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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

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

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CPSC 689: Discrete Algorithms for Mobile and Wireless Systems

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

  2. 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 Discrete Algs for Mobile Wireless Sys

  3. 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? Discrete Algs for Mobile Wireless Sys

  4. Locality 1 rounds 2 rounds 3 rounds k communication rounds means being restricted to a locality radius k. Discrete Algs for Mobile Wireless Sys

  5. 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? Discrete Algs for Mobile Wireless Sys

  6. 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 Discrete Algs for Mobile Wireless Sys

  7. Overview of [KMW] Results • Any k-round MVC algorithm has an approximation ratio that is (nc/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 Discrete Algs for Mobile Wireless Sys

  8. 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 Discrete Algs for Mobile Wireless Sys

  9. 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… Discrete Algs for Mobile Wireless Sys

  10. 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 Discrete Algs for Mobile Wireless Sys

  11. 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 Discrete Algs for Mobile Wireless Sys

  12. Two-Round Lower Bound m nodes m3 … … … … complete m m … … m2 m2 m2-1 m nodes … same view Discrete Algs for Mobile Wireless Sys

  13. C1, n1 = 8, d1 = 2 C0, n0 = 4, d0 = 4 Hint of Proof • Construct graph Gk for each k > 0 containing a bipartite subgraph S with node set C0 U C1 • C0 has n0 nodes, each with d0 neighbors in C1 • C1 has n1 nodes, each with d1 neighbors in C0 • n1 = n0*d0/d1 Discrete Algs for Mobile Wireless Sys

  14. Hint of Proof • In a globally optimal solution, all edges of S (the bipartite graph) can be covered by choosing all nodes of C1, and none of C0, to be in the VC • But in a local algorithm, decision can only be made based on k-neighborhood • Construct Gk so that two adjacent nodes (one in C0 and one in C1) 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 Discrete Algs for Mobile Wireless Sys

  15. Constructing Gk • The heart of the paper is recursive construction of Gk with high degree of symmetry • See Appendix for G3 • What do we do with Gk? Have to consider what happens with node IDs Discrete Algs for Mobile Wireless Sys

  16. 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 Discrete Algs for Mobile Wireless Sys

  17. 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 Discrete Algs for Mobile Wireless Sys

  18. Hints on Rest of Proof • Lemma: Any (randomized or deterministic) k-round distributed algorithm, when run on Gk, puts at least half the nodes of C0 into the VC. • Proof is based on constructed properties of Gk and previous discussion about handling IDs. • So approx ratio is at least (n0/2) / (n – n0), since optimal solution does not need any node in C0 • Do some math to show that the construction of Gk can be tweaked to ensure that n0 is sufficiently large relative to n to show the claimed lower bounds w.r.t. n and . Discrete Algs for Mobile Wireless Sys

  19. 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: (nc/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 ADS as a subroutine in algorithm AVC to approximate MVC with only a constant number of extra rounds • Analyze the resulting approx ratio for the MVC problem Discrete Algs for Mobile Wireless Sys

  20. Reduction Here is algorithm AVC: • Suppose input graph for MVC is G' • Simulate another graph G • see next slide • Call MDS approx alg ADS on G • ADS returns some set of nodes S • Return S as an approx MVC for G' Discrete Algs for Mobile Wireless Sys

  21. a b c a b c d Reduction • Transform G' into G: ab ad bd bc d cd Discrete Algs for Mobile Wireless Sys

  22. a c a c VC to DS • Any VC of G' is a DS of G: b ab b ad bd bc d red nodes cover all edges cd d red nodes cover all nodes Discrete Algs for Mobile Wireless Sys

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

  24. Relating Quality of Approximations • Algorithm ADS 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 … Discrete Algs for Mobile Wireless Sys

  25. 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. Discrete Algs for Mobile Wireless Sys

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