The FloydWarshall Algorithm

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The Floyd-Warshall Algorithm

• Or There and Back Again, a Directed, Weighted
  Graphs Tale
• By Dale Earnest
• MET CS566

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The Floyd-Warshall Algorithm

• Graph analysis algorithm for finding shortest
  paths in a weighted, directed graph
• Graph can have negative weights, but not negative
  weight cycles
• Example of dynamic programming, a method of
  solving problems where one needs to find the best
  decision one after another

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Designers

• Robert Floyd
• Eminent computer scientist
• Appointed associate Prof at Carnegie Mellon at
  27, full professor at Stanford at 33, never
  earned a PhD
• Worked closely with Knuth and was a major
  reviewer of the Art of Programming
• Earned Turing award in 1978
• Stephen Warshall
• Founder of Applied Data Research, now part of
  Computer Associates

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Initialization

• Find all the minimal distances for pairs without
  using intermediate vertices
• Load a matrix with the values

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Pseudo code

• M matrix
• n number of nodes
• k n loops
• i vertical axis (start point)
• j horizontal axis (end point)

for ( k0 k lt n k ) for ( i0 i lt n
i ) for ( j0 j lt n j )
Mij min( Mij, Mik Mkj)

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First Loop

• n 6
• k 0
• i 0
• j 0

• Mij min( Mij, Mik Mkj)
• M00 min( M00, M00 M00 )
• M00 min (0, 0) 0
• So MAA 0

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First Loop, cont

• n 6
• k 0
• i 1
• j 3

• Mij min( Mij, Mik Mkj)
• M13 min( M13, M10 M03 )
• M13 min (999, 5 3) 8
• So MBD 8

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First Loop, final

• n 6
• k 0
• i 1
• j 5

• Mij min( Mij, Mik Mkj)
• M15 min( M15, M10 M05 )
• M15 min (999, 5 2) 7
• So MBF 7

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Matrix Final State (all shortest paths)
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Efficiency

• Three for-loops
• The matrix is examined n n n
• O(n3)

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Drawbacks

• Negative cycles

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Applications

• Shortest path in directed graphs
• Transitive closure of directed graphs
• Regular expressions
• Matrix inversion
• Optimal routing
• Determining if an undirected graph is bipartite