Captures the Complexity of Sampling Edges The Task Given query - - PowerPoint PPT Presentation

The Arboricity Captures the Complexity of Sampling Edges

Talya Eden Dana Ron Will Rosenbaum

The Task

 Given query access to a graph 𝐻  Design a sublinear algorithm for sampling an edge from a

distribution that is 𝜗−pointwise close to uniform ∀ 𝑣, 𝑤 ∈ 𝐹, Pr 𝑣, 𝑤 is returned ∈

1−𝜗 𝑛

,

1 𝑛 G Sample Edge

𝑤 𝑦 𝑧 𝑨 𝑣 𝑥

w.p. ≈

1 𝑛

w.p. ≈

1 𝑛

w.p. ≈ 1

𝑛

𝑛 – sum of degrees

Pointwise vs. Total Variational Distance (TVD)

𝜗-TVD close:

An 𝜗-fraction of the edges can be ignored Might be exactly the edges of interest

𝜗-Pointwise close: Every edge is (almost) equally likely to be the returned edge

𝐻

𝜗𝑛 -clique

Counting arbitrary subgraphs in sublinear time given uniform edge samples [Assadi Kapralov Khanna 19] 𝑛𝜍(𝐼) #𝐼

𝑈𝑊𝐸 𝑄, 𝑅 = ෍

𝑦

|𝑄 𝑌 − 𝑅(𝑌)|

The Query Access Model

 Vertices are labeled arbitrarily [1..n]  Query access to the graph:  Degree queries on the 𝑗th vertex  Uniform neighbor queries

𝑤𝑗 𝑒 𝑤𝑗 =? 𝑤𝑘 𝑤𝑘′ 𝑒 𝑤𝑗 = 5 Query Answer G w.p.

1 𝑒(𝑤𝑗)

w.p.

1 𝑒(𝑤𝑗)

Previous results

 Sampling an edge almost uniformly [E Rosenbaum 18]:

𝑃∗ 𝑛 𝑒𝑏𝑤𝑕

 Lower bound Ω

𝑛 𝑒𝑏𝑤𝑕

Can we improve for sparse graphs? Yes, for sparse “everywhere” graphs

𝑛 – sum of degrees 𝑒𝑏𝑤𝑕 - avg. degree 𝑃∗ - poly(log 𝑜, 1/𝜗) factors

𝛽 − Maximal average degree over all subgraphs

∀𝐼 ⊆ 𝐻,

𝑒𝑏𝑤𝑕(𝐼) ≤ 𝛽 If 𝛽 is small then:

No dense subgraphs Sparse “everywhere”

For all graphs, 𝛽 ≤ 𝑛

Arboricity – A Measure of Sparseness

Rich family of graphs with bounded arboricity: □ Planar graphs □ Excluded-minor graphs □ ‘Real world’ graphs □ …

Results

Sampling an edge from an 𝜗-pointwise close to uniform dist.: 𝑃

𝛽 𝑒𝑏𝑤𝑕 ⋅ log3 𝑜 𝜗 [E Rosenbaum 18]

𝑃∗ 𝑛 𝑒𝑏𝑤𝑕

Results

Sampling an edge from an 𝜗-pointwise close to uniform dist.: 𝑃

𝜷 𝒆𝒃𝒘𝒉 ⋅ log3 𝑜 𝜗

 Lower Bounds:

Ω

𝛽 𝑒𝑏𝑤𝑕

Ω

log 𝑜 log log 𝑜 for any graph with arboricity 𝑃(𝛽) for graphs with 𝛽 = Θ(1)

≤

𝛽 ≤ 𝑛

𝑃∗ 𝒏 𝒆𝒃𝒘𝒉

Ex Expon

• nen

ential tial imp mproveme ment nt for graphs with constant arboricity 𝛽 = Θ(1)

[E Rosenbaum 18]

The Task

 Given query access to a graph 𝐻  Design a sublinear algorithm for sampling an edge from a

distribution that is 𝜗−pointwise close to uniform ∀ 𝑣, 𝑤 ∈ 𝐹, Pr 𝑣, 𝑤 is returned ∈

1−𝜗 𝑛

,

1 𝑛

֞

∀ 𝑤 ∈ 𝑊, Pr 𝑤 is returned ∈

1−𝜗 𝒆 𝒘 𝑛

,

𝒆 𝒘 𝑛 ≈ 𝑒(𝑤) 𝑛 ⋅ 1 𝑒 𝑤 = 1 𝑛 𝑤 𝑣 𝑛 – sum of degrees Uniform neighbor

The Algorithm

The algorithm

Try-to-sample-a-vertex(𝛽, 𝜗): Returns each vertex w.p. ≈

1 ℓ ⋅ 𝑒 𝑤 𝑜⋅෥ 𝛽 or fa

fails (for ℓ = log 𝑜, ෤ 𝛽 ≈ 𝛽) Total success prob. of a single execution of Try-to-sample: ≈ ෍

𝑤∈𝑊

𝑒 𝑤 ℓ𝑜 ෤ 𝛽 = 𝑛 ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 Expected number of sufficient attempts: ≈ ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 𝑛 = ෤ 𝛽 𝑒𝑏𝑤𝑕 ⋅ ℓ

Sample ple-a-ver ertex (𝜷, 𝝑)

• 1. While tru

rue:

a) 𝑤 ← Try-to-sample-a-vertex(𝛽, 𝜗) b) If succeeded, retu eturn 𝑤

𝛽 𝑒𝑏𝑤𝑕 ⋅ log3 𝑜 𝜗

A Relative Decomposition

An inductive definition:

෤ 𝛽 ≈

𝛽⋅4𝑚𝑝𝑕 𝑜 𝜗

,෥ 𝜗 ≈

𝜗 log 𝑜

𝑀0

𝑀0 = {𝑤 ∣ 𝑒 𝑤 ≤ ෤ 𝛽}

A Relative Decomposition

An inductive definition:

෤ 𝛽 ≈

𝛽⋅4𝑚𝑝𝑕 𝑜 𝜗

,෥ 𝜗 ≈

𝜗 log 𝑜

𝑀1 𝑀𝑘 𝑀ℓ

𝑀0 = {𝑤 ∣ 𝑒 𝑤 ≤ ෤ 𝛽} 𝑀1 = {𝑤 ∉ 𝑀0 ∣ 𝑒0 𝑤 ≥ 1 − ෥ 𝜗 𝑒(𝑤)} 𝑀𝑘 = {𝑤 ∉ 𝑀<𝑘 ∣ 𝑒<𝑘 𝑤 ≥ 1 − ෥ 𝜗 𝑒(𝑤)}

𝑀0

ℓ = log 𝑜

𝑀0

(Try to) Sample a vertex w.p.

𝑒(𝑤) ℓ⋅𝑜⋅෥ 𝛽

Try-to to-Sam Sampl ple-a-vertex

• 1. Choose 𝑘 ∈ 0, ℓ
• 2. 𝑤 ← Sa

Sampl ple-𝑴𝟏 (if failed abort)

3.

• 3. Rando

dom m wa walk k for 𝑘 steps

• 4. (If returned to 𝑀0, abort)

5.

• 5. Retu

eturn rn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝛽

𝑀1 𝑀𝑘 𝑀ℓ

Correctness

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abort)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Retu eturn rn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

1−෤ 𝜗 𝑘𝑒(𝑤) ℓ⋅𝑜⋅෥ 𝛽

𝑀1 𝑀0 𝑀𝑘 𝑀ℓ

𝑀1 𝑀0 𝑀𝑘 𝑀ℓ

Correctness - 𝑘 = 0

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abor

• rt)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Ret eturn urn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

𝑤 1 ℓ ⋅ 1 𝑜 ⋅ 𝑒 𝑤 ෤ 𝛽 = 𝑒(𝑤) ℓ𝑜 ෤ 𝛽

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

𝟐−෤ 𝝑 𝒌𝒆(𝒘) ℓ⋅𝒐⋅෥ 𝜷

𝑀1 𝑀0 𝑀𝑘 𝑀ℓ

Correctness - 𝑘 = 0

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abor

• rt)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Ret eturn urn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

𝑤 𝑒(𝑤) ℓ ⋅ 𝑜 ⋅ ෤ 𝛽

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

𝟐−෤ 𝝑 𝒌𝒆(𝒘) ℓ⋅𝒐⋅෥ 𝜷

Correctness - 𝑘 = 1

𝑀1 𝑀0

𝑤 𝑒(𝑤) ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 𝑒(𝑤) 𝑜 ෤ 𝛽 ⋅ 1 𝑒(𝑤) = 1 𝑜 ෤ 𝛽

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abor

• rt)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Ret eturn urn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

𝑣 1 ℓ ⋅ 𝑒0 𝑣 𝑜 ෤ 𝛽 ≥ (1 − ǁ 𝜗)𝑒(𝑣) ℓ𝑜 ෤ 𝛽

𝑀𝑘 𝑀ℓ

1 𝑜 ෤ 𝛽 1 𝑜 ෤ 𝛽 1 𝑜 ෤ 𝛽 1 𝑜 ෤ 𝛽

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

𝟐−෤ 𝝑 𝒌𝒆(𝒘) ℓ⋅𝒐⋅෥ 𝜷

𝑀1 𝑀0 𝑀𝑘 𝑀ℓ

Correctness - 𝑘 = 1

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abor

• rt)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Ret eturn urn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

𝑤 𝑣 𝑒(𝑤) ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 (1 − ǁ 𝜗)𝑒(𝑣) ℓ𝑜 ෤ 𝛽

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

𝟐−෤ 𝝑 𝒌𝒆(𝒘) ℓ⋅𝒐⋅෥ 𝜷

𝑀1 𝑀0 𝑀𝑘 𝑀ℓ

Correctness - general 𝑘

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abor

• rt)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Ret eturn urn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

1 − ǁ 𝜗 𝑘𝑒(𝑥) ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 ≥ (1 − 𝜗)𝑒(𝑥) ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 𝑤 𝑣 𝑥 𝑒(𝑤) ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 (1 − ǁ 𝜗)𝑒(𝑣) ℓ𝑜 ෤ 𝛽

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

𝟐−෤ 𝝑 𝒌𝒆(𝒘) ℓ⋅𝒐⋅෥ 𝜷

𝑀1 𝑀0 𝑀𝑘 𝑀ℓ

Recap

Try-to to-Sa Sample le-a-ver ertex

1.

Choose 𝑘 ∈ 0, ℓ

2.

𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀0 (if failed abor

• rt)

3. 3.

Rando dom m walk k for 𝑘 steps

4.

If returned to 𝑀0, abort

5. 5.

Ret eturn urn the 𝑘th vertex

Sample ple-𝑴𝟏

• 1. Choose 𝑤 ∈ 𝑊 u.a.r
• 2. If 𝑒 𝑤 ≤ ෤

𝛽, keep 𝑤 w.p.

𝑒(𝑤) ෥ 𝜷

𝑀2

Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀𝑘, Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥

𝟐−෤ 𝝑 𝒌𝒆(𝒘) ℓ⋅𝒐⋅෥ 𝜷

Future Work

Sampling vs. Counting Counting edges (approximately): 𝑃∗ 𝑛 𝑒𝑏𝑤𝑕 ՜ 𝑃∗ 𝛽 𝑒𝑏𝑤𝑕 For triangles: Separation between counting and sampling Exists graphs s.t. counting is 𝑃∗(1) but sampling Ω(n Τ

1 4) [Feige 06, Goldreich Ron 08] [E Ron Seshadhri 17]

Future Work

Sampling vs. Counting Sampling other subgraphs Additional uses for the Relative Decomposition

Future Work

Sampling vs. Counting Sampling other subgraphs Other uses for the Relative Decomposition?

Thanks!