Graph Sparsifiers by Edge-Connectivity and - PowerPoint Presentation

1. Graph Sparsifiers byEdge-Connectivity andRandom Spanning TreesNick HarveyU. Waterloo C&OJoint work with Isaac FungTexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. What are sparsifiers?Approximating all cutsSparsifiers: number of edges = O(n log n /²2) ,every cut approximated within 1+². [BK’96]O~(m) time algorithm to construct themSpectral approximationSpectral sparsifiers: number of edges = O(n log n /²2), “entire spectrum” approximated within 1+². [SS’08]O~(m) time algorithm to construct them[BSS’09]Poly(n)n = # verticesLaplacian matrix of GLaplacian matrix of SparsifierWeighted subgraphs that approximately preserve some propertiesm = # edgesPoly(n)[BSS’09]

3. Why are sparsifiers useful?Approximating all cutsSparsifiers: fast algorithms for cut/flow problemProblemApproximationRuntimeReferenceMin st Cut1+²O~(n2)BK’96Sparsest CutO(log n)O~(n2)BK’96Max st Flow1O~(m+nv)KL’02Sparsest CutO~(n2)AHK’05Sparsest CutO(log2 n)O~(m+n3/2)KRV’06Sparsest CutO~(m+n3/2+²)S’09Perfect Matching in Regular Bip. Graphsn/aO~(n1.5)GKK’09Sparsest CutO~(m+n1+²)M’10v = flow valuen = # verticesm = # edges

4. Our MotivationBSS algorithm is very mysterious, and“too good to be true”Are there other methods to get sparsifiers with only O(n/²2) edges?Wild Speculation: Union of O(1/²2) random spanning trees gives a sparsifier (if weighted appropriately)True for complete graph [GRV ‘08]We prove: Speculation is false, butUnion of O(log2 n/²2) random spanning trees gives a sparsifier

5. Formal problem statementDesign an algorithm such thatInput: An undirected graph G=(V,E)Output: A weighted subgraph H=(V,F,w),where FµE and w : F ! RGoals:| |±G(U)| - w(±H(U)) | · ² |±G(U)| 8U µ V|F| = O(n log n / ²2)Running time = O~( m / ²2 )# edges between U and V\U in Gweight of edges between U and V\U in Hn = # verticesm = # edges | |±(U)| - w(±(U)) | · ² |±(U)| 8U µ V

6. Sparsifying Complete GraphSampling: Construct H by sampling every edge of Gwith prob p=100 log n/n. Give each edge weight 1/p.Properties of H:# sampled edges = O(n log n)|±G(U)| ¼ |±H(U)| 8U µ VSo H is a sparsifier of G

7. Generalize to arbitrary G?Can’t sample edges with same probability!Idea [BK’96]Sample low-connectivity edges with high probability, and high-connectivity edges with low probabilityKeep thisEliminate most of these

8. Non-uniform sampling algorithm [BK’96]Input: Graph G=(V,E), parameters pe 2 [0,1]Output: A weighted subgraph H=(V,F,w),where FµE and w : F ! RFor i=1 to ½ For each edge e2E With probability pe, Add e to F Increase we by 1/(½pe)Main Question: Can we choose ½ and pe’sto achieve sparsification goals?

9. Non-uniform sampling algorithm [BK’96]Claim: H perfectly approximates G in expectation!For any e2E, E[ we ] = 1) For every UµV, E[ w(±H(U)) ] = |±G(U)|Goal: Show every w(±H(U)) is tightly concentratedInput: Graph G=(V,E), parameters pe 2 [0,1]Output: A weighted subgraph H=(V,F,w),where FµE and w : F ! RFor i=1 to ½ For each edge e2E With probability pe, Add e to F Increase we by 1/(½pe)

10. Prior WorkBenczur-Karger ‘96Set ½ = O(log n), pe = 1/“strength” of edge e(max k s.t. e is contained in a k-edge-connected vertex-induced subgraph of G)All cuts are preservede pe · n ) |F| = O(n log n) (# edges in sparsifier)Running time is O(m log3 n)Spielman-Srivastava ‘08Set ½ = O(log n), pe = “effective resistance” of edge e(view G as an electrical network where each edge is a 1-ohm resistor)H is a spectral sparsifier of G ) all cuts are preservede pe = n-1 ) |F| = O(n log n) (# edges in sparsifier)Running time is O(m log50 n)Uses “Matrix Chernoff Bound”Assume ² is constantO(m log3 n)[Koutis-Miller-Peng ’10]Similar to edge connectivity

11. Our WorkFung-Harvey ’10 (independently Hariharan-Panigrahi ‘10)Set ½ = O(log2 n), pe = 1/edge-connectivity of edge eAll cuts are preservede pe · n ) |F| = O(n log2 n)Running time is O(m log2 n)Advantages:Edge connectivities natural, easy to computeFaster than previous algorithmsImplies sampling by edge strength, effective resistances,or random spanning trees worksDisadvantages:Extra log factor, no spectral sparsificationAssume ² is constant(min size of a cut that contains e)Why?Pr[ e 2 T ] = effective resistance of eand edges are negatively correlated

12. Our WorkFung-Harvey ’10 (independently Hariharan-Panigrahi ‘10)Set ½ = O(log2 n), pe = 1/edge-connectivity of edge eAll cuts are preservede pe · n ) |F| = O(n log2 n)Running time is O(m log2 n)Advantages:Edge connectivities natural, easy to computeFaster than previous algorithmsImplies sampling by edge strength, effective resistances…Extra trick: Can shrink |F| to O(n log n) by using Benczur-Karger to sparsify our sparsifier!Running time is O(m log2 n) + O~(n)Assume ² is constant(min size of a cut that contains e)O(n log n)

13. Our WorkFung-Harvey ’10 (independently Hariharan-Panigrahi ‘10)Set ½ = O(log2 n), pe = 1/edge-connectivity of edge eAll cuts are preservede pe · n ) |F| = O(n log2 n)Running time is O(m log2 n)Advantages:Edge connectivities natural, easy to computeFaster than previous algorithmsImplies sampling by edge strength, effective resistances…Panigrahi ’10A sparsifier with O(n log n /²2) edges, with running timeO(m) in unwtd graphs and O(m)+O~(n/²2) in wtd graphsAssume ² is constant(min size of a cut that contains e)

14. Notation: kuv = min size of a cut separating u and vMain ideas:Partition edges into connectivity classesE = E1 [ E2 [ ... Elog n where Ei = { e : 2i-1·ke<2i }

15. Notation: kuv = min size of a cut separating u and vMain ideas:Partition edges into connectivity classesE = E1 [ E2 [ ... Elog n where Ei = { e : 2i-1·ke<2i }Prove weight of sampled edges that each cuttakes from each connectivity class is about rightThis yields a sparsifierU

16. Prove weight of sampled edges that each cuttakes from each connectivity class is about rightNotation:C = ±(U) is a cut Ci = ±(U) Å Ei is a cut-induced setNeed to prove:C1C2C3C4

17. Notation: Ci = ±(U) Å Ei is a cut-induced setC1C2C3C4 Prove 8 cut-induced set CiKey IngredientsChernoff bound: Prove smallBound on # small cuts: Prove #{ cut-induced sets Ci induced by a small cut |C| }is small.Union bound: sum of failure probabilities is small, so probably no failures.

18. Counting Small Cut-Induced SetsTheorem: Let G=(V,E) be a graph. Fix any BµE. Suppose ke¸K for all e in B. (kuv = min size of a cut separating u and v) Then, for every ®¸1, |{ ±(U) Å B : |±(U)|·®K }| < n2®.Corollary: Counting Small Cuts [K’93] Let G=(V,E) be a graph. Let K be the edge-connectivity of G. (i.e., global min cut value) Then, for every ®¸1, |{ ±(U) : |±(U)|·®K }| < n2®.

19. ComparisonTheorem: Let G=(V,E) be a graph. Fix any BµE. Suppose ke¸K for all e in B. (kuv = min size of a cut separating u and v) Then |{ ±(U) Å B : |±(U)|·c }| < n2c/K 8c¸1.Corollary [K’93]: Let G=(V,E) be a graph. Let K be the edge-connectivity of G. (i.e., global min cut value) Then, |{ ±(U) : |±(U)|·c }| < n2c/K 8c¸1.How many cuts of size 1? Theorem says < n2, taking K=c=1. Corollary, says < 1, because K=0.(Slightly unfair)

20. ConclusionsGraph sparsifiers important for fast algorithms and some combinatorial theoremsSampling by edge-connectivities gives a sparsifierwith O(n log2 n) edges in O(m log2 n) timeImprovements: O(n log n) edges in O(m) + O~(n) time[Panigrahi ‘10]Sampling by effective resistances also works ) sampling O(log2 n) random spanning trees gives a sparsifierQuestionsImprove log2 n to log n?Sampling o(log n) random trees gives a sparsifier with o(log n) approximation?