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

Talya Eden Dana Ron Will Rosenbaum The Arboricity Captures the Complexity of Sampling Edges

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 1−𝜗 1 ∀ 𝑣, 𝑤 ∈ 𝐹, Pr 𝑣, 𝑤 is returned ∈ , 𝑛 𝑛 𝑣 1 w.p. ≈ G 𝑛 𝑤 Sample 1 w.p. ≈ 𝑦 𝑧 Edge 𝑛 w.p. ≈ 1 𝑨 𝑛 𝑥 𝑛 – sum of degrees

Pointwise vs. Total Variational Distance (TVD) 𝜗 -TVD close: 𝑈𝑊𝐸 𝑄, 𝑅 = ෍ |𝑄 𝑌 − 𝑅(𝑌)| An 𝜗 -fraction of the edges can be ignored 𝑦 Might be exactly the edges of interest 𝜗 -Pointwise close: Every edge is (almost) equally likely to be the returned edge 𝐻 Counting arbitrary subgraphs 𝜗𝑛 -clique in sublinear time given uniform edge samples [Assadi Kapralov Khanna 19] 𝑛 𝜍(𝐼) #𝐼

The Query Access Model  Vertices are labeled arbitrarily [1..n]  Query access to the graph: G Query  Degree queries on the 𝑗 th vertex Answer  Uniform neighbor queries 1 𝑤 𝑘 w.p. 𝑒(𝑤 𝑗 ) 𝑤 𝑗 𝑒 𝑤 𝑗 = 5 𝑒 𝑤 𝑗 =? 𝑤 𝑘′ 1 w.p. 𝑒(𝑤 𝑗 )

Previous results  Sampling an edge almost uniformly [E Rosenbaum 18]: 𝑛 𝑃 ∗ 𝑛 – sum of degrees 𝑒 𝑏𝑤𝑕 𝑒 𝑏𝑤𝑕 - avg. degree 𝑃 ∗ - poly(log 𝑜, 1/𝜗) factors 𝑛  Lower bound Ω 𝑒 𝑏𝑤𝑕 Can we improve for sparse graphs? Yes, for sparse “ everywhere ” graphs

Arboricity – A Measure of Sparseness 𝛽 − Maximal average degree over all subgraphs 𝑒 𝑏𝑤𝑕 (𝐼) ≤ 𝛽 ∀𝐼 ⊆ 𝐻, If 𝛽 is small then: Rich family of graphs No dense subgraphs with bounded arboricity: □ Planar graphs Sparse “ everywhere ” □ Excluded-minor graphs □ ‘ Real world ’ graphs For all graphs, 𝛽 ≤ □ … 𝑛

Results Sampling an edge from an 𝜗 -pointwise close to uniform dist.: 𝑛 log 3 𝑜 [E Rosenbaum 18] 𝛽 𝑃 ∗ 𝑃 𝑒 𝑏𝑤𝑕 ⋅ 𝑒 𝑏𝑤𝑕 𝜗

Results Sampling an edge from an 𝜗 -pointwise close to uniform dist.: 𝒏 log 3 𝑜 [E Rosenbaum 18] 𝜷 𝑃 ∗ 𝑃 𝒆 𝒃𝒘𝒉 ⋅ ≤ 𝒆 𝒃𝒘𝒉 𝜗 𝛽 ≤ 𝑛 Expon Ex onen ential tial imp mproveme ment nt for graphs with constant arboricity 𝛽 = Θ(1)  Lower Bounds: 𝛽 log 𝑜 Ω Ω 𝑒 𝑏𝑤𝑕 log log 𝑜 for any graph with for graphs with 𝛽 = Θ(1) arboricity 𝑃(𝛽)

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 1−𝜗 1 ∀ 𝑣, 𝑤 ∈ 𝐹, Pr 𝑣, 𝑤 is returned ∈ , 𝑛 𝑛 ֞ 1−𝜗 𝒆 𝒘 𝒆 𝒘 ∀ 𝑤 ∈ 𝑊, Pr 𝑤 is returned ∈ , 𝑛 𝑛 Uniform 𝑤 𝑣 neighbor ≈ 𝑒(𝑤) 𝑒 𝑤 = 1 1 ⋅ 𝑛 – sum of degrees 𝑛 𝑛

The Algorithm

The algorithm 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 Try-to-sample-a-vertex ( 𝛽, 𝜗 ): 1 𝑒 𝑤 Returns each vertex w.p. ≈ 𝛽 or fa fails (for ℓ = log 𝑜, ෤ 𝛽 ≈ 𝛽 ) ℓ ⋅ 𝑜⋅෥ Total success prob. of a single execution of Try-to-sample: 𝑒 𝑤 𝑛 ≈ ෍ 𝛽 = ℓ𝑜 ෤ ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 𝑤∈𝑊 Expected number of sufficient attempts: ⋅ log 3 𝑜 𝛽 ≈ ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 𝛽 ෤ = ⋅ ℓ 𝑒 𝑏𝑤𝑕 𝜗 𝑛 𝑒 𝑏𝑤𝑕

A Relative Decomposition An inductive definition: 𝛽⋅4𝑚𝑝𝑕 𝑜 𝜗 , ෥ 𝛽 ≈ ෤ 𝜗 ≈ 𝜗 log 𝑜 𝑀 0 = {𝑤 ∣ 𝑒 𝑤 ≤ ෤ 𝛽} 𝑀 0

A Relative Decomposition An inductive definition: 𝑀 ℓ ℓ = log 𝑜 𝛽⋅4𝑚𝑝𝑕 𝑜 𝜗 , ෥ 𝛽 ≈ ෤ 𝜗 ≈ 𝜗 log 𝑜 𝑀 𝑘 𝑀 𝑘 = {𝑤 ∉ 𝑀 <𝑘 ∣ 𝑒 <𝑘 𝑤 ≥ 1 − ෥ 𝜗 𝑒(𝑤)} 𝑀 1 𝑀 1 = {𝑤 ∉ 𝑀 0 ∣ 𝑒 0 𝑤 ≥ 1 − ෥ 𝜗 𝑒(𝑤)} 𝑀 0 = {𝑤 ∣ 𝑒 𝑤 ≤ ෤ 𝛽} 𝑀 0

𝑒(𝑤) (Try to) Sample a vertex w.p. ℓ⋅𝑜⋅෥ 𝛽 Try-to to-Sam Sampl ple-a-vertex 𝑀 ℓ 1. Choose 𝑘 ∈ 0, ℓ 2. 𝑤 ← Sa Sampl ple- 𝑴 𝟏 (if failed abort) 3. Rando dom m wa walk k for 𝑘 steps 3. 𝑀 𝑘 4. (If returned to 𝑀 0 , abort) rn the 𝑘 th vertex 5. Retu eturn 5. Sample ple- 𝑴 𝟏 1. Choose 𝑤 ∈ 𝑊 u.a.r 𝑀 1 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. ෥ 𝛽 𝑀 0

Correctness Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abort) 2. Rando dom m walk k for 𝑘 steps 3. 3. 𝑀 𝑘 If returned to 𝑀 0 , abort 4. rn the 𝑘 th vertex Retu eturn 5. 5. Sample ple- 𝑴 𝟏 1. Choose 𝑤 ∈ 𝑊 u.a.r 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑀 1 𝜷 ෥ Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝜗 𝑘 𝑒(𝑤) 1−෤ Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ ℓ⋅𝑜⋅෥ 𝛽

Correctness - 𝑘 = 0 Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abor ort) 2. Rando dom m walk k for 𝑘 steps 3. 3. 𝑀 𝑘 If returned to 𝑀 0 , abort 4. urn the 𝑘 th vertex Ret eturn 5. 5. Sample ple- 𝑴 𝟏 1. Choose 𝑤 ∈ 𝑊 u.a.r 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑀 1 𝜷 ෥ Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝝑 𝒌 𝒆(𝒘) 𝑤 1 ℓ ⋅ 1 𝑜 ⋅ 𝑒 𝑤 = 𝑒(𝑤) 𝟐−෤ Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ 𝛽 ෤ ℓ𝑜 ෤ 𝛽 ℓ⋅𝒐⋅෥ 𝜷

Correctness - 𝑘 = 0 Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abor ort) 2. Rando dom m walk k for 𝑘 steps 3. 3. 𝑀 𝑘 If returned to 𝑀 0 , abort 4. urn the 𝑘 th vertex Ret eturn 5. 5. Sample ple- 𝑴 𝟏 1. Choose 𝑤 ∈ 𝑊 u.a.r 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑀 1 𝜷 ෥ Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝝑 𝒌 𝒆(𝒘) 𝑤 𝟐−෤ 𝑒(𝑤) Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ ℓ⋅𝒐⋅෥ 𝜷 ℓ ⋅ 𝑜 ⋅ ෤ 𝛽

Correctness - 𝑘 = 1 Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abor ort) 2. Rando dom m walk k for 𝑘 steps 3. 3. 𝑀 𝑘 If returned to 𝑀 0 , abort 4. urn the 𝑘 th vertex Ret eturn 5. 5. Sample ple- 𝑴 𝟏 ℓ ⋅ 𝑒 0 𝑣 1 ≥ (1 − ǁ 𝜗)𝑒(𝑣) 1. Choose 𝑤 ∈ 𝑊 u.a.r 𝑜 ෤ 𝛽 ℓ𝑜 ෤ 𝛽 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑣 𝑀 1 𝜷 ෥ 𝑒(𝑤) 𝑒(𝑤) = 1 1 𝛽 ⋅ Claim: aim: 1 𝑜 ෤ 𝑜 ෤ 𝛽 1 1 𝑜 ෤ 𝛽 ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝑜 ෤ 𝛽 1 𝑜 ෤ 𝛽 𝝑 𝒌 𝒆(𝒘) 𝑤 𝟐−෤ 𝑜 ෤ 𝛽 𝑒(𝑤) Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ ℓ⋅𝒐⋅෥ 𝜷 ℓ ⋅ 𝑜 ⋅ ෤ 𝛽

Correctness - 𝑘 = 1 Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abor ort) 2. Rando dom m walk k for 𝑘 steps 3. 3. 𝑀 𝑘 If returned to 𝑀 0 , abort 4. urn the 𝑘 th vertex Ret eturn 5. 5. Sample ple- 𝑴 𝟏 (1 − ǁ 𝜗)𝑒(𝑣) 1. Choose 𝑤 ∈ 𝑊 u.a.r ℓ𝑜 ෤ 𝛽 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑣 𝑀 1 𝜷 ෥ Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝝑 𝒌 𝒆(𝒘) 𝑤 𝟐−෤ 𝑒(𝑤) Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ ℓ⋅𝒐⋅෥ 𝜷 ℓ ⋅ 𝑜 ⋅ ෤ 𝛽

Correctness - general 𝑘 Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abor ort) 2. 𝜗 𝑘 𝑒(𝑥) Rando dom m walk k for 𝑘 steps 1 − ǁ ≥ (1 − 𝜗)𝑒(𝑥) 3. 3. 𝑥 𝑀 𝑘 If returned to 𝑀 0 , abort ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 ℓ ⋅ 𝑜 ⋅ ෤ 𝛽 4. urn the 𝑘 th vertex Ret eturn 5. 5. Sample ple- 𝑴 𝟏 (1 − ǁ 𝜗)𝑒(𝑣) 1. Choose 𝑤 ∈ 𝑊 u.a.r ℓ𝑜 ෤ 𝛽 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑣 𝑀 1 𝜷 ෥ Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝝑 𝒌 𝒆(𝒘) 𝑤 𝟐−෤ 𝑒(𝑤) Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ ℓ⋅𝒐⋅෥ 𝜷 ℓ ⋅ 𝑜 ⋅ ෤ 𝛽

Recap Try-to to-Sa Sample le-a-ver ertex 𝑀 ℓ Choose 𝑘 ∈ 0, ℓ 1. 𝑤 ← 𝑇𝑏𝑛𝑞𝑚𝑓 − 𝑀 0 (if failed abor ort) 2. Rando dom m walk k for 𝑘 steps 3. 3. 𝑀 𝑘 If returned to 𝑀 0 , abort 4. urn the 𝑘 th vertex Ret eturn 5. 5. Sample ple- 𝑴 𝟏 𝑀 2 1. Choose 𝑤 ∈ 𝑊 u.a.r 𝑒(𝑤) 2. If 𝑒 𝑤 ≤ ෤ 𝛽, keep 𝑤 w.p. 𝑀 1 𝜷 ෥ Claim: aim: ∀𝑘 ∈ 0, ℓ , ∀𝑤 ∈ 𝑀 𝑘 , 𝑀 0 𝝑 𝒌 𝒆(𝒘) 𝟐−෤ Pr 𝑤 is 𝐬𝐟𝐮𝐯𝐬𝐨𝐟𝐞 ≥ ℓ⋅𝒐⋅෥ 𝜷