Download - Graph Sparsifiers by Edge-Connectivity and Random Spanning Trees

Graph Sparsifiers byEdge-Connectivity and

Random Spanning Trees

Nick Harvey University of Waterloo

Department of Combinatorics and Optimization

Joint work with Isaac Fung

What are sparsifiers?

• Approximating all cuts– Sparsifiers: number of edges = O(n log n /²2) ,

every cut approximated within 1+². [BK’96]– O~(m) time algorithm to construct them

• Spectral approximation– Spectral 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 = # vertices

Laplacian matrix of GLaplacian matrix of Sparsifier

Weighted subgraphs that approximately preserve some properties

m = # edges

Poly(n)

[BSS’09]

Why are sparsifiers useful?• Approximating all cuts– Sparsifiers: fast algorithms for cut/flow problem

Problem Approximation Runtime ReferenceMin st Cut 1+² O~(n2) BK’96Sparsest Cut O(log n) O~(n2) BK’96Max st Flow 1 O~(m+nv) KL’02Sparsest Cut O~(n2) AHK’05Sparsest Cut O(log2 n) O~(m+n3/2) KRV’06Sparsest Cut O~(m+n3/2+²) S’09Perfect Matching in Regular Bip. Graphs

n/a O~(n1.5) GKK’09

Sparsest Cut O~(m+n1+²) M’10

v = flow value

n = # verticesm = # edges

Our Motivation• BSS 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]• Corollary of our Main Result:

The Wild Speculation is false, but the union of O(log2 n/²2) random spanning trees gives a sparsifier

Formal problem statement

• Design an algorithm such that• Input: An undirected graph G=(V,E)• Output: A weighted subgraph H=(V,F,w),

where FµE and w : F ! R• Goals:• | |±G(U)| - w(±H(U)) | · ² |±G(U)| 8U µ V

(We only want to preserve cuts)• |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 H

n = # verticesm = # edges

| |±(U)| - w(±(U)) | · ² |±(U)| 8U µ V

Sparsifying Complete Graph

• Sampling: 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 V

• So H is a sparsifier of G

• Consider any cut ±G(U) with |U|=k. Then |±G(U)|¸kn/2.

• Let Xe = 1 if edge e is sampled. Let X = e2C Xe = |±H(U)|.

• Then ¹ = E[X] = p |±(U)| ¸ 50 k log n.

• Say cut fails if |X-¹| ¸ ¹/2.

• So Pr[ cut fails ] · 2 exp( - ¹/12 ) · n-4k.

• # of cuts with |U|=k is .

• So Pr[ any cut fails ] · k n-4k < k n-3k < n-2.

• Whp, every U has ||±H(U)| - p |±(U)|| < p |±(U)|/2

Chernoff Bound

Bound on # small cuts

Key Ingredients

Union bound

Proof Sketch

Exponentially increasing # of bad events

Exponentially decreasingprobability of failure

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 probability

Keep this

Eliminate most of these

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 e2EWith probability pe,

Add e to F Increase we by 1/(½pe)

• Main Question: Can we choose ½ and pe’sto achieve sparsification goals?

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 concentrated

• 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 e2EWith probability pe,

Add e to F Increase we by 1/(½pe)

Prior Work• Benczur-Karger ‘96– Set 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 ‘08– Set ½ = O(log n), pe = 1/“effective conductance” 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 constant

O(m log3 n)[Koutis-Miller-Peng ’10]

Similar to edge connectivity

Our Work• Fung-Harvey ’10 (independently Hariharan-Panigrahi

‘10)

– Set ½ = O(log2 n), pe = 1/edge-connectivity of edge e

– Edge-connectivity ¸ max { strength, effective conductance }– e pe · n ) |F| = O(n log2 n)– Running time is O(m log2 n)– Advantages:• Edge connectivities natural, easy to compute• Faster than previous algorithms• Implies sampling by edge strength, effective resistances,

or random spanning trees works– Disadvantages:• Extra log factor, no spectral sparsification


Why?Pr[ e 2 T ] = effective resistance of e

edges are negatively correlated) Chernoff bound still works

(min size of a cut that contains e)


‘10)


– Edge-connectivity ¸ max { strength, effective conductance }– e pe · n ) |F| = O(n log2 n)– Running time is O(m log2 n)– Advantages:• Edge connectivities natural, easy to compute• Faster than previous algorithms• Implies 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)



O(n log n)


‘10)


– Edge-connectivity ¸ max { strength, effective conductance }– e pe · n ) |F| = O(n log2 n)– Running time is O(m log2 n)– Advantages:• Edge connectivities natural, easy to compute• Faster than previous algorithms• Implies sampling by edge strength, effective resistances…

• Panigrahi ’10– A sparsifier with O(n log n /²2) edges, with running time

O(m) in unwtd graphs and O(m)+O~(n/²2) in wtd graphs



• Notation: kuv = min size of a cut separating u and v• Main ideas:– Partition edges into connectivity classes

E = E1 [ E2 [ ... Elog n where Ei = { e : 2i-1·ke<2i }– Prove weight of sampled edges that each cut

takes from each connectivity class is about right

– Key point: Edges in ±(U)ÅEi have nearly same weight– This yields a sparsifier

U

Prove weight of sampled edges that each cuttakes from each connectivity class is about right

• Notation:• C = ±(U) is a cut • Ci = ±(U) Å Ei is a cut-induced set

• Need to prove:

C1 C2 C3 C4

• Notation: Ci = ±(U) Å Ei is a cut-induced set

C1 C2 C3 C4

Prove 8 cut-induced set Ci

• Key Ingredients• Chernoff bound: Prove small• Bound 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.

Counting Small Cut-Induced Sets• Theorem: 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®.

Comparison• Theorem: Let G=(V,E) be a graph. Fix any BµE.


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)

Comparison• Theorem: Let G=(V,E) be a graph. Fix any BµE.


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.

• Important point: A cut-induced set is a subset of edges. Many cuts can induce the same set.

(Slightly unfair)

±(U’)±(U)

Algorithm for Finding a Min Cut [K’93]

• Input: A graph• Output: A minimum cut (maybe)• While graph has 2 vertices–Pick an edge at random–Contract it

• End While• Output remaining edges

• Claim: For any min cut, this algorithm outputs it with probability ¸ 1/n2.

• Corollary: There are · n2 min cuts.

Finding a Small Cut-Induced Set• Input: A graph G=(V,E), and BµE• Output: A cut-induced subset of B• While graph has 2 vertices– If some vertex v has no incident edges in B• Split-off all edges at v and delete v

–Pick an edge at random–Contract it

• End While• Output remaining edges in B• Claim: For any min cut-induced subset of B, this

algorithm outputs it with probability >1/n2.• Corollary: There are <n2 min cut-induced subsets of B

Splitting OffReplace edges {u,v} and {u’,v} with {u,u’}while preserving edge-connectivityBetween all vertices other than v Wolfgang Mader

vu u’

vu u’

Sparsifiers from Random Spanning Trees• Let H be union of ½=log2 n uniform random spanning trees,

where we is 1/(½¢(effective resistance of e))• Then all cuts are preserved and |F| = O(n log2 n)

• Why does this work?– PrT[ e 2 T ] = effective resistance of edge e [Kirchoff 1847]

– Similar to usual independent sampling algorithm,with pe = effective resistance of e

– Key difference: edges in a random spanning tree arenot independent, but they are negatively correlated![BSST 1940]

– Chernoff bounds still work. [Panconesi, Srinivasan 1997]

Sparsifiers from Random Spanning Trees• Let H be union of ½=log2 n uniform random spanning trees,

where we is 1/(½¢(effective resistance of e))• Then all cuts are preserved and |F| = O(n log2 n)

• How is this different than independent sampling?– Consider an n-cycle. There are n/2 disjoint cuts of size 2.– When ½=1, each cut has constant prob of having no edges) need ½=(log n) to get a connected graph– With random trees, get connectivity after just one tree– Are O(1) trees are enough to preserve all cuts?– No! ( log n ) trees are required

Conclusions• Graph sparsifiers important for fast algorithms and some

combinatorial theorems• Sampling by edge-connectivities gives a sparsifier

with O(n log2 n) edges in O(m log2 n) time– Improvements: 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 sparsifier

Questions• Improve log2 n to log n?• Sampling o(log n) random trees gives a sparsifier with

o(log n) approximation?

Download - Graph Sparsifiers by Edge-Connectivity and Random Spanning Trees

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