By…By… Sukanta beheraSukanta behera

Reg. No. 07SBSCA048Reg. No. 07SBSCA048

Warshall’s and Floyd’s Warshall’s and Floyd’s algorithmalgorithm

OutlineOutline IntroductionIntroduction Warshall's AlgorithmWarshall's Algorithm Example of Warshall's AlgorithmExample of Warshall's Algorithm Floyd’s algorithmFloyd’s algorithm Example for Floyds AlgorithmExample for Floyds Algorithm ConclusionConclusion

IntroductionIntroduction Dynamic programming is an algorithm design Dynamic programming is an algorithm design

technique.technique. Dynamic programming is a technique for solving Dynamic programming is a technique for solving

problems with overlapping subproblems.problems with overlapping subproblems. Warshall’s and Floyd’s Algorithms are based on Warshall’s and Floyd’s Algorithms are based on

essentially the same idea, which we can interpret as essentially the same idea, which we can interpret as an application of the dynamic programming an application of the dynamic programming technique.technique.

Warshall’s algorithm for computing the transitive Warshall’s algorithm for computing the transitive closure of a directed graph and Floyd’s algorithm for closure of a directed graph and Floyd’s algorithm for the all-pairs shortest-paths problem.the all-pairs shortest-paths problem.

Warshall’s AlgorithmsWarshall’s Algorithms The transitive closure of a direct graph with n The transitive closure of a direct graph with n

vertices can be defined as the n-by-n Boolean vertices can be defined as the n-by-n Boolean matrix T={tmatrix T={tijij}, in which the element in the ith row }, in which the element in the ith row (1≤i≤n) and the jth column (1≤j≤n)is 1(1≤i≤n) and the jth column (1≤j≤n)is 1

If there exists a nontrivial directed path from the If there exists a nontrivial directed path from the ith vertex to the jth vertex; otherwise, tith vertex to the jth vertex; otherwise, t ijij is 0. is 0.

Since this method traverses the same digraph Since this method traverses the same digraph several times, we should hope that a better several times, we should hope that a better algorithm can be found.algorithm can be found.

Such a algorithm is called Warshall’s algorithm.Such a algorithm is called Warshall’s algorithm.

ALGORITHMALGORITHMALGORITHM ALGORITHM Warshall(A[1…n,1……n)Warshall(A[1…n,1……n)//Implements Warshall’s algorithm for computing the //Implements Warshall’s algorithm for computing the

transitive closuretransitive closure//Input: The adjacency matrix A of a diagraph with n vertices//Input: The adjacency matrix A of a diagraph with n vertices//Output: The transitive closure of the digraph//Output: The transitive closure of the digraphR ← A⁽⁰⁾R ← A⁽⁰⁾for k ← 1 to n dofor k ← 1 to n do for i ← 1 to n dofor i ← 1 to n do for j ← 1 to n dofor j ← 1 to n do R(k)[i,j] ← R(k-1)[i,j] or R(k-1)[i,k] and R(k-1)[k,j]R(k)[i,j] ← R(k-1)[i,j] or R(k-1)[i,k] and R(k-1)[k,j]return R ⁿ⁽ ⁾return R ⁿ⁽ ⁾

Example for Warshall’s AlgorithmExample for Warshall’s Algorithma b

c d

0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0

a b c d

1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1

A =

T =

a b c d

Application of Warshall’s Application of Warshall’s algorithmalgorithm

ba

c d

R⁽⁰⁾ =

a b c d 0 1 0 00 0 0 10 0 0 01 0 1 0

0 1 0 00 0 0 10 0 0 01 0 1 0

R⁽¹⁾ =

0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0

a b c d

R⁽²⁾ =

0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0

a b c d

Application of Warshall’s Application of Warshall’s algorithm(con…)algorithm(con…)

0 1 0 1 0 0 0 1 0 0 0 0 1 1 1 1

a b c d

R⁽³⁾ =

1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1

a b c d

R⁽⁴⁾ =

Floyd’s AlgorithmFloyd’s Algorithm To find the distances from each vertex to all other To find the distances from each vertex to all other

vertices is called vertices is called all-pairs shortest paths all-pairs shortest paths problemproblem

To record the length of the shortest paths in an n-To record the length of the shortest paths in an n-by-n matrix D called the by-n matrix D called the distance matrix distance matrix

The element dThe element dijij in the ith row and the jth column of in the ith row and the jth column of this matrix indicates the length of the shortest path this matrix indicates the length of the shortest path from the ith vertex (1≤i,j≤n)from the ith vertex (1≤i,j≤n)

It is called Floyd’s algorithm, after its inventor It is called Floyd’s algorithm, after its inventor R.FloydR.Floyd

Example for Floyd’s algorithmExample for Floyd’s algorithma b

c d1

6

2

3 7

0 ∞ 3 ∞ 2 0 ∞ ∞ ∞ 7 0 1 6 ∞ ∞ 0

a b c d

W=

0 10 3 4 2 0 5 6 7 7 0 1 6 16 9 0

a b c d

D =

ALGORITHMALGORITHMALGORITHM Floyd(W[1….n,1….n])ALGORITHM Floyd(W[1….n,1….n])//Implements Floyd’s algorithm for the all-pairs shortest-//Implements Floyd’s algorithm for the all-pairs shortest-

paths problempaths problem//Input : The weight matrix W of a graph//Input : The weight matrix W of a graph//Output : The distance matrix of the shortest paths //Output : The distance matrix of the shortest paths

lengthslengthsD ← W //is not necessary if W can be D ← W //is not necessary if W can be

overwrittenoverwrittenfor k ← 1 to n dofor k ← 1 to n do for i ← 1 to n dofor i ← 1 to n do for j ← 1 to n dofor j ← 1 to n do D[i,j] ← min(D[i,j]+D[k,j])D[i,j] ← min(D[i,j]+D[k,j])return Dreturn D

Application of Floyd’s algorithmApplication of Floyd’s algorithma b

c d

2

3

1

6 70 ∞ 3 ∞2 0 ∞ ∞∞ 7 0 16 ∞ ∞ 0

a b c d

D⁽⁰⁾ =

0 ∞ 3 ∞20 5 ∞∞ 7 0 16 ∞ 9 0

a b c d

D⁽¹⁾ =

0 ∞ 3 ∞ 2 0 5 ∞ 9 7 0 1 6 ∞ 9 0

a b c d

D⁽²⁾ =

Application of Floyd’s Application of Floyd’s algorithm(cont…)algorithm(cont…)

0 10 3 4 2 0 5 6 9 7 0 1 6 16 9 0

D⁽³⁾ =

a b c d

0 10 3 4 2 0 5 4 7 7 0 1 6 16 9 0

a b c d

D⁽⁴⁾ =

ConclusionConclusion It deals with general principle that gives It deals with general principle that gives

dynamic programming algorithms for dynamic programming algorithms for optimization problems.optimization problems.

Richard Bellman called it the principle of Richard Bellman called it the principle of optimalityoptimality

The optimal solution to any instance of The optimal solution to any instance of optimization problem is composed of optimal optimization problem is composed of optimal solution to its sub instances.solution to its sub instances.