken in only k rounds, suboptimal local decisions are made and a suboptimal approximation ratio achieved

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Discrete Algs for Mobile Wireless Sys15 Constructing G k The heart of the paper is recursive construction of G k with high degree of symmetry See Appendix for G 3 What do we do with G k ? Have to consider what happens with node IDs

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Discrete Algs for Mobile Wireless Sys16 Handling Node IDs Assume random node ID assignment with IDs from {1,…,N} If nodes u and v see same topology up to distance k: Every possible ID assignment is equally probable Probability to see a particular ID assignment equal for u and v u and v make the same decision with the same probability

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Discrete Algs for Mobile Wireless Sys17 Handling Node IDs Deterministic algorithms: there exists a node assignment for which solution is at least as bad as expected value with random IDs Randomized algorithms: Same bound using Yao’s principle

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Discrete Algs for Mobile Wireless Sys18 Hints on Rest of Proof Lemma: Any (randomized or deterministic) k- round distributed algorithm, when run on G k, puts at least half the nodes of C 0 into the VC. Proof is based on constructed properties of G k and previous discussion about handling IDs. So approx ratio is at least (n 0 /2) / (n – n 0 ), since optimal solution does not need any node in C 0 Do some math to show that the construction of G k can be tweaked to ensure that n 0 is sufficiently large relative to n to show the claimed lower bounds w.r.t. n and .

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Discrete Algs for Mobile Wireless Sys19 Relationship to Dominating Sets Theorem: Every (randomized or deterministic) k- round distributed algorithm for MDS has same asymptotic lower bounds on approx ratio as for MVC: (n c/k*k /k) and ( 1/k /k). Proof: By reduction. Let A be a k-round alg for MDS with approx ratio R. Show how to use A DS as a subroutine in algorithm A VC to approximate MVC with only a constant number of extra rounds Analyze the resulting approx ratio for the MVC problem

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Discrete Algs for Mobile Wireless Sys20 Reduction Here is algorithm A VC : 1.Suppose input graph for MVC is G' 2.Simulate another graph G see next slide 3.Call MDS approx alg A DS on G 4.A DS returns some set of nodes S 5.Return S as an approx MVC for G'

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Discrete Algs for Mobile Wireless Sys21 Reduction Transform G' into G: ab dc a b d c ab bc cd adbd

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Discrete Algs for Mobile Wireless Sys22 VC to DS Any VC of G' is a DS of G: ab dc a b d c ab bc cd adbd red nodes cover all edges red nodes cover all nodes

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Discrete Algs for Mobile Wireless Sys23 DS to VC Take any DS of G, replace any green node with a non- green neighbor; result is a VC of G' ab dc a b d c ab bc cd adbd red nodes cover all nodes red nodes cover all edges

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Discrete Algs for Mobile Wireless Sys24 Relating Quality of Approximations Algorithm A DS returns S, a DS of G that is at most R times as large as an optimal DS of G Size of optimal DS of G is ≤ size of optimal VC on G' since every VC of G' forms a DS of G Thus S is a VC of G' that is at most R times as large as an optimal VC of G' By MVC lower bound R must be at least …

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Discrete Algs for Mobile Wireless Sys25 Summary: Lower Bound Lower bound shows that the time-approximation trade-off of the existing algorithm is not too far off the optimum (there still is a significant gap…) By a reduction, the time lower bound for polylog approximations also holds for the apparently unrelated problem of computing a maximal independent set Remark: The lower bound is obtained by using very special graphs. This is definitely not how wireless network graphs look! In fact, for special graph classes, we can do better.

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