1. Distributed Computing Group Locality and the Hardness of Distributed Approximation Thomas Moscibroda Joint work with: Fabian Kuhn, Roger Wattenhofer

2. 2Thomas Moscibroda @ DANSS 2004 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!

3. 3Thomas Moscibroda @ DANSS 2004 Locality... Many modern networks are large-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!

4. 4Thomas Moscibroda @ DANSS 2004 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)  Nodes have only local information  Nodes have to optimize a global goal Crucial for fast Algorithms!

5. 5Thomas Moscibroda @ DANSS 2004 k-Neighborhood What does „local“ mean ? How far does „locality“ go? Neighborhood, 2-Hop Neighborhood,... or something else...??? In communication round: 12 3

6. 6Thomas Moscibroda @ DANSS 2004 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!

7. 7Thomas Moscibroda @ DANSS 2004 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. TRADE-OFF LOCALITY Communication Rounds GLOBAL GOAL Approximation Upper Bounds:  Constant-Time Approximation Algorithms Lower Bounds:  Hardness of Distributed Approximation

8. 8Thomas Moscibroda @ DANSS 2004 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 v 2 V is in S or has at least one neighbor in S) –Minimum Vertex Cover (Choose minimum S µ V, s.t. each e 2 E has one node in S) Both problems appear to be local in nature!

9. 9Thomas Moscibroda @ DANSS 2004 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!

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

11. 11Thomas Moscibroda @ DANSS 2004 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!

12. 12Thomas Moscibroda @ DANSS 2004 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.

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

14. 14Thomas Moscibroda @ DANSS 2004 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

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

16. 16Thomas Moscibroda @ DANSS 2004 Each node stores a value x i Each edge stores a value y j –In a real network: edge is simulated by incident node! Idea: –Compute a feasible solution for MVC –While doing so: Distribute the dual values y j among incident, uncovered edges, hence –We show that  This yields an O(  1/k ) approximation! Distributed MVC Algorithm y1y1 y2y2 y3y3 xixi =1 =1/3 y4y4

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

18. 18Thomas Moscibroda @ DANSS 2004 Analysis Lemma: At the end of the algorithm, for all nodes v i 2 V: Y i · 3+  1/k Idea: Bound the sum of the incident dual variables for each node i. Proof: Let  i denote the i th iteration of the loop Case 1: Consider a node which does not join the vertex cover! 1)Until  0,, Y i is smaller than 1 2)In  0,  i must be 0 3)All neighbors must have joined VC before  0  Y i · 1

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

20. 20Thomas Moscibroda @ DANSS 2004 Summary VC-Algorithm 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 Can we do better?

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

22. 22Thomas Moscibroda @ DANSS 2004 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

23. 23Thomas Moscibroda @ DANSS 2004 Basic Idea S 0 and S 1 contain n 0 and n 0 /  nodes, resp. Optimal VC does not contain nodes of S 0 Basic Structure of our proof: 1.Construct graph such that nodes in S 0 and S 1 have same view 2.Algorithm has to take nodes of S 0 in order to cover edges between S 0 and S 1 3.VC ALG >> VC OPT because |S 0 | >> |S 1 | S 0 S 1

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

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

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

27. 27Thomas Moscibroda @ DANSS 2004 S 0 S 1 77777777 321341121 4422411 View of node in S 0 View of node in S 1 Two Round Lower Bound: Views

28. 28Thomas Moscibroda @ DANSS 2004 The Cluster Tree I

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

30. 30Thomas Moscibroda @ DANSS 2004 Construction of G k How can we achieve high girth? G k is a bipartite graph (even level clusters / odd level clusters) For prime power q, D(r,q) is bipartite graph with 2q r 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, G k can be constructed as sub-graph of D(2k-4,q) for q=  O(k)  G k has n=  O(k 2 ) nodes This is according to Intuition......because every node in S 0 must see a tree up to distance k.

31. 31Thomas Moscibroda @ DANSS 2004 Bounding the Optimum The number of nodes decreases by factor at least  /k on each level. If  >2k, n < n 0 +2n 0 k/  All nodes V \ S 0 form a feasible vertex cover, hence |VC OPT | < 2n 0 k/  n 0: · n 0 k/  : · n 0 k 2 /  2 : · n 0 k 3 /  3 : geometric series

32. 32Thomas Moscibroda @ DANSS 2004 Bounding any distributed algorithm Assume that the labeling (IDs) is chosen uniformly at random: Nodes v 0 in S 0 and v 1 in S 1 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 S 0 join VC! For all algorithms, there is labeling with |VC ALG | ¸ n 0 /2 Randomized: |VC ALG | ¸ n 0 /2 by Yao’s minimax principle v0v0 v1v1

33. 33Thomas Moscibroda @ DANSS 2004 Approximation Lower Bound We have |VC ALG | ¸ n 0 /2 and |VC OPT |  /(4k) n =  O(k 2 ),  =  k+1 In k communication rounds, no algorithm can approximate MVC better than  n c/k 2 /k) or  (  1/k /k)

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

35. 35Thomas Moscibroda @ DANSS 2004 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!

36. 36Thomas Moscibroda @ DANSS 2004 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....! Time lower bounds for MIS and Maximal Matching Compare with  log*n) lower bound on ring and O(log n) upper bound

37. 37Thomas Moscibroda @ DANSS 2004 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)

38. 38Thomas Moscibroda @ DANSS 2004 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

39. 39Thomas Moscibroda @ DANSS 2004 Distributed Computing Group Questions? Comments?