An Algorithm for Graph Bisection

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?An Algorithm for Graph Bisection

• Bruno Gonçalves
• Gabriel Istrate

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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Load Balancing
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Circuit Boards
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Spin Systems with m0
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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Erdös-Rényi Graphs - G(N,p)

• N nodes
• Each pair of nodes connnected with probability p
• Poissonian degree distribution
• No self loops
• No multiple edges
• Average connectivity ltkgtp(N-1)gtpltkgt/(N-1)
• Two points of interest
• Np2log2
• Np2

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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Core Peeling

• Start by making all nodes BLACK.
• Designate the largest component as the Giant
  Connected Component'' (GCC) and mark all its
  nodes as being RED
• Distinguish 2 different possibilities
• 1. If the size S of the GCC is less than N/2
  mark as RED exactly (N/2-S) nodes from the
  smallest components and return the solution
  found.
• 2. If the size S is larger than N/2 reduce it to
  its 2-core .

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Core Peeling
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Quality of Solution
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Cost
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Tree Size
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Tree Size Distribution
Np1.4
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Tree Size Distribution
Np2.0
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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Core Cutting

• Color a randomly chosen node RED
• Calculate the number of broken links
• Repeat N/2 times
• Repeat 10 times and choose the minimum at each
  step

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Core Size
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Core Cut Cost
Np1.4
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Core Cut Cost
Np2.0
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Tree Size Distribution
Np1.4
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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Discussion

• Core peeling obtains perfect bisections up to
  Np2
• Tree size distribution is exponential
• Tree cutting is cheap but not always possible
• Cutting the core is expensive
• The expander ratio decreases linearly with the
  partition size but is always greater than one

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Run Time
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Contents

• Motivation
• Mathematical Background
• Core Peeling
• Core Cutting
• Discussion
• Future Work

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Future Work

• Spectral Gap
• Unicyclical Components
• Improve time complexity
• Compare solutions found with other algorithms
• Theoretical Analysis

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The End

• Any Questions?

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The End

• Any Questions?

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The End

• Any Questions?

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Dangling Unicyclical Components
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Expander

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Spectral Gap