An mn GomoryHu tree Construction Algorithm for Unweighted Graphs

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An Õ(mn) Gomory-Hu tree Construction Algorithm
for Unweighted Graphs

• Debmalya Panigrahi
• Bell Laboratories, India
• June 2007

(with Anand Bhalgat, Ramesh Hariharan
Telikepalli Kavitha)
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Gomory-Hu tree/Cut tree

• Weighted tree T on vertices of (un)weighted
  undirected graph G capturing pair-wise
  connectivity
• Lightest edge euv on u-v path in T
• wt(euv) in T f(u, v), the u-v maxflow in G
• euv defines a u-v mincut

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Cut tree construction algorithm (1 of 3)

• Max-flow based Gomory Hu 61
• At any stage, maintain a partial tree T on a
  partition of vertices
• In each iteration
• Choose vertices u, v in the same set S of the
  partition in G, contract each neighbor sub-tree
  (in T) of S
• Find the max-flow f(u, v)
• Modify T according to the (u, v)-connectivity cut
  by splitting S and connect them with an edge of
  weight f(u, v)

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Cut tree construction algorithm (2 of 3)
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Cut tree construction algorithm (3 of 3)

• Time complexity dominated by n-1 max-flows
• Time complexity has improved as a side-effect of
  improved algorithms for max-flow
• Current best (n-1) Õ(n2.2) Õ(n3.2)
  randomized complexity Karger-Levine 97

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Steiner min-cut based approach (1 of 2)

• Steiner min-cut smallest cut splitting a subset
  S of vertices
• In cut-tree algorithm, replace max-flow by
  Steiner min-cut of a set S in the partition

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Steiner min-cut based approach (2 of 2)

• Correctness
• Special case of max-flow based algo
• Complexity
• n-1 Steiner min-cut computations
• Fastest Steiner min-cut algo Õ(mk2)
• (k Steiner min-cut) Cole-Hariharan, 03
• Cut tree complexity Õ(n5) (for simple graphs)
• Recall that we are aiming to better Õ(n3.2)!!

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Improvements

• Decrease Steiner min-cut complexity from Õ(mk2)
  to Õ(mk)
• Amortize calls to the Steiner min-cut sub-routine
  to show that n-1 calls have a cost of just log n
  calls!
• Õ(mn) algo for cut tree construction

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Cole-Hariharan Steiner min-cut algorithm

• Generalizes Gabows round-robin algorithm for
  finding global connectivity
• Trees no longer span all vertices
• white vertices (appear in all trees) and black
  super-vertices (hierarchical structure for
  vertices not appearing in all trees)
• Identify maximal black super-vertices and balance
  degree of each occurrence to 2
• Apply round-robin on trees of white vertices
  with paths of compressed black super-vertices
  connecting white vertices

O(im)
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Our contribution

• Use the round-robin process itself to identify
  black super-vertices- Õ(m) rather than Õ(im)
• Challenge To continue round-robin with
  non-maximal black super-vertices, their degrees
  have to be balanced
• Solution Use a local degree balancing
  sub-routine also has Õ(m) complexity
• Once maximal black super-vertices have been
  found, use global degree balancing again, Õ(m)
• Overall complexity Õ(mk)

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Amortization (1 of 3)

• Target Amortize n-1 calls to the Steiner min-cut
  sub-routines to get cost of log n calls
• Computation tree

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Amortization (2 of 3)

• The side of the Steiner cut not containing the
  root is a black super-vertex
• These trees can be re-used in one
• of the 2 sub-problems spawned
• Sub-problem containing root comes
• for free!

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Amortization (3 of 3)

• (Suppose) S1 gt S2 and no repeated edge
• Each edge pays only when it goes into the smaller
  sub-problem at most log n times
• Edges repeated in both sub-problems?
• Lemma Total edges lt 3m
• S1, S2 unknown when choosing root
• Choose root uniformly at random
• S1 S2 doesnt matter where root is
• S1 gtgt S2 root likely to be in S1

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Open Problems

• Derandomize?
• Improve Steiner min-cut further through
  randomization?
• Extend to the weighted case?

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Thank You