1 Chandler 18.304lecture1

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Floyd-Warshall Algorithm

Chandler Burfield

February 20, 2013

Chandler Burfield Floyd-Warshall February 20, 2013 1 / 15


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All-Pairs Shortest Paths

Problem

To find the shortest pathbetween all vertices vVfor a weighted graphG = (V,E).



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Dynamic-Programming

How to Develop a Dynamic-Programming Algorithm

1. Characterize the structure of an optimal solution.

2. Recursively define the value of an optimal solution.

3. Compute the value of an optimal solution in a bottom-up manner.



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Weight Matrix

wij

The weight of an edge between vertex iand vertex j in graph G = (V,E),where

wij=

0 if i=j

the weight of a directed edge (i,j) if i=jand (i,j) E if i=jand (i,j) /E

W

An n x n matrix representing the edge weights of an n-vertex graph,where W = (wij).



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Path Matrix

d(k)ij

The weight of the shortest path from vertex i to vertex jfor which allintermediate vertices are in the set {1, 2, ...,k}.

D(k)

An n x n matrix representing the path distances between vertices in a

directed n-vertex graph, where D(k) = (d(k)ij ).



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Observations

A shortest path does not contain the same vertex more than once.

For a shortest path from i to jsuch that any intermediate vertices onthe path are chosen from the set {1, 2, ..., k}, there are twopossibilities:

1. k is not a vertex on the path,so the shortest such path has length dk1ij

2. k is a vertex on the path,so the shortest such path is dk1ik +d

k1kj

So we see that we can recursively define d(k)ij as

d(k)ij =

wij ifk= 0

min(d(k1)ij , d

(k1)ik +d

(k1)kj ) ifk1



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Predecessor Matrix

(k)

ijThe predecessor of vertex jon a shortest path from vertex iwith allintermediate vertices in the set {1, 2, ...,k}. Where

(0)

ij

= NIL if i=j orwij=i if i=j and wijd

(k1)ik +d

(k1)kj

(k)

The predecessor matrix, where (k) = ((k)ij ).



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Algorithm

Floyd-Warshall(W)n=W.rowsD(0) =W

fork

= 1 ton

let D(k) = (d(k)ij ) be a new matrixfor i= 1 to n

for j= 1 to n

d(k)ij = min(d

(k1)ij , d

(k1)ik +d

(k1)kj )

return D(n)



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Analysis of Algorithm

Floyd-Warshall(W)n=W.rowsD(0) =Wfork= 1 to n

let D(k) = (d(k)ij ) be a new matrixfor i= 1 to n

for j= 1 to n

d(k)ij = min(d

(k1)ij , d

(k1)ik +d

(k1)kj )

return D(n)

Runtime

(n3)

Space

(n3) here,but if we reuse spacethis can be done in(n2).



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Analysis of Improved Algorithm

Floyd-Warshall(W)n=W.rowsD=W initialization

fork= 1 tonfor i= 1 to n

for j= 1 to nifdij>dik+dkjthen dij=dik+dkj

ij=kjreturn D

Analysis

The shortest pathcan be constructed,not just the lengthsof the paths.

Runtime: (n3).

Space: (n2).



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Example

Floyd-Warshall(W)n=W.rowsD=W initialization

fork= 1 to nfor i= 1 to n

for j= 1 to nifdij>dik+dkjthen dij=dik+dkj

ij=kjreturn D



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Proof of Correctness

Inductive Hypothesis

Suppose that prior to the kth iteration it holds that for i,jV, dijcontains the length of the shortest path Q from i to j in Gcontaining onlyvertices in the set {1, 2, ..., k 1}, and ijcontains the immediatepredecessor of jon path Q.



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Applications

Detecting the Presence of a Negative Cycle

Transitive Closure of a Directed Graph



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Other All-Pairs Shortest Paths Algorithms

Dynamic Programming Approach Based on Matrix Multiplication

Johnsons Algorithm for Sparse Graphs


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