[PPT] - Latest on Linear Sketches for Large Graphs: Lots of Problems, Little PowerPoint Presentation

SLIDE 1

Latest on Linear Sketches for Large Graphs: Lots of Problems, Little Space, and Loads of Handwaving

Andrew McGregor

University of Massachusetts

SLIDE 2

Latest on Linear Sketches for Large Graphs: Lots of Problems, Little Space, and Loads of Handwaving

Andrew McGregor

University of Massachusetts

Vertex Connectivity and Sparsification

Guha, McGregor, Tench [PODS 2015]

Densest Subgraphs

McGregor, Tench, Vorotnikova, Vu [MFCS 2015]

Matching, Vertex Cover, Hitting Set

Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh, Vorotnikova [TBA 2016]

SLIDE 3

Motivation: Dynamic Graph Streams. Want to analyze a

massive graph defined by a long sequence of edge insertions and deletions. Don’t want to have to store the entire graph.

Background

SLIDE 4

Motivation: Dynamic Graph Streams. Want to analyze a

massive graph defined by a long sequence of edge insertions and deletions. Don’t want to have to store the entire graph.

Main

Technique: Linear Sketches. Maintain a random linear projections of vectors and matrices representing the graph.

Background

SLIDE 5

Motivation: Dynamic Graph Streams. Want to analyze a

massive graph defined by a long sequence of edge insertions and deletions. Don’t want to have to store the entire graph.

Main

Technique: Linear Sketches. Maintain a random linear projections of vectors and matrices representing the graph.

What’s Known: Lots and lots! Edge and vertex connectivity,

spectral sparsification, matching, vertex cover, hitting set, correlation clustering, triangles, spanners, densest subgraph…

Background

SLIDE 6

Motivation: Dynamic Graph Streams. Want to analyze a

massive graph defined by a long sequence of edge insertions and deletions. Don’t want to have to store the entire graph.

Main

Technique: Linear Sketches. Maintain a random linear projections of vectors and matrices representing the graph.

What’s Known: Lots and lots! Edge and vertex connectivity,

spectral sparsification, matching, vertex cover, hitting set, correlation clustering, triangles, spanners, densest subgraph…

Background

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Graph Streaming Survey

McGregor [SIGMOD Record 2014]

SLIDE 7

Preliminary

L0 Sampling Primitive There’s a distribution over matrices

M∈ℜpolylog(N) x N such that for any x∈ℜN, a random non-zero element of x can be reconstructed from Mx whp.

Jowhari, Saglam,

Tardos [PODS 2011]

SLIDE 8

Preliminary

L0 Sampling Primitive There’s a distribution over matrices

M∈ℜpolylog(N) x N such that for any x∈ℜN, a random non-zero element of x can be reconstructed from Mx whp.

Jowhari, Saglam,

Tardos [PODS 2011]

Corollary Can sample a uniform edge from a graph in the

dynamic graph stream model using O(polylog n) bits of space.

SLIDE 9

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.

Densest Subgraph

SLIDE 10

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]

Densest Subgraph

SLIDE 11

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]
Our Result Single pass (1+ε)-approx. using Õ(ε-2 n) space:

Densest Subgraph

SLIDE 12

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]
Our Result Single pass (1+ε)-approx. using Õ(ε-2 n) space:
Sample of t=O(n log n) edges using L0 sampling. Let ĎS be

density among sampled edge scaled by m/t. Return maxS ĎS

Densest Subgraph

SLIDE 13

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]
Our Result Single pass (1+ε)-approx. using Õ(ε-2 n) space:
Sample of t=O(n log n) edges using L0 sampling. Let ĎS be

density among sampled edge scaled by m/t. Return maxS ĎS

Analysis With probability 1-n-2k for any subset S of size k,

Densest Subgraph

SLIDE 14

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]
Our Result Single pass (1+ε)-approx. using Õ(ε-2 n) space:
Sample of t=O(n log n) edges using L0 sampling. Let ĎS be

density among sampled edge scaled by m/t. Return maxS ĎS

Analysis With probability 1-n-2k for any subset S of size k,
ĎS ≈ε DS if DS ≈ D* and ĎS ≪ D* if DS ≪ D*

Densest Subgraph

SLIDE 15

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]
Our Result Single pass (1+ε)-approx. using Õ(ε-2 n) space:
Sample of t=O(n log n) edges using L0 sampling. Let ĎS be

density among sampled edge scaled by m/t. Return maxS ĎS

Analysis With probability 1-n-2k for any subset S of size k,
ĎS ≈ε DS if DS ≈ D* and ĎS ≪ D* if DS ≪ D*
Use union bound over O(nk) subsets of size k for each k.

Densest Subgraph

SLIDE 16

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximations using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 2015], Bahmani et al. [PVLDB 2012]
Our Result Single pass (1+ε)-approx. using Õ(ε-2 n) space:
Sample of t=O(n log n) edges using L0 sampling. Let ĎS be

density among sampled edge scaled by m/t. Return maxS ĎS

Analysis With probability 1-n-2k for any subset S of size k,
ĎS ≈ε DS if DS ≈ D* and ĎS ≪ D* if DS ≪ D*
Use union bound over O(nk) subsets of size k for each k.
see also Mitzenmacher et al. [KDD 2015], Esfandiari et al. [ArXiv 2015]

Densest Subgraph

SLIDE 17

What other types of sampling are there that a) are useful for solving graph problems and b) can be supported on dynamic graph streams?

SLIDE 18

1. Graph Matching

via SNAPE Sampling

II. Graph Connectivity

via DEALS Sampling

SLIDE 19

1st Result If max matching has size ≤k, can find optimal

matching in dynamic stream model using Õ(k2) space.

Optimal & Simple. Extends to hypergraph matching, vertex

cover, hitting set… but gets a lot more complicated.

Basic Idea: “SNAPE” sampling primitive.

Graph Matchings

SLIDE 20

1st Result If max matching has size ≤k, can find optimal

matching in dynamic stream model using Õ(k2) space.

Optimal & Simple. Extends to hypergraph matching, vertex

cover, hitting set… but gets a lot more complicated.

Basic Idea: “SNAPE” sampling primitive.
2nd Result If max matching has size ≥k, can find matching of

size Ω(k/t) in the dynamic stream model using Õ(k2/t3) space.

Application: Guessing k gives O(t)-approx for max matching

using Õ(n2/t3) space. This is also optimal; see Sanjeev’s talk.

Graph Matchings

SLIDE 21

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 22

SAMPLE each node with prob. ϴ(1/k) and DELETE the rest

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 23

SAMPLE each node with prob. ϴ(1/k) and DELETE the rest

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 24

SAMPLE each node with prob. ϴ(1/k) and DELETE the rest

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 25

SAMPLE each node with prob. ϴ(1/k) and DELETE the rest
RETURN a random edge amongst those that remain. If no

edges remain, return NULL.

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 26

SAMPLE each node with prob. ϴ(1/k) and DELETE the rest
RETURN a random edge amongst those that remain. If no

edges remain, return NULL.

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 27

SAMPLE each node with prob. ϴ(1/k) and DELETE the rest
RETURN a random edge amongst those that remain. If no

edges remain, return NULL.

Theorem If G has max matching size ≤k then O(k2 log k)

SNAPE samples will include a max matching from G.

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 28

Let G have max matching of size ≤k. Say node is heavy if degree

is ≥10k and edge is shallow if both endpoints aren’t heavy.

Small Matching Analysis: Basic Idea

SHALLOW EDGE HEAVY NODE

SLIDE 29

Let G have max matching of size ≤k. Say node is heavy if degree

is ≥10k and edge is shallow if both endpoints aren’t heavy.

Small Matching Analysis: Basic Idea

SHALLOW EDGE HEAVY NODE

SLIDE 30

Let G have max matching of size ≤k. Say node is heavy if degree

is ≥10k and edge is shallow if both endpoints aren’t heavy.

Lemma Let G’ contains a max matching of G if:

G’ includes all shallow edges in G. Every heavy node in G has degree at least 5k in G’.

Small Matching Analysis: Basic Idea

SHALLOW EDGE HEAVY NODE

SLIDE 31

Let G have max matching of size ≤k. Say node is heavy if degree

is ≥10k and edge is shallow if both endpoints aren’t heavy.

Lemma Let G’ contains a max matching of G if:

G’ includes all shallow edges in G. Every heavy node in G has degree at least 5k in G’.

Small Matching Analysis: Basic Idea

SHALLOW EDGE HEAVY NODE

SLIDE 32

Let G have max matching of size ≤k. Say node is heavy if degree

is ≥10k and edge is shallow if both endpoints aren’t heavy.

Lemma Let G’ contains a max matching of G if:

G’ includes all shallow edges in G. Every heavy node in G has degree at least 5k in G’.

Proof Each missing edge is incident to some heavy node but you

still have plenty of other edges on that node.

Small Matching Analysis: Basic Idea

SHALLOW EDGE HEAVY NODE

SLIDE 33

Let G have max matching of size ≤k. Say node is heavy if degree

is ≥10k and edge is shallow if both endpoints aren’t heavy.

Lemma Let G’ contains a max matching of G if:

G’ includes all shallow edges in G. Every heavy node in G has degree at least 5k in G’.

Proof Each missing edge is incident to some heavy node but you

still have plenty of other edges on that node.

Useful Fact G has a vertex cover W of size at most 2k.

Small Matching Analysis: Basic Idea

SHALLOW EDGE HEAVY NODE

SLIDE 34

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 35

If we delete nodes (other than u and v) in hitting set W and

neighbors of u, v leaves exactly the edge uv if u and v sampled.

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 36

If we delete nodes (other than u and v) in hitting set W and

neighbors of u, v leaves exactly the edge uv if u and v sampled.

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 37

If we delete nodes (other than u and v) in hitting set W and

neighbors of u, v leaves exactly the edge uv if u and v sampled.

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 38

If we delete nodes (other than u and v) in hitting set W and

neighbors of u, v leaves exactly the edge uv if u and v sampled.

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 39

If we delete nodes (other than u and v) in hitting set W and

neighbors of u, v leaves exactly the edge uv if u and v sampled.

Hence, if uv is shallow:

Small Matching Analysis: Shallow Edges

Pr[uv is only remaining edge] ≥ p2(1 − p)|Γ(u)|+|Γ(v)|+|W | = Ω(k−2)

u v

SHALLOW EDGE

SLIDE 40

If we delete nodes (other than u and v) in hitting set W and

neighbors of u, v leaves exactly the edge uv if u and v sampled.

Hence, if uv is shallow:
After O(k2 log k) repetitions, have sampled edge uv whp.

Small Matching Analysis: Shallow Edges

Pr[uv is only remaining edge] ≥ p2(1 − p)|Γ(u)|+|Γ(v)|+|W | = Ω(k−2)

u v

SHALLOW EDGE

SLIDE 41

Small Matching Analysis: Edges on Heavy Nodes

HEAVY NODE

SLIDE 42

For heavy u, deleting W\{u} leaves star on u with ≥ 8k leaves.

Small Matching Analysis: Edges on Heavy Nodes

HEAVY NODE

SLIDE 43

For heavy u, deleting W\{u} leaves star on u with ≥ 8k leaves.

Small Matching Analysis: Edges on Heavy Nodes

HEAVY NODE

SLIDE 44

For heavy u, deleting W\{u} leaves star on u with ≥ 8k leaves.
Hence,

Small Matching Analysis: Edges on Heavy Nodes

Pr[edge incident to u is sampled] ≥ p(1 − p)|W | = Ω(k−1)

HEAVY NODE

SLIDE 45

For heavy u, deleting W\{u} leaves star on u with ≥ 8k leaves.
Hence,
After O(k2 log k) repetitions, have sampled 5k edges on u.

Small Matching Analysis: Edges on Heavy Nodes

Pr[edge incident to u is sampled] ≥ p(1 − p)|W | = Ω(k−1)

HEAVY NODE

SLIDE 46

Theorem If G has matching ≥k then O(k/t3) SNAPE samples

with p=ϴ(t/k) has matching of size Ω(k/t) with high probability.

Approximate Matching: Basic Idea

SLIDE 47

Theorem If G has matching ≥k then O(k/t3) SNAPE samples

with p=ϴ(t/k) has matching of size Ω(k/t) with high probability.

Proof
Let e1, e2, e3, e4,… be sequence of SNAPE samples and

consider constructing greedy matching M.

Approximate Matching: Basic Idea

SLIDE 48

Theorem If G has matching ≥k then O(k/t3) SNAPE samples

with p=ϴ(t/k) has matching of size Ω(k/t) with high probability.

Proof
Let e1, e2, e3, e4,… be sequence of SNAPE samples and

consider constructing greedy matching M.

Assuming |M|=o(k/t) then

Approximate Matching: Basic Idea

Pr[ei added to M] ≈ Pr[ei isn’t a NULL] · Pr[all endpoints in M are deleted] = Ω(kp2) · (1 − p)o(k/t) = Ω(t2/k)

SLIDE 49

Theorem If G has matching ≥k then O(k/t3) SNAPE samples

with p=ϴ(t/k) has matching of size Ω(k/t) with high probability.

Proof
Let e1, e2, e3, e4,… be sequence of SNAPE samples and

consider constructing greedy matching M.

Assuming |M|=o(k/t) then
After O(k/t2) SNAPE samples we have |M|= Ω(k/t)

Approximate Matching: Basic Idea

Pr[ei added to M] ≈ Pr[ei isn’t a NULL] · Pr[all endpoints in M are deleted] = Ω(kp2) · (1 − p)o(k/t) = Ω(t2/k)

SLIDE 50

1. Graph Matching

via SNAPE Sampling

II. Graph Connectivity

via DEALS Sampling

SLIDE 51

Graph Connectivity

SLIDE 52

1st Result Test if graph is k-edge-connected in Õ(kn) space.
Basic Idea: “DEALS” sampling primitive.

Graph Connectivity

SLIDE 53

1st Result Test if graph is k-edge-connected in Õ(kn) space.
Basic Idea: “DEALS” sampling primitive.
2nd Result Distinguish node connectivity ≤k from ≥(1+ε)k

using Õ(ε-1kn) space.

Basic Idea: Combine node sampling and DEALS sampling.
Open: Testing exactly exact node connectivity?

Graph Connectivity

SLIDE 54

1st Result Test if graph is k-edge-connected in Õ(kn) space.
Basic Idea: “DEALS” sampling primitive.
2nd Result Distinguish node connectivity ≤k from ≥(1+ε)k

using Õ(ε-1kn) space.

Basic Idea: Combine node sampling and DEALS sampling.
Open: Testing exactly exact node connectivity?
3rd Result (1+ε)-approx every cut using Õ(ε-2n) space.
Basic Idea: Combine edge sampling and DEALS sampling.
Hypergraph Sparsifiers: Extends Kogan, Krauthgamer [ITCS 2015]

Graph Connectivity

SLIDE 55

DEALS Sampling

Direct-Edges-Add-L0-Sketches

SLIDE 56

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

SLIDE 57

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

SLIDE 58

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

1 2 3 5 4

SLIDE 59

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

1 2 3 5 4

{1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}

a1 = 1 1 a1 = ( 1 1 0)

SLIDE 60

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

1 2 3 5 4

{1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}

a1 = 1 1 a1 = ( 1 1 0) a2 = ( − 1 1 0)

SLIDE 61

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

1 2 3 5 4

{1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}

a1 = 1 1 a1 = ( 1 1 0) a2 = ( − 1 1 0)

SLIDE 62

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

1 2 3 5 4

{1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}

a1 = 1 1 a1 = ( 1 1 0) a2 = ( − 1 1 0) a1 + a2 = ( 1 1 0)

SLIDE 63

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

Lemma Non-zero entries of ∑i∈S ai = edges across (S,V\S) and

hence, ∑i∈S Mai = M(∑i∈S ai) yields random edge across (S,V\S).

1 2 3 5 4

{1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}

a1 = 1 1 a1 = ( 1 1 0) a2 = ( − 1 1 0) a1 + a2 = ( 1 1 0)

SLIDE 64

DEALS Sampling

Direct-Edges-Add-L0-Sketches

Problem Sample edge across cut (S,V\S) where cut is specified

at end of the stream. May use Õ(n) space.

Algorithm Construct Ma1, Ma2, … , Man where M is L0-sampling

sketch and ai encodes neighborhood of node i.

Lemma Non-zero entries of ∑i∈S ai = edges across (S,V\S) and

hence, ∑i∈S Mai = M(∑i∈S ai) yields random edge across (S,V\S).

Application Find spanning trees and edges in light cuts.

1 2 3 5 4

{1,2} {1,3} {1,4} {1,5} {2,3} {2,4} {2,5} {3,4} {3,5} {4,5}

a1 = 1 1 a1 = ( 1 1 0) a2 = ( − 1 1 0) a1 + a2 = ( 1 1 0)

SLIDE 65

Application to Node Connectivity…

SLIDE 66

Simplified Result Can answer queries of form “are u and v

connected after removal of set of k nodes S” using Õ(kn) space.

Application to Node Connectivity…

SLIDE 67

Simplified Result Can answer queries of form “are u and v

connected after removal of set of k nodes S” using Õ(kn) space.

Algorithm
Sample some edges and answer “no” iff there’s no S-avoiding

path between u and v amongst sampled edges.

Application to Node Connectivity…

SLIDE 68

Simplified Result Can answer queries of form “are u and v

connected after removal of set of k nodes S” using Õ(kn) space.

Algorithm
Sample some edges and answer “no” iff there’s no S-avoiding

path between u and v amongst sampled edges.

How to sample: Pick each node with probability1/k and find

spanning forest on these nodes. Repeat Õ(k2) times.

Application to Node Connectivity…

SLIDE 69

Simplified Result Can answer queries of form “are u and v

connected after removal of set of k nodes S” using Õ(kn) space.

Algorithm
Sample some edges and answer “no” iff there’s no S-avoiding

path between u and v amongst sampled edges.

How to sample: Pick each node with probability1/k and find

spanning forest on these nodes. Repeat Õ(k2) times.

Analysis Let u-x1-x2-….-xt-v be S-avoiding path in input graph.

Application to Node Connectivity…

SLIDE 70

Simplified Result Can answer queries of form “are u and v

connected after removal of set of k nodes S” using Õ(kn) space.

Algorithm
Sample some edges and answer “no” iff there’s no S-avoiding

path between u and v amongst sampled edges.

How to sample: Pick each node with probability1/k and find

spanning forest on these nodes. Repeat Õ(k2) times.

Analysis Let u-x1-x2-….-xt-v be S-avoiding path in input graph.
Spanning forest on sampled nodes contains an S-avoiding path

between xi and xi+1 with prob. p2(1-p)k≈k-2. After Õ(k2) repeats we have S-avoiding path in E’ with high probability.

Application to Node Connectivity…

SLIDE 71

Result Can (1+ε) approximate all cuts using O(ε-2n) space.

Application to Cut Sparsification…

SLIDE 72

Result Can (1+ε) approximate all cuts using O(ε-2n) space.
Basic Idea
Sampling edges with probability ≥ (cε-2 log n)/λe preserves all

cut sizes where λe is the edge connectivity. Fung et al. [STOC 2011]

Application to Cut Sparsification…

SLIDE 73

Result Can (1+ε) approximate all cuts using O(ε-2n) space.
Basic Idea
Sampling edges with probability ≥ (cε-2 log n)/λe preserves all

cut sizes where λe is the edge connectivity. Fung et al. [STOC 2011]

Use DEALS sampling to pick all edges with λe≤2cε-2 log n and

sample each remaining edge with probably 1/2.

Application to Cut Sparsification…

SLIDE 74

Result Can (1+ε) approximate all cuts using O(ε-2n) space.
Basic Idea
Sampling edges with probability ≥ (cε-2 log n)/λe preserves all

cut sizes where λe is the edge connectivity. Fung et al. [STOC 2011]

Use DEALS sampling to pick all edges with λe≤2cε-2 log n and

sample each remaining edge with probably 1/2.

Recurse O(log n) times in parallel until we have sparse graph.

Application to Cut Sparsification…

SLIDE 75

Thanks!

Graph Streaming Survey

McGregor [SIGMOD Record 2014]

Vertex Connectivity and Sparsification.

Guha, McGregor, Tench [PODS 2015]

Densest Subgraphs.

McGregor, Tench, Vorotnikova, Vu [MFCS 2015]

Matching, Vertex Cover, Hitting Set.

Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh, Vorotnikova [TBA 2016]

SLIDE 76

Thanks!

Graph Streaming Survey

McGregor [SIGMOD Record 2014]

Vertex Connectivity and Sparsification.

Guha, McGregor, Tench [PODS 2015]

Densest Subgraphs.

McGregor, Tench, Vorotnikova, Vu [MFCS 2015]

Matching, Vertex Cover, Hitting Set.

Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh, Vorotnikova [TBA 2016]