Graph Sparsifiers Smaller graph that (approximately) preserves the - - PowerPoint PPT Presentation

Graph Sparsifiers

Graph Sparsification

“Smaller” graph that (approximately) preserves the values of some set of graph parameters

Graph Sparsifiers

Graph Sparsification

• Spanners
• Emulators
• Small stretch spanning trees
• Vertex sparsifiers
• …
• Spectral sparsifiers
• Cut sparsifiers

Spectral Sparsification

Graph Sparsification

• Undirected graph G = (V, E); error parameter ε
• Goal: Gε = (V, Eε) with Õ(n/ε2) edges such that

for all n-dimensional vectors x, (1–ε) xT L(G) x ≤ xT L(Gε) x ≤ (1+ε) xT L(G) x

• Graph Laplacian L = D – A, where

– D = Diagonal Degree Matrix of the graph – A = Adjacency Matrix of the graph

Spectral Sparsification: Previous work

Graph Sparsification

Running time of the sparsification algorithm Number of edges in the sparsifier O(n3m) O(n/ε2) [Batson-Spielman-Srivastava ’09] O(n2m log3n + n4 log n) [Zouzias ’12] O(m logO(1) n) O(n logO(1) n/ε2) [Spielman-Teng ’04] O(m logO(1) n) O(n log n/ε2) [Spielman-Srivastava ’08] O(m log3 n) SS + [Koutis-Miller-Peng ’10, ’11] O(m log2 n) [Koutis-Levin-Peng ’12] O(m log n) O(n log3 n/ε2) [Koutis-Levin-Peng ’12] O(m) ??? ???

Spectral to Cut Sparsifiers

Graph Sparsification

• Gε = (V, Eε) is a spectral sparsifier of G = (V, E)

if for all n-dimensional vectors x, (1–ε) xT L(G) x ≤ xT L(Gε) x ≤ (1+ε) xT L(G) x

• xT L x = Σ(i, j) ϵ E (xi - xj)2
• Suppose x ϵ {0, 1}n; S = {i ϵ V: xi = 1}. Then,

xT L x = Σ(i, j) ϵ E (xi - xj)2 = Σ(i, j) ϵ (S, V - S) 1 = E(S)

Cut Sparsification

Graph Sparsification

Weight of every cut is preserved up to a multiplicative error of (1 ± Ɛ)

Cut Sparsification

• Undirected (unweighted) graph G = (V, E); error

parameter ε

• Goal: Gε = (V, Eε) with O(n log n/ε2) edges such

that for all cuts (S, V – S), (1–ε) E(S) ≤ Eε(S) ≤ (1+ε) E(S)

• Introduced by Benczur-Karger ’96

– O(m log2n)-time algorithm to find a cut sparsifier (with high probability) containing O(n log n/ε2) edges in expectation

Graph Sparsification

Graph Sparsification

Fung-Hariharan-Harvey-P.: A linear-time, i.e. O(m), algorithm that produces a cut sparsifier (whp) containing O(n log n/ε2) edges in expectation

Cut Sparsification by Sampling

• Uniformly sample all edges with prob p ≈ n/m

– Selected edge is given weight 1/p

Graph Sparsification

p ≈ 1/n; graph gets disconnected Non n

edge e with prob pe 1/pe

Sampling Probabilities

Graph Sparsification

Edge Connectivity λe = size of smallest cut containing e Belongs to a small cut Belong only to large cuts pe = log n/λe

Sampling by Edge Connectivity

• Sample edge e independently (of other edges)

with probability pe ≈ log n/λe

• If edge e is selected, it is given a weight of

1/pe in the sparsifier

• Sparsifier has O(n log n) edges in expectation

λe ≥ 1/re ΣeϵE 1/λe ≤ ΣeϵE re = n - 1

• Pr[Eε(S) ∈ (1±ε) E(S) for all cuts (S, V - S)]?

Graph Sparsification

Bounding Deviation

• Expected number of

edges in the cut ≥ log n

• Chernoff bounds:

Probability of εΔ error ≤ 1/poly(n)

• Exponential number of

cuts!

Graph Sparsification

∆ edges λe ≤ ∆, i.e. pe ≥ log n/∆

…

Bounding Deviation

• Error probability for

single cut ≤ 1/poly(n) but exp(n) cuts

• Cut projections

Categorize edges in a cut according to the value of λe (i.e., pe)

Graph Sparsification

pe = log n/n pe = 1

Bounding Deviation

• For λe ≈ Δ/k cut projection, pe = k log n/Δ
• Probability of εΔ error ≤ exp(-k log n) = n-Ω(k)

Graph Sparsification

∆ edges λe ≈ ∆ λe ≈ ∆/2 λe ≈ ∆/4

Cut Projections

Graph Sparsification

Lemma: There are ≤ nO(k) distinct (Δ, k) cut projections in cuts of size Δ

union bound on k, Δ

Theorem: Sampling edge e with probability log2 n / λe produces a cut sparsifier

Graph Sparsification

Difficulty: Edge connectivities (λe) are time-consuming to calculate (Gomory-Hu tree takes Õ(mn) time [Bhalgat-Hariharan-Kavitha-P., ’07])

Greedy Spanning Forest packing

Graph Sparsification

a b d f e g h c a b d f e g h c a b d f e g h c

T1 T2

Sampling by NI Index

Nagamochi-Ibaraki (NI) index of edge e ye = index of e in an arbitrary but fixed greedy spanning forest packing Proposed Cut Sparsification Algorithm

• Sample edge e with probability pe ≈ log2 n/ ye
• If edge e is selected, it is given a weight of 1/pe in the

sparsifier

Graph Sparsification

Sampling by NI Index: Cut preservation

Lemma: The graph Gε = (V, Eε) produced by sampling using NI indices is a cut sparsifier, i.e., with high probability, for all cuts (S, V-S) (1–ε) E(S) ≤ Eε(S) ≤ (1+ε) E(S)

– For each edge e, ye ≤ λe (if edge e is in ith forest, then its endpoints are connected by disjoint paths in the previous i-1 forests) – Now piggyback on the proof for sampling using edge connectivities

Graph Sparsification

Sampling by NI Index: Sparsification

Lemma: The sparsifier has O(n log3 n) edges in expectation

Σe∈E 1/ye = Σk |Tk|/k = (n-1) Σk 1/k = O(n log n)

Graph Sparsification

Sampling by NI Index: Running time

Lemma [Nagamochi-Ibaraki ’92]: The running time of the sampling algorithm (i.e., time taken to estimate the NI indices of all edges) is O(m)

Graph Sparsification

Graph Sparsification

We have shown: An O(m)-time algorithm that produces a cut sparsifier containing O(n log3 n) edges We had promised (see the paper): An O(m)-time algorithm that produces a cut sparsifier containing O(n log n) edges We will now show: An O(m)-time algorithm that produces a cut sparsifier containing O(n log2 n) edges

Sampling by NI Index: New Algorithm

Previous Algorithm

• Sample edge e with probability pe ≈ log2 n/ ye
• If edge e is selected, it is given a weight of 1/pe in the

sparsifier New Algorithm

• Sample edge e with probability pe ≈ log n/ ye
• If edge e is selected, it is given a weight of 1/pe in the

sparsifier

Graph Sparsification

Sampling by NI Index: New Algorithm

New Algorithm

• Sample edge e with probability pe ≈ log n/ ye
• If edge e is selected, it is given a weight of 1/pe in the

sparsifier

• Running time remains O(m)
• The expected number of edges is O(n log2 n)
• Is the sample a cut sparsifier?

[Note: We can no longer piggyback on the analysis for sampling with edge connectivity]

Graph Sparsification

Bucketing the forests

Graph Sparsification

T1 T2 T2

i-1

T2

i

T2

i+1

… … …

Fi Gi = Fi-1 + Fi

…

Properties of the bucketing

• Similarity property: All edges in Fi have sampling

probability pe ≈ log n / 2i-1 (up to a factor of 2)

• Overlap property: Every edge appears in Gi for at

most two values of i

• Connectivity property: Every edge in Fi has edge

connectivity ≥ 2i-1 in Gi

– The endpoints of the edge have 2i-1 disjoint paths between them, one in each forest, in Gi

Graph Sparsification

Analysis of a cut

Graph Sparsification

S V - S

C C XC,1 XC,2

…

XC,i

… C ∩ F1 C ∩ F2 C ∩ Fi

C YC,1 YC,2

…

YC,i

… C ∩ G1 C ∩ G2 C ∩ Gi

Input Graph G

Tail Bounds on Deviation

Graph Sparsification

S V - S

Cε Cε ZC,1 ZC,2

…

ZC,i

… Cε ∩ F1 Cε ∩ F2 Cε ∩ Fi

Sampled graph G ε

Tail Bounds on Deviation

• Need to show: whp, |C – Cε| < ε C for all cuts C
• whp, |XC,i – ZC,i| < ε XC,i for all cuts C and all i

Graph Sparsification

Lemma: whp, |XC,i – ZC,i| < εYC,i for all cuts C and all i

∑i YC,i = 2C by the

• verlap property

Tail Bounds on Deviation

• Lemma: whp, |XC,i – ZC,i| < εYC,i for all cuts C and all i
• Let Ck be cuts for which YC,i = |C ∩ Gi|= 2i+k
• By the connectivity property, every edge in XC,i is

2i-1-connected in YC,i

• By Cut Projection Counting Lemma,

There are at most n2^(i+k)/2^i = n2^k distinct XC,i in Ck

A General Framework for Graph Sparsification 31

Tail Bounds on Deviation

• Lemma: whp, |XC,i – ZC,i| < εYC,i for all cuts C and all i
• There are at most n2^k distinct XC,i in Ck
• By the similarity property + Chernoff bounds,

Pr[|XC,i - ZC,i | > ε YC,i] < exp(- 2i+k (log n / 2i)) = n–2^k union bound over distinct XC,i in Ck , all values of k and i

A General Framework for Graph Sparsification 32

Open Problems

• Linear-time spectral sparsification algorithm
• (Near)-linear time construction of O(n/ε2)-sized

cut/spectral sparsifiers

– Edge sampling has fundamental limitations (connectivity of Erdos-Renyi random graph has a probability threshold of log n/n) – Cut/spectral sparsifiers from spanning trees? [Goyal-Rademacher-Vempala ’09, Fung-Harvey ’10] – Cut/spectral sparsifiers from spanners? [Kapralov-Panigrahy ’12, Koutis ’14]

Graph Sparsification

Graph Sparsification