Transitive-Closure Spanner of Directed Graphs

1. 2009 Combinatorics Workshop Transitive-Closure Spanner of Directed Graphs Kyomin Jung KAIST Joint work with Arnab Bhattacharyya MIT Elena GrigorescuMIT SofyaRaskhodnikovaPenn State U. David Woodruff IBM Almaden Appeared in SODA ‘09

2. Graph Spanners [Awerbuch85,Peleg Schäffer89] A subgraph H of a graph G is a k-spanner if for all pairs of vertices u, v in G, dH(u,v) ≤ k dG(u,v) Goal: find a sparsest k-spanner dense graph G sparse subgraph H

3. Transitive-Closure Spanners Transitive closure TC(G) has an edge from u to v iff G has a path from u to v k-TC-spannerH of G has TC(H) = TC(G) and dH(u,v) ≤k iff G has a path from u to v Alternatively: k-TC-spanner of G is a k-spanner of TC(G) G TC(G)

4. Example: Directed Line on n Vertices • 2-TC-spanner O(n log n) edges • 3-TC-spanner O(n log log n) edges • 4-TC-spanner O(n log* n) edges • k-TC-spanner O(n(k,n)) edges where (k,n) is the kth-row inverse Ackermann function … …

5. Previous work Structural results on TC-spanners (what is a sparsest k-TC-spanner for a given graph family?) • Shortcut graphs (special case when |E(H)|≤2 |E(G)|) [Thorup 92, 95, Hesse 03] implicit in • data structures [Yao 82, Alon Schieber 87, Chazelle 87] • property testing [Dodis Goldreich Lehman Raskhodnikova Ron Samorodnitsky 99] • access control [Attalah Frikken Blanton 05] Computational results on directed spanners (given a graph, compute a sparsest k-spanner) LetOPTbe number of edges in sparsest k-TC-spanner. • O(log n)-approximation of OPT for k=2[Kortsarz Peleg 94] • O(n2/3 polylog n)-approximation of OPT for k=3[Elkin Peleg 99]

6. Our Contributions • Common abstraction for several applications • property testing • access control • data structures • Structural results on TC-spanners • Sparse TC-spanners for planar graphs and other path-separable graph families • Computational results k-TC-Spanner: Given a graph, compute a sparsest k-TC-spanner • Algorithms • Inapproximability

7. Application : Testing if a List is Sorted • Is a list of n numbers sorted? Requires reading entire list. • Is a list of n numbers sorted or ²-far from sorted? (An ² fraction of list entries have to be changed to make it sorted.) [Ergün Kannan Kumar Rubinfeld Viswanathan 98]: O((log n)/²) time

8. Generalization: Monotonicity over PO domains 1 1 • 4 5 6 • 4 5 6 • 3 4 5 • 1 2 3 4 Graph = partially ordered domain; node labels = values of the function • A function is monotone if there are no violated edges (along which labels decrease): 1 0 • A function is ²-far from monotone if ¸² fraction of labels need to be changed to make it monotone. [GGLRS 00], [DDGLR 99], [FLNRRS 02] 1 1 0 0 1 2 5 4 3 6 7 0 0

9. Monotonicity Testers via Sparse 2-TC-spanners [Jung et al] Sparse 2-TC-spanner for graphG Fast monotonicity tester with respect to poset given by G Size:s(n) Query complexity: O(s(n)/(² n)) Tester checks for violations on randomly chosen edges of the 2-TC-spanner

10. Our Contributions • Common abstraction for several applications • Structural results on TC-spanners [Jung et al] • Path-separable graphs O(1)-path-separable graphs have k-TC-spanners of size O(nlog n ¢(k,n)) • e.g., improves run time of monotonicity testers on planar graphs from O((n1/2 log n)/²)[FLNRRS02] to O((log2n)/²) • Computational results k-TC-Spanner: Given a graph, compute a sparsest k-TC-spanner • Algorithms • Inapproximability

11. Graph Separators (for Undirected Graphs) Used in recursive constructions S-separators[Lipton Tarjan 79] • Removing S nodes disconnects a graph G on n nodes into connected components with ≤n/2 nodes each r-path-separators[Abraham Gavoille 06] • Removing nodes on rpaths from any spanning tree of Gdisconnects G into connected components with ≤n/2 nodes each Gain • For some graphs r << S e.g., planar graphs are £(n1/2)-separable but 3-path-separable [Thorup 04]

12. Our Contributions • Common abstraction for several applications • Structural results on TC-spanners • Computational results on k-TC-Spanner Algorithms

13. Our Contributions • Common abstraction for several applications • Structural results on TC-spanners • Computational results on k-TC-Spanner Hardness

14. Approximation Algorithms • For any k > 2, we achieve an O((n lg n)1-1/k)approximation algorithm for the Directed Spanner Problem in polynomial time • Gives the same ratio for Transitive-Closure spanners • Yields the first sublinear ratio for k > 3 • Solves the main open question of [Elkin Peleg 05] • Our technique is a new balancing of a linear program with a sampling-based approach • Greatly simplifies the previous O(n2/3 polylog n)-approximation algorithm [Elkin Peleg 05] for k = 3

15. Approximation Algorithm: Linear Program • For each edge e2G, introduce binary variable xe, which indicates whether or not we include e in the k-Spanner H • For each path p of length at most k in G, introduce binary variable yp indicating whether or not we include p in H. For constant k, the number of such paths is O(nk+1) = poly(n). • Integer programming formulation: Minimize total number of edges included, subject to: (i) having at least 1 path of length at most k between every pair of vertices adjacent in G and (ii) for every path p, for every edge e in p, having xe¸ yp • Linear programming relaxation: xe, yp2 [0,1]

16. Approximation Algorithm: LP + Sampling Sampling solves the “many paths” case LP solves the “few paths” case … • Solve the linear program, let the solution variables bexe*, yp* • Define a rounding scheme: include e in thek-Spanner H if and only ifxe*¸1/n1-1/k Problem for LP if lots of paths present between a pair of vertices • Sample O(n1-1/k ) vertices and grow BFS trees around them. • Claim: H is a k-Spanner of G with prob. at least 1-1/n.

17. Conclusion Our contributions • Common abstraction for several applications • monotonicity testing, access control, data structures • Structural results on TC-spanners • path-separable graphs • Computational results on k-TC-Spanner • new algorithms and inapproximability results Open question • For which k, does k-TC-Spanner admit efficient approximation algorithms? • Showed strong inapproximability for any constant k > 2 • What about restricted class of graphs, ex. Hyper-cube ? Thank you.