The Sketching Complexity of Graph Cuts - PowerPoint Presentation

1. The Sketching Complexity of Graph CutsRobert Krauthgamer, Weizmann Institute of ScienceBertinoro workshop, May 2014Joint work with Alexandr Andoni and David WoodruffTexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA

2. Compressing GraphsVast literature on “compression” (succinct representation) of graphsWe focus on preserving specific features – distances, cuts, etc.Edge sparsification:Cut and spectral sparsifiers [Benczur-Karger, …, Batson-Spielman-Srivastava, …]Spanners and distance oracles [Peleg-Schaffer, …, Thorup-Zwick,…]The Sketching Complexity of Graph Cutsdata structure (w/fast query)graphical representationexactly/approximatelyLossy Compression2

3. Sketching CutsNetwork G with edge weights . (“huge” network)Care about cuts: = weight of edges between and . The Sketching Complexity of Graph CutsG:SSGoal: Small data structure that can answer “cut queries” (quickly) preprocessdata structure   query3

4. Known SolutionsDenote . Naive: store a table of cut values Fast query time O(1), but prohibitive storageGraphical: store itselfSimple, but not fast or succinctCompute cut-sparsifier (has approximately same cuts) Introduced by [Benczur-Karger’96], several followups [Spielman-Teng’04, Spielman-Srivastava’08, Batson-Spielman-Srivastava’09, Fung-Hariharan-Harvey-Panigrahi’11, Kapralov-Panigrahy’12]Storage: words (using best sparsifier)Negative evidence: is complete graph, is d-regular [Alon’97]Also for spectral sparsifiers [Batson-Spielman-Srivastava] The Sketching Complexity of Graph CutsQ: Improve dependence on ? 4

5. Our ResultsWe achieve storage – linear dependence on Theorem 1. There is a randomized sketch of size , that given any , outputs w.h.p. -approximation to Can be viewed as a relaxation of “usual” cut-sparsifierA sketch, not a graphA “for each” guarantee, not “for all”We further show the second relaxation is necessaryTheorem 2. Every randomized sketching achieving w.h.p. -approximation for all cuts, must have size bitsGeneralizes known lower bounds: For cut-sparsifiers (even if is not regular)For spectral-sparsifier (even if sketch is not graphical) The Sketching Complexity of Graph Cuts5

6. UB: First Attempt – Edge SamplingSuppose is the complete graphStandard approach: subsample edges with probability .Smaller probability fails:Even for “singleton cuts” , we now expect Concrete difficult case: For a random graph , Above argument for singleton cuts still holdsThey seem “most difficult” Because for large sets, relative deviation (from expectation) is But vertex degrees can be stored using O(n) wordsAnd this info handles all small sets (whenever ) The Sketching Complexity of Graph Cuts6

7. UB: Core IdeaSuppose is unweighted, and let’s handle cuts of weight Repeatedly cut the graph along cuts of sparsity Store all cut edgesTheir contribution to any desired is easy to computeFocus now on cuts inside one part Sample edges out of every vertexStore additionally for all vertices Assuming w.l.o.g. , estimate by the differenceObservation: Total sketch size is Observation: The estimator is unbiased, need to analyze variance… The Sketching Complexity of Graph Cuts7

8. Variance AnalysisLet be the degree of into , and be the sample of edges out of , and write The variance comes from sampling of edges Plugging and ,  The Sketching Complexity of Graph Cuts8

9. General Case (Polynomial Weights)Compute -cut-sparsifer graph Proceed “in parallel” for every “guess” = power of 2Assume for normalization , thus Importance samplingDiscard edges of weight Surely not relevantSample other edges with probability and assign them new weight An unbiased estimator, with variance W.h.p. the cut contains edgesBreak edges into levels where , estimate each level separately (using sparse-cuts), and sum up In each level, use weights – they are similar and thus our variance analysis still applies The Sketching Complexity of Graph Cuts9

10. Further ExtensionsPolynomial time: use -approximation of sparse cuts [Arora-Rao-Vazirani’04] (or polylog-approximation w/faster runtime)Two passesUnbounded weights: Compute a maximum-weight spanning tree (or Gomory-Hu tree) This tree yields a -approximation of Proceed “in parallel” for each such guess, by contracting very heavy edges and discarding very light ones, and applying the “basic” sketch  The Sketching Complexity of Graph Cuts10

11. Example ApplicationTheorem 3. A -approximation of minimum-cut by a 2-pass algorithm using space Known: one pass (streaming/incremental) using spaceIdea: Compute cut-sparsifier (using known algorithm), and identify all candidates approximate min-cuts [Karger’00]Space In parallel, apply our sketching algorithm, implemented in passesSuccess probability amplified by repetitionsSpace Use our sketch to -approximate each of the candidates, and report minimum one The Sketching Complexity of Graph Cuts11

12. LB: First Attempt – One-way Comm.Theorem 2. Every randomized sketching achieving w.h.p. -approximation for all cuts, must have size bitsNatural attempt: But we’ve just seen Alice can send only bits Must then “use” multiple cuts (sets )Assume for simplicity  The Sketching Complexity of Graph Cuts12Alice Bob 

13. LB: OutlineAlice is given a random bipartite graph with and edge probability ½ Bob is given a random vertex and a random subset , and has to decide whether is or Essentially a Gap-Hamming Distance, even for small constant , thus requires communicationSuppose Alice sends to Bob a sketch of all cuts of Bob enumerates over all of size , and estimates , and finds a maximizer is factor larger than typical , and its estimator should “stand out”More precisely, will “mostly agree” with , hence Bob can just test whether  The Sketching Complexity of Graph Cuts13

14. Concluding RemarksConcrete: One pass? Avoid sparse-cut computations? Handle an (adaptive) sequence of queries?A sketch for spectral queries, i.e., quadratic forms ?Abstract: Understand tradeoffs between different representations (graphical vs. data structure)Connections between distances/cuts/flows?Preserve other combinatorial features (graphs)? The Sketching Complexity of Graph CutsThank You! 14