Parallel construction of Euler Cycle

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Parallel construction of Euler Cycle
An Euler cycle in a graph is a cyclic path which
traverses each edge of the graph exactly once

• Lecture 13

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Euler cycle in a directed graph

• Given graph G
• Find an Euler cycle of G
• (1) Creation of a set of edge disjoint cycles
• (2) Creation of a special biconncetd graph B
  between cycles and nodes of G
• (3) Parallel construction of a spanning tree T
  for the graph B
• (4) Construction an Euler tour C of a tree T by
  replacing each edge of T by two directed
  antiparallel edges
• (5) Converting the cycle C into an Euler cycle
  of G

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Graph G
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Step 1 Creation of a set of edge disjoint cycles
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Step 2 Creating an auxiliary graph G, nodes of
G are edge-disjoint cycles of G, and vertices of
G. Each cycle is connected to vertices which lie
on it
For our example A PART of the graph G looks as
follows
Step 3 Parallel construction of a spanning tree
T for the graph B
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Step 4 Construction an Euler tour C of a tree T
by replacing each edge of T by two directed
antiparallel edges
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Step 5 Converting the cycle C into an Euler
cycle of G
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Euler paths on arbitrary graphs
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• Euler paths on arbitrary graphs
• Complexity O(log(n))
• Sort using p processors O(log n)
• with p gt n

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Euler paths on arbitrary graphs
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Euler paths on arbitrary graphs
// input n num-nodes, m num edges // output
D each edge labeled with representative // of
each cycle // note succin(v,k)
out(k,v) par_for x 1 ... m Dex
ex // every edge is a potential
cycle-representative nextex
succex // scratch space for contraction for k
1 ... ceil(log(n)) par_for x 1 ... m
Dex min(Dex, Dnextex)
nextex nextnextex
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