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

## Presentation on theme: "CPSC 689: Discrete Algorithms for Mobile and Wireless Systems Spring 2009 Prof. Jennifer Welch."— Presentation transcript:

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

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

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?

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

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?

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

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

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

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…

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

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

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

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

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

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

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

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

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 .

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

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'

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

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

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

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 …

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.

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

Similar presentations