Finding Graph Matchings in Data Streams Andrew McGregor, UPenn The - - PowerPoint PPT Presentation

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Finding Graph Matchings in Data Streams Andrew McGregor, UPenn The - - PowerPoint PPT Presentation

Nov 25, 2023 32 likes •919 views

Finding Graph Matchings in Data Streams Andrew McGregor, UPenn The Streaming Model The Streaming Model Classic Problem: Median Finding [Munro & Paterson] The Streaming Model Classic Problem: Median Finding [Munro & Paterson]

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Finding Graph Matchings in Data Streams

Andrew McGregor, UPenn

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The Streaming Model

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The Streaming Model

  • Classic Problem: Median Finding [Munro & Paterson]
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The Streaming Model

  • Classic Problem: Median Finding [Munro & Paterson]
  • Parameters of the Model:
  • How much memory?
  • How many passes?
  • How much computation time between data elements?
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The Streaming Model

  • Classic Problem: Median Finding [Munro & Paterson]
  • Parameters of the Model:
  • How much memory?
  • How many passes?
  • How much computation time between data elements?
  • Statistics, Norms and Histograms…
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The Streaming Model

  • Classic Problem: Median Finding [Munro & Paterson]
  • Parameters of the Model:
  • How much memory?
  • How many passes?
  • How much computation time between data elements?
  • Statistics, Norms and Histograms…
  • What about graph problems?
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Graph Streaming

  • Instance of graph problem G = (V, E)
  • Edges arrive in arbitrary order: e1, e2, e3, …, em
  • Memory limit O(n polylog n) where n = |V|
  • Spanner Construction, Bipartite Matching, Lower Bounds

[Feigenbaum, Kannan, M. , Suri, Zhang ’04 &’05]

  • “Annotation” Stream Model [Aggarwal, Datar, Rajagopalan,

Ruhl ’04, Demetrescu, Finocchi, Ribichini ’05]

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Matching

  • A matching - set of edges with no two edges sharing an end point.
  • Problems:

Find the matching of maximum cardinality (MCM) Find the matching of maximum weight (MWM)

  • (Non-streamable) Algorithms:

Exact polytime algorithm for both [Gabow ’90] Linear-time 1+ε approx for MCM [Kalantari & Shokoufandeh ’95] Linear-time 3/2+ε approx for MWM [Drake & Hougardy ’03]

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Results

  • Unweighted Matchings:

1+ε approximation in constant passes.

  • Weighted Matchings:

3+2√2 approximation in single pass. 2+ε approximation in constant passes.

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Unweighted Matchings.

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An Easy 2 Approximation

  • Greedy Algorithm:

Store an edge if it is not adjacent to stored edge

  • Construct a maximal matching - 2 Approximation
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Augmenting Paths

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Augmenting Paths

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Augmenting Paths

Matching M

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Augmenting Paths

  • Augmenting Path: simple path starting and ending

at unmatched nodes such that edges alternate between M and E\M.

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Augmenting Paths

  • Augmenting Path: simple path starting and ending

at unmatched nodes such that edges alternate between M and E\M.

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Augmenting Paths

  • Augmenting Path: simple path starting and ending

at unmatched nodes such that edges alternate between M and E\M.

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Augmenting Paths

  • Consider augmenting paths defined by taking the

symmetric difference between current (maximal) matching and optimum matching.

  • Let Pi be the number of length i augmenting paths

|M| +

  • 1≤i≤k

Pi ≥ OPT(1 − 1/k)

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Algorithm Outline

  • 1. Find a maximal matching
  • 2. For 1 ≤ i ≤ k:

Find a set, Si, of length i augmenting paths

  • 3. Augment current matching with Sj where j = argmax Si
  • 4. Repeat from 2 unless Sj is small
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Projecting to Layered Graphs

G

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Projecting to Layered Graphs

G L(G)

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Lemma: If there are Pi length i augmenting paths in G then we expect Pi / 2(2i)i node disjoint paths in L(G).

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Lemma: If there are Pi length i augmenting paths in G then we expect Pi / 2(2i)i node disjoint paths in L(G). Lemma: A maximal set of node disjoint paths in L(G), is an i+2 approximation to the maximum set of node disjoint paths in L(G).

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Lemma: If there are Pi length i augmenting paths in G then we expect Pi / 2(2i)i node disjoint paths in L(G). Lemma: A maximal set of node disjoint paths in L(G), is an i+2 approximation to the maximum set of node disjoint paths in L(G). To find a constant fraction of length i augmenting paths Pi, create layered graph and greedily find node disjoint paths.

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Limiting Backtracking

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Limiting Backtracking

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Limiting Backtracking

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Limiting Backtracking

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Limiting Backtracking

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Limiting Backtracking

  • Solution: If number of paths being grown falls below threshold

δn then delete and backtrack.

Good: Only backtrack a constant number of times Bad: Don’t find a maximal set of node disjoint paths

  • In a constant number of passes, we find a constant fraction of

length i node disjoint paths/augmenting paths.

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Weighted Matching.

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Single Pass 3+2√2 Approximation

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Single Pass 3+2√2 Approximation

  • At all times we store some matching M
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Single Pass 3+2√2 Approximation

  • At all times we store some matching M
  • For each edge e:

Compute total weight W of edges e1, e2 in M incident to e If w(e) > (1+γ) W then M ← M ∪ {e} \ {e1,e2}

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Single Pass 3+2√2 Approximation

  • At all times we store some matching M
  • For each edge e:

Compute total weight W of edges e1, e2 in M incident to e If w(e) > (1+γ) W then M ← M ∪ {e} \ {e1,e2}

  • We say e is “born” and “killed” e1 and e2
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Proof (Sketch)

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Proof (Sketch)

  • We say an edge e is a survivor if it is born and was never killed.
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Proof (Sketch)

  • We say an edge e is a survivor if it is born and was never killed.
  • Let S = all survivors.
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Proof (Sketch)

  • We say an edge e is a survivor if it is born and was never killed.
  • Let S = all survivors.
  • For survivor e we define the trail of the dead T(e) to be the

transitive closure of edges killed by e.

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Proof (Sketch)

  • We say an edge e is a survivor if it is born and was never killed.
  • Let S = all survivors.
  • For survivor e we define the trail of the dead T(e) to be the

transitive closure of edges killed by e.

  • Claim 1: w(T(e)) ≤ w(e)/γ
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Proof (Sketch)

  • We say an edge e is a survivor if it is born and was never killed.
  • Let S = all survivors.
  • For survivor e we define the trail of the dead T(e) to be the

transitive closure of edges killed by e.

  • Claim 1: w(T(e)) ≤ w(e)/γ
  • Claim 2: Can charge the weights of edges in OPT such that:
  • At most (1+ γ) w(T(e)) is charged to T(e)
  • At most 2(1+ γ) w(e) is charged to e
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Proof (Sketch)

  • We say an edge e is a survivor if it is born and was never killed.
  • Let S = all survivors.
  • For survivor e we define the trail of the dead T(e) to be the

transitive closure of edges killed by e.

  • Claim 1: w(T(e)) ≤ w(e)/γ
  • Claim 2: Can charge the weights of edges in OPT such that:
  • At most (1+ γ) w(T(e)) is charged to T(e)
  • At most 2(1+ γ) w(e) is charged to e
  • Hence w(OPT) ≤ (1+ γ) w(T(S)) + 2(1+ γ) w(S)< (3+2√2) w(S)
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Multi-pass 2+ε Approximation

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Multi-pass 2+ε Approximation

  • First pass: find a constant approximate M1
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Multi-pass 2+ε Approximation

  • First pass: find a constant approximate M1
  • Subsequent passes: create Mi from Mi-1 by running the

previous algorithm with γ(ε)

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Multi-pass 2+ε Approximation

  • First pass: find a constant approximate M1
  • Subsequent passes: create Mi from Mi-1 by running the

previous algorithm with γ(ε)

  • Repeat if |Mi|/ |Mi-1|> 1+κ(ε)
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Multi-pass 2+ε Approximation

  • First pass: find a constant approximate M1
  • Subsequent passes: create Mi from Mi-1 by running the

previous algorithm with γ(ε)

  • Repeat if |Mi|/ |Mi-1|> 1+κ(ε)
  • Claim 1: A constant number of passes suffices
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Multi-pass 2+ε Approximation

  • First pass: find a constant approximate M1
  • Subsequent passes: create Mi from Mi-1 by running the

previous algorithm with γ(ε)

  • Repeat if |Mi|/ |Mi-1|> 1+κ(ε)
  • Claim 1: A constant number of passes suffices
  • Claim 2: When |Mi|/ |Mi-1| ≤ 1+κ we have a 2+ε approx.
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Conclusions

  • Unweighted Matchings:

1+ε approximation in constant passes.

  • Weighted Matchings:

3+2√2 approximation in single pass. 2+ε approximation in constant passes.

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Thanks.