CSE 780: Design and Analysis of Algorithms

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CSE 780 Design and Analysis of Algorithms

• Lecture 20
• All-pairs SPs
• DP algorithm
• Floyd-Warshall alg.

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All-pairs Shortest Paths

• Use Bellman-Ford V times
• O ( E )
• Non-negative weights
• Use Dijkstras alg. V times
• O( log V VE)
• Can we do better for general graphs
• i.e., weighted directed graph with possible
  negative weights

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Terminologies

• G (V, E) where V are numbered 1, 2, .. nV
• Input , n ? n matrix
• Output , n ? n matrix

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Dynamic Programming Approach

• Optimal substructure property holds
• Define subproblem
• minimum weight of any path from vertex i
  to j that contains at most m edges.

• Recursive solution
• Base case m 0

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Recursive Solution cont.

• Goal

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Build DP-table

• Compute in
  order

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Pseudo-code cont.
Can we do better ?

• Time complexity

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Matrix Multiplication

• Extend procedure
• same as matrix multiplication (w/ a twist) !
• L L W !

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Faster Algorithm
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Floyd-Warshall Algorithm

• Yet another DP algorithm
• Why was previous algorithm slow?
• In the recursive solution, needs to spend O(n)
  time for each new entry
• Floyd-Warshall algorithm
• Use a different recursive solution O(1) time to
  compute each entry

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Problem Definition

• Subproblem definition
• the weight of shortest path from
  vertex i to j where all intermediate vertices are
  from set 1, 2, , k.

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Pseudo-code