[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 15]

Densest Subgraphs

McGregor, Tench, Vorotnikova, Vu [MFCS 15]

Matching, Vertex Cover, Hitting Set

Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh, Vorotnikova [SODA 16]

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 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 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

Graph Stream Algorithms: A Survey A n d r e w M c G r e g

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Graph Streaming Survey

McGregor [SIGMOD Record 14]

SLIDE 6

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 11]

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 11]

Corollary Can uniformly sample an edge in the dynamic graph

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

SLIDE 8

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

Densest Subgraph

SLIDE 9

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]

Densest Subgraph

SLIDE 10

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]
Our Result One pass 1+ε approximation using Õ(ε-2 n) space:

Densest Subgraph

SLIDE 11

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]
Our Result One pass 1+ε approximation using Õ(ε-2 n) space:
Uniformly sample of Õ(ε-2 n) edges. Let ĎS be estimate of DS

based on sampled edges. Return maxS ĎS.

Densest Subgraph

SLIDE 12

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]
Our Result One pass 1+ε approximation using Õ(ε-2 n) space:
Uniformly sample of Õ(ε-2 n) edges. Let ĎS be estimate of DS

based on sampled edges. Return maxS ĎS.

Analysis For any subset S of size k, with probability 1-n-2k,

Densest Subgraph

SLIDE 13

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]
Our Result One pass 1+ε approximation using Õ(ε-2 n) space:
Uniformly sample of Õ(ε-2 n) edges. Let ĎS be estimate of DS

based on sampled edges. Return maxS ĎS.

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

Densest Subgraph

SLIDE 14

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]
Our Result One pass 1+ε approximation using Õ(ε-2 n) space:
Uniformly sample of Õ(ε-2 n) edges. Let ĎS be estimate of DS

based on sampled edges. Return maxS ĎS.

Analysis For any subset S of size k, with probability 1-n-2k,
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 15

Density of node set S is DS =|ES|/|S|. Estimate D*=maxS DS.
Previous Result 2+ε approximation using Õ(ε-2 n) space.
Bhattycharya et al. [STOC 15], Bahmani et al. [PVLDB 12]
Our Result One pass 1+ε approximation using Õ(ε-2 n) space:
Uniformly sample of Õ(ε-2 n) edges. Let ĎS be estimate of DS

based on sampled edges. Return maxS ĎS.

Analysis For any subset S of size k, with probability 1-n-2k,
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 15], Esfandiari et al. [ArXiv 15]

Densest Subgraph

SLIDE 16

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 17

1. Graph Matching

via SNAPE Sampling

II. Graph Connectivity

via DEALS Sampling

SLIDE 18

1st Result If max matching has size ≤k, can solve exactly exact

max 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 19

1st Result If max matching has size ≤k, can solve exactly exact

max 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; ask Grigory!

Graph Matchings

SLIDE 20

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 21

Sample each node with probability ϴ(1/k) and delete the rest

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 22

Sample each node with probability ϴ(1/k) and delete the rest

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 23

Sample each node with probability ϴ(1/k) and delete the rest

SNAPE Sampling

Sample-Nodes-And-Pick-Edge

SLIDE 24

Sample each node with probability 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 25

Sample each node with probability 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 probability 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 27

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 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.

Lemma G’ includes 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 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 G’ includes 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 G’ includes 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 Idea 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 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 G’ includes 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 Idea 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 33

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 34

If u and v are sampled but no other nodes in vertex cover W or

Γ(u) or Γ(v) are sampled then uv is only edge remaining!

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 35

If u and v are sampled but no other nodes in vertex cover W or

Γ(u) or Γ(v) are sampled then uv is only edge remaining!

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 36

If u and v are sampled but no other nodes in vertex cover W or

Γ(u) or Γ(v) are sampled then uv is only edge remaining!

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 37

If u and v are sampled but no other nodes in vertex cover W or

Γ(u) or Γ(v) are sampled then uv is only edge remaining!

Small Matching Analysis: Shallow Edges

u v

SHALLOW EDGE

SLIDE 38

If u and v are sampled but no other nodes in vertex cover W or

Γ(u) or Γ(v) are sampled then uv is only edge remaining!

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 39

If u and v are sampled but no other nodes in vertex cover W or

Γ(u) or Γ(v) are sampled then uv is only edge remaining!

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 40

Small Matching Analysis: Edges on Heavy Nodes

HEAVY NODE

SLIDE 41

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

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.
Hence,

Small Matching Analysis: Edges on Heavy Nodes

HEAVY NODE

Pr[edge incident to u is sampled] ≈ Ω(kp2) · (1 − p)|W | = Ω(k−1)

SLIDE 44

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

HEAVY NODE

Pr[edge incident to u is sampled] ≈ Ω(kp2) · (1 − p)|W | = Ω(k−1)

SLIDE 45

Theorem If G has matching ≥k then O(k2/t3) SNAPE samples

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

Approximate Matching: Basic Idea

SLIDE 46

Theorem If G has matching ≥k then O(k2/t3) SNAPE samples

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

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

constructing greedy matching M.

Approximate Matching: Basic Idea

SLIDE 47

Theorem If G has matching ≥k then O(k2/t3) SNAPE samples

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

Proof
Let e1, e2, e3,… 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 48

Theorem If G has matching ≥k then O(k2/t3) SNAPE samples

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

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

constructing greedy matching M.

Assuming |M|=o(k/t) then
After O(k2/t3) 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 49

1. Graph Matching

via SNAPE Sampling

II. Graph Connectivity

via DEALS Sampling

SLIDE 50

Graph Connectivity

SLIDE 51

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

Graph Connectivity

SLIDE 52

1st Result Test if k-edge-connected using 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 exact node connectivity?

Graph Connectivity

SLIDE 53

1st Result Test if k-edge-connected using 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 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 15]

Graph Connectivity

SLIDE 54

DEALS Sampling

Direct-Edges-Add-L0-Sketches

SLIDE 55

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 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.

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

sketch and ai is vector encoding neighborhood of node i.

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 is vector encoding neighborhood of node i.

1 2 3 5 4

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 is vector encoding 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 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 is vector encoding 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 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 is vector encoding 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 is vector encoding 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 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 is vector encoding 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 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 is vector encoding 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 64

Application to Node Connectivity…

SLIDE 65

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 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.

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 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.

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 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.

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

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.
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 70

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

Application to Cut Sparsification…

SLIDE 71

Result Can 1+ε approximate all cuts using Õ(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 11]

Application to Cut Sparsification…

SLIDE 72

Result Can 1+ε approximate all cuts using Õ(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 11]

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 73

Result Can 1+ε approximate all cuts using Õ(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 11]

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 74

Thanks!

Graph Streaming Survey

McGregor [SIGMOD Record 14]

Vertex Connectivity and Sparsification

Guha, McGregor, Tench [PODS 15]

Densest Subgraphs

McGregor, Tench, Vorotnikova, Vu [MFCS 15]

Matching, Vertex Cover, Hitting Set

Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh, Vorotnikova [SODA 16]

SLIDE 75

Thanks!

Graph Streaming Survey

McGregor [SIGMOD Record 14]

Vertex Connectivity and Sparsification

Guha, McGregor, Tench [PODS 15]

Densest Subgraphs

McGregor, Tench, Vorotnikova, Vu [MFCS 15]

Matching, Vertex Cover, Hitting Set

Chitnis, Cormode, Esfandiari, Hajiaghayi, McGregor, Monemizadeh, Vorotnikova [SODA 16]