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D istributed C omputing G roup

Locality and the Hardness of Distributed Approximation Thomas Moscibroda Joint work with: Fabian Kuhn, Roger Wattenhofer. D istributed C omputing G roup. Locality. Communication in multi-hop networks is inherently local ! Issue of locality is crucial in distributed systems!

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D istributed C omputing G roup

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  1. Locality and the Hardness of Distributed ApproximationThomas MoscibrodaJoint work with: Fabian Kuhn, Roger Wattenhofer DistributedComputing Group

  2. Locality... • Communication in multi-hop networks is inherently local ! • Issue of locality is crucial in distributed systems! • Direct communication only between neighbors • Obtaining information about distant nodes requires multi-hop communication Obtaining information from entire network requires plenty of communication! Thomas Moscibroda @ DANSS 2004

  3. Locality... • Many modern networks arelarge-scale and highly complex • Internet • Peer-to-Peer Networks • Wireless Sensor Networks • Or even dynamic... • Wireless Ad Hoc Networks • No node has global information • Each node has information from its vicinity only (local information)  Yet, nodes have to come up with a global goal! Thomas Moscibroda @ DANSS 2004

  4. Nodes have only local information • Nodes have to optimize a global goal Example: Global Goal – Local Information • Clustering in Wireless Sensor Networks • Choose Clusterheads such that • Every node is either a clusterhead or... • ...has a clusterhead in its neighborhood. • Idea: Clusterhead sense environment Non-clusterheads can go to sleep mode  Save energy! • Goal: We want as few clusterheads as possible! (Minimum Dominating Set Problem) Crucial for fast Algorithms! Thomas Moscibroda @ DANSS 2004

  5. k-Neighborhood • What does „local“ mean ? • How far does „locality“ go? • Neighborhood, 2-Hop Neighborhood,... or something else...??? 3 1 2 In communication round: Thomas Moscibroda @ DANSS 2004

  6. k-Neighborhood In k rounds of communication... ... each node can gather information only from k-neighborhood! • If message size is unbounded:  Entire information from k-neighborhood (IDs, topology, edge-weights,...) can be collected! • If message size is bounded:  Only subset of this information can be gathered. Strongest model for lower bounds on locality! Thomas Moscibroda @ DANSS 2004

  7. What can be computed locally? [Naor,Stockmeyer;1993] • We want to establish a trade-off between amount of communication and quality of the global solution. LOCALITY Communication Rounds GLOBAL GOAL Approximation TRADE-OFF • Upper Bounds:  Constant-Time Approximation Algorithms • Lower Bounds:  Hardness of Distributed Approximation Thomas Moscibroda @ DANSS 2004

  8. What can be computed locally? [Naor,Stockmeyer;1993] • How well can global tasks be locally approximated? • Minimum Dominating Set (Choose minimum S µ V, s.t. each v2V is in S or has at least one neighbor in S) • Minimum Vertex Cover (Choose minimum S µ V, s.t. each e2E has one node in S) Both problems appear to be local in nature! Thomas Moscibroda @ DANSS 2004

  9. What can be computed locally? [Naor,Stockmeyer;1993] • An answer to the previous question... ...helps in answering the following question about exact variants of the problems. How large must the locality be in order to compute a maximal independent set or maximal matching? • An answer to this question implies important time-lower bounds for distributed algorithms! Thomas Moscibroda @ DANSS 2004

  10. Overview • Introduction to Locality • Related Work • Vertex Cover Upper Bound • Vertex Cover Lower Bound • Conclusions & Open Problems Thomas Moscibroda @ DANSS 2004

  11. Related Work On Locality • Naor, Stockmeyer 1993: Which locally checkable labelings can be computed in constant time? [Naor,Stockmeyer;1993] • O(log n) algorithm for maximal independent set [Luby;1986] O(log n) algorithm for maximal matching [Israeli, Itai;1986] • 3-coloring in a ring in O(log*n) time [Cole, Vishkin;1986] • O(log*n) was shown to be optimal by Linial [Linial;1992]  Only previous lower bound on locality! Thomas Moscibroda @ DANSS 2004

  12. Related Work On Distributed Approximation Upper Bounds (examples...) • Minimum Dominating Set Problem [Jia, Rajaraman, Suel; 2001] [Kuhn, Wattenhofer; 2003],... • Minimum Edge-Coloring [Panconesi, Srinivasan; 1997],... • General covering and packing problems [Bartal, Byers, Raz; 1997][Kuhn, Moscibroda, Wattenhofer; submitted] • Facility Location Lower Bounds • Results based on Linials (log*n) lower bound • Recently, strong lower bound on the distributed approximability for the MST. [Elkin; 2004] For general graphs, we drastically improve this result. Thomas Moscibroda @ DANSS 2004

  13. Overview • Introduction to Locality • Related Work • Vertex Cover Upper Bound • Vertex Cover Lower Bound • Conclusions & Open Problems Thomas Moscibroda @ DANSS 2004

  14. Minimum Vertex Cover • We consider the most basic coordination Problem  Minimum Vertex Cover (MVC) Choose as few nodes as possible to cover all edges • We give an approximation algorithm with...  O(k) communication rounds  O(1/k) approximation  O(log n) bits message size • General idea: Consider the integer linear program of MVC. • Distributed Primal-Dual Approach Thomas Moscibroda @ DANSS 2004

  15. Minimum Vertex Cover (Fractional) MVC is captured by the following linear program Its dual is the fractional maximum matching (MM) problem Thomas Moscibroda @ DANSS 2004

  16. Distributed MVC Algorithm • Each node stores a value xi • Each edge stores a value yj • In a real network: edge is simulated by incident node! • Idea: • Compute a feasible solution for MVC • While doing so: Distribute the dual values yj among incident, uncovered edges, hence • We show that  This yields an O(1/k) approximation! y4 =1/3 y3 xi =1 y2 =1/3 y1 =1/3 Thomas Moscibroda @ DANSS 2004

  17. Distributed MVC Algorithm Number of incident, uncovered edges Maximum i in neighborhood If relative number of uncovered edges is high  join VC It can be shown that: i·(l+1)/k If sum of yj in neighborhood is ¸ 1: • Pick node and distribute yj proportionally! Thomas Moscibroda @ DANSS 2004

  18. Analysis Lemma: Idea: Bound the sum of the incident dual variables for each node i. At the end of the algorithm, for all nodes vi2 V: Yi· 3+1/k Proof: Let i denote the ith iteration of the loop Case 1: Consider a node which does not join the vertex cover! • Until 0 ,, Yi is smaller than 1 • In 0 , i must be 0 • All neighbors must have joined VC before 0  Yi· 1 Thomas Moscibroda @ DANSS 2004

  19. Analysis Case 3 is similar beforel : duringl : Case 2: Consider a node that joins VC in line 5 of iteration l. 0 0.33 vi 0.33 0.62 0 0.3 • Before l: Yi· 1 • During l: • xi := 1  Yi· 2 • neighbors vk may join VC, too  increase Yi • All these nodes have k¸i(1) l/(l+1) ¸il/(l+1) • We get at most 1/kfrom each of the v uncovered neighbors!  v / k·1/k vk 0 0.33 Additional nodes joining VC can increase Yi by at most 1 Yi· 3+1/k Thomas Moscibroda @ DANSS 2004

  20. Summary VC-Algorithm Can we do better? • Algorithm runs in O(k) rounds  Equivalent: Locality is O(k) hops! • Message size is O(log n) bits  Approximation quality is 1/k+O(1) How many rounds are necessary for a O(1) or O(polylog ) approximation? O(log ) time  O(1) approximation O(log /loglog ) time  O(polylog ) approximation Thomas Moscibroda @ DANSS 2004

  21. Overview • Introduction to Locality • Related Work • Vertex Cover Upper Bound • Vertex Cover Lower Bound • Conclusions & Open Problems Thomas Moscibroda @ DANSS 2004

  22. Model • Network graph = graph on which we compute VC • Nodes have unique identifiers • Message size and local computation are unbounded • Strongest possible model for lower bounds Lower bounds are consequence of locality alone Thomas Moscibroda @ DANSS 2004

  23. S S 0 1 Basic Idea • S0 and S1 contain n0 and n0/ nodes, resp. • Optimal VC does not contain nodes of S0 • Basic Structure of our proof: • Construct graph such that nodes in S0 and S1 have same view • Algorithm has to take nodes of S0 in order to cover edges between S0 and S1 • VCALG >> VCOPT because |S0| >> |S1| Thomas Moscibroda @ DANSS 2004

  24. S S 1 0 One Round Lower Bound Thomas Moscibroda @ DANSS 2004

  25. S S 0 1 One Round Lower Bound Thomas Moscibroda @ DANSS 2004

  26. S S 0 1 Two Round Lower Bound Thomas Moscibroda @ DANSS 2004

  27. View of node in S1 7 7 7 7 1 4 1 4 2 4 2 7 7 7 7 S S 1 0 Two Round Lower Bound: Views View of node in S0 1 4 1 1 2 3 1 2 3 Thomas Moscibroda @ DANSS 2004

  28. The Cluster Tree I Thomas Moscibroda @ DANSS 2004

  29. The Cluster Tree II • Cluster Tree = a tree of clusters of nodes • Recursively defined for k>0 • Defines the structure of a graph Gk • Each link on the tree is a bipartite sub-graph of Gk • If girth of Gk is at least 2k+1, nodes in S0 and S1 have the same view up to distance k Thomas Moscibroda @ DANSS 2004

  30. Construction of Gk • How can we achieve high girth? • Gk is a bipartite graph (even level clusters / odd level clusters) • For prime power q, D(r,q) is bipartite graph with 2qr nodes and girth at least r+5[Lazebnik,Ustimenko; Explicit Construction of Graphs With an Arbitrary Large Girth and of Large Size; 1995] • If >k, Gk can be constructed as sub-graph of D(2k-4,q) for q=O(k)  Gk has n=O(k2) nodes This is according to Intuition... ...because every node in S0 must see a tree up to distance k. Thomas Moscibroda @ DANSS 2004

  31. Bounding the Optimum • The number of nodes decreases by factor at least /k on each level. • If >2k, n <n0+2n0k/ • All nodes V \ S0 form a feasible vertex cover, hence geometric series · n0k3/3: |VCOPT| < 2n0k/ · n0k2/2: · n0k/: n0: Thomas Moscibroda @ DANSS 2004

  32. v1 v0 Bounding any distributed algorithm • Assume that the labeling (IDs) is chosen uniformly at random: • Nodes v0 in S0 and v1 in S1 see • Same topology • Same probability distribution of labels • Both have same probability for being in VC (probability p) • p is at least ½,  otherwise there is a probability that VC is not feasible!  Therefore: At least half of the nodes in S0 join VC! • For all algorithms, there is labeling with |VCALG|¸n0/2 • Randomized: |VCALG|¸n0/2 by Yao’s minimax principle Thomas Moscibroda @ DANSS 2004

  33. Approximation Lower Bound • We have |VCALG| ¸ n0/2 and |VCOPT| < 2n0k/Approximation ¸ |VCALG|/|VCOPT| > /(4k) • n = O(k2),  = k+1 In k communication rounds, no algorithm can approximate MVC better than (nc/k2/k) or (1/k/k) Thomas Moscibroda @ DANSS 2004

  34. Time Lower Bound • For constant/polylog approximation, we need • Recall our vertex cover algorithm Can we do better? O(log ) time  O(1) approximation O(log /loglog ) time  O(polylog ) approximation Algorithm tight for polylog approximation and tight up to O(loglog ) for constant approximation! Thomas Moscibroda @ DANSS 2004

  35. Hardness of Approximation  Exact Problems • Approximation Theory is very active area of research (see STOC, FOCS, SODA,...) • Study of lower bounds on approximation  Hardness of Approximation! • This has lead to new insight in complexity theory (PCP,...) Study of Approximability Better understanding of exact problems! Thomas Moscibroda @ DANSS 2004

  36. Hardness of Approximation  Exact Problems • Maximal matching (MM) is 2-approximation for MVC… • (MM) is maximal independent set (MIS) on line graph, … Does the same hold in distributed computing? To some degree, it does....! Compare with (log*n) lower bound on ring and O(log n) upper bound Time lower bounds for MIS and Maximal Matching Thomas Moscibroda @ DANSS 2004

  37. What about Dominating Sets ? • For each VC instance, there is graph on which dominating set is the same MVC bounds also hold for MDS • Approximation lower bound can also be extended to maximum matching (more than just a reduction) Thomas Moscibroda @ DANSS 2004

  38. Conclusions • Locality is vital in distributed systems. • Not much is known so far... • In this talk, lower bounds on local computation  tight up to a factor of and  Vertex Cover, Dominating Set, Maximum Matching, MIS, Maximal Matching,... • The hardness of distributed approximation is an interesting research topic Thomas Moscibroda @ DANSS 2004

  39. Questions?Comments? DistributedComputing Group Thomas Moscibroda @ DANSS 2004

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