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<!DOCTYPE html>

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<title>Floyd Warshall Algorithm</title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

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<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

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<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

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<h1>Floyd-Warshall Algorithm</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn how floyd-warshall algorithm works. Also, you will find working examples of floyd-warshall algorithm in C, C++, Java and Python.</p>

<p id="definition">Floyd-Warshall Algorithm is an algorithm for finding the shortest path between all the pairs of vertices in a weighted graph. This algorithm works for both the directed and undirected weighted graphs. But, it does not work for the graphs with negative cycles (where the sum of the edges in a cycle is negative).</p>

<p>A weighted graph is a graph in which each edge has a numerical value associated with it.</p>

</div>

<p>Floyd-Warhshall algorithm is also called as Floyd's algorithm, Roy-Floyd algorithm, Roy-Warshall algorithm, or WFI algorithm.</p>

<p>This algorithm follows the <a href="/dsa/dynamic-programming.html">dynamic programming</a> approach to find the shortest paths.</p>

<hr><h2 id="working">How Floyd-Warshall Algorithm Works?</h2>

<p>Let the given graph be:</p>

<figure><img alt="graph" src="//cdn.programiz.com/sites/tutorial2program/files/fw-Graph.png" title="initial graph" width="300" height="268"><figcaption><span id="docs-internal-guid-4e2d692b-7fff-15f4-ff34-1a76da71f46f" style="font-size:11pt;font-family:Arial;color:#000000;background-color:transparent;font-weight:400;font-style:normal;font-variant:normal;text-decoration:none;vertical-align:baseline;white-space:pre;white-space:pre-wrap;">Initial graph</span></figcaption></figure><p></p><div class="clearfix"></div><p>Follow the steps below to find the shortest path between all the pairs of vertices.</p>

<ol><li>Create a matrix <code>A<sup>0</sup></code> of dimension <code>n*n</code> where n is the number of vertices. The row and the column are indexed as <var>i</var> and <var>j</var> respectively. <var>i</var> and <var>j</var> are the vertices of the graph.<br><br>

Each cell <var>A[i][j]</var> is filled with the distance from the <code>i<sup>th</sup></code> vertex to the <code>j<sup>th</sup></code> vertex. If there is no path from <code>i<sup>th</sup></code> vertex to <code>j<sup>th</sup></code> vertex, the cell is left as infinity.

<figure><img alt="matrix floyd warshall algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/fw-Matrix-1.png" title="matrix" width="250" height="186"><figcaption>Fill each cell with the distance between ith and jth vertex</figcaption></figure></li>

<li>Now, create a matrix <code>A<sup>1</sup></code> using matrix <code>A<sup>0</sup></code>. The elements in the first column and the first row are left as they are. The remaining cells are filled in the following way.<br><br>

Let <var>k</var> be the intermediate vertex in the shortest path from source to destination. In this step, <var>k</var> is the first vertex. <code>A[i][j]</code> is filled with <code>(A[i][k] + A[k][j]) if (A[i][j] > A[i][k] + A[k][j])</code>.<br><br>

That is, if the direct distance from the source to the destination is greater than the path through the vertex <var>k</var>, then the cell is filled with <code>A[i][k] + A[k][j]</code>.<br><br>

In this step, k is vertex 1. We calculate the distance from source vertex to destination vertex through this vertex k.

<figure><img alt="matrix floyd warshall algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/fw-Matrix-2.png" title="matrix A1" width="500" height="197"><figcaption>Calculate the distance from the source vertex to destination vertex through this vertex k</figcaption></figure><br>

For example: For <code>A<sup>1</sup>[2, 4]</code>, the direct distance from vertex 2 to 4 is 4 and the sum of the distance from vertex 2 to 4 through vertex (ie. from vertex 2 to 1 and from vertex 1 to 4) is 7. Since <code>4 < 7</code>, <code>A<sup>0</sup>[2, 4]</code> is filled with 4.</li>

<li>Similarly, <code>A<sup>2</sup></code> is created using <code>A<sup>1</sup></code>. The elements in the second column and the second row are left as they are.<br><br>

In this step, <var>k</var> is the second vertex (i.e. vertex 2). The remaining steps are the same as in <strong>step 2</strong>.

<figure><img alt="matrix floyd warshall algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/fw-Matrix-3.png" title="matrix A2" width="500" height="197"><figcaption>Calculate the distance from the source vertex to destination vertex through this vertex 2</figcaption></figure></li>

<li>Similarly, <code>A<sup>3</sup></code> and <code>A<sup>4</sup></code> is also created.

<figure><img alt="matrix floyd warshall algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/fw-Matrix-4.png" title="matrix A3" width="500" height="197"><figcaption>Calculate the distance from the source vertex to destination vertex through this vertex 3</figcaption></figure>

<figure><img alt="matrix floyd warshall algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/fw-Matrix-5.png" title="matrix A4" width="500" height="197"><figcaption>Calculate the distance from the source vertex to destination vertex through this vertex 4</figcaption></figure></li>

<li><code>A<sup>4</sup></code> gives the shortest path between each pair of vertices.</li>

</ol><hr><h2 id="algortihm">Floyd-Warshall Algorithm</h2>

<pre>n = no of vertices

A = matrix of dimension n*n

for k = 1 to n

for i = 1 to n

for j = 1 to n

A<sup>k</sup>[i, j] = min (A<sup>k-1</sup>[i, j], A<sup>k-1</sup>[i, k] + A<sup>k-1</sup>[k, j])

return A</pre>

<hr><h2 id="code">Python, Java and C/C++ Examples</h2>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"><span class="hljs-comment"># Floyd Warshall Algorithm in python</span>

<span class="hljs-comment"># The number of vertices</span>

nV = <span class="hljs-number">4</span>

INF = <span class="hljs-number">999</span>

<span class="hljs-comment"># Algorithm implementation</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">floyd_warshall</span><span class="hljs-params">(G)</span>:</span>

distance = list(map(<span class="hljs-keyword">lambda</span> i: list(map(<span class="hljs-keyword">lambda</span> j: j, i)), G))

<span class="hljs-comment"># Adding vertices individually</span>

<span class="hljs-keyword">for</span> k <span class="hljs-keyword">in</span> range(nV):

<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(nV):

<span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> range(nV):

distance[i][j] = min(distance[i][j], distance[i][k] + distance[k][j])

print_solution(distance)

<span class="hljs-comment"># Printing the solution</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">print_solution</span><span class="hljs-params">(distance)</span>:</span>

<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(nV):

<span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> range(nV):

<span class="hljs-keyword">if</span>(distance[i][j] == INF):

<span class="hljs-keyword">print</span>(<span class="hljs-string">"INF"</span>, end=<span class="hljs-string">" "</span>)

<span class="hljs-keyword">else</span>:

<span class="hljs-keyword">print</span>(distance[i][j], end=<span class="hljs-string">" "</span>)

<span class="hljs-keyword">print</span>(<span class="hljs-string">" "</span>)

G = [[<span class="hljs-number">0</span>, <span class="hljs-number">3</span>, INF, <span class="hljs-number">5</span>],

[<span class="hljs-number">2</span>, <span class="hljs-number">0</span>, INF, <span class="hljs-number">4</span>],

[INF, <span class="hljs-number">1</span>, <span class="hljs-number">0</span>, INF],

[INF, INF, <span class="hljs-number">2</span>, <span class="hljs-number">0</span>]]

floyd_warshall(G)</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs"><span class="hljs-comment">// Floyd Warshall Algorithm in Java</span>

<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">FloydWarshall</span> </span>{

<span class="hljs-keyword">final</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">int</span> INF = <span class="hljs-number">9999</span>, nV = <span class="hljs-number">4</span>;

<span class="hljs-comment">// Implementing floyd warshall algorithm</span>

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">floydWarshall</span><span class="hljs-params">(<span class="hljs-keyword">int</span> graph[][])</span> </span>{

<span class="hljs-keyword">int</span> matrix[][] = <span class="hljs-keyword">new</span> <span class="hljs-keyword">int</span>[nV][nV];

<span class="hljs-keyword">int</span> i, j, k;

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < nV; i++)

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < nV; j++)

matrix[i][j] = graph[i][j];

<span class="hljs-comment">// Adding vertices individually</span>

<span class="hljs-keyword">for</span> (k = <span class="hljs-number">0</span>; k < nV; k++) {

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < nV; i++) {

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < nV; j++) {

<span class="hljs-keyword">if</span> (matrix[i][k] + matrix[k][j] < matrix[i][j])

matrix[i][j] = matrix[i][k] + matrix[k][j];

}

printMatrix(matrix);

}

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printMatrix</span><span class="hljs-params">(<span class="hljs-keyword">int</span> matrix[][])</span> </span>{

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < nV; ++i) {

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < nV; ++j) {

<span class="hljs-keyword">if</span> (matrix[i][j] == INF)

System.out.print(<span class="hljs-string">"INF "</span>);

System.out.print(matrix[i][j] + <span class="hljs-string">" "</span>);

}

System.out.println();

}

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">void</span> <span class="hljs-title">main</span><span class="hljs-params">(String[] args)</span> </span>{

<span class="hljs-keyword">int</span> graph[][] = { { <span class="hljs-number">0</span>, <span class="hljs-number">3</span>, INF, <span class="hljs-number">5</span> }, { <span class="hljs-number">2</span>, <span class="hljs-number">0</span>, INF, <span class="hljs-number">4</span> }, { INF, <span class="hljs-number">1</span>, <span class="hljs-number">0</span>, INF }, { INF, INF, <span class="hljs-number">2</span>, <span class="hljs-number">0</span> } };

FloydWarshall a = <span class="hljs-keyword">new</span> FloydWarshall();

a.floydWarshall(graph);

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp"><span class="hljs-comment">// Floyd-Warshall Algorithm in C</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdio.h></span></span>

<span class="hljs-comment">// defining the number of vertices</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> nV 4</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> INF 999</span>

<span class="hljs-comment">// Implementing floyd warshall algorithm</span>

<span class="hljs-keyword">int</span> matrix[nV][nV], i, j, k;

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < nV; i++)

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < nV; j++)

matrix[i][j] = graph[i][j];

<span class="hljs-comment">// Adding vertices individually</span>

<span class="hljs-keyword">for</span> (k = <span class="hljs-number">0</span>; k < nV; k++) {

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < nV; i++) {

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < nV; j++) {

<span class="hljs-keyword">if</span> (matrix[i][k] + matrix[k][j] < matrix[i][j])

matrix[i][j] = matrix[i][k] + matrix[k][j];

}

printMatrix(matrix);

}

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < nV; i++) {

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < nV; j++) {

<span class="hljs-keyword">if</span> (matrix[i][j] == INF)

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%4s"</span>, <span class="hljs-string">"INF"</span>);

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%4d"</span>, matrix[i][j]);

}

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"\n"</span>);

}

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-keyword">int</span> graph[nV][nV] = {{<span class="hljs-number">0</span>, <span class="hljs-number">3</span>, INF, <span class="hljs-number">5</span>},

{<span class="hljs-number">2</span>, <span class="hljs-number">0</span>, INF, <span class="hljs-number">4</span>},

{INF, <span class="hljs-number">1</span>, <span class="hljs-number">0</span>, INF},

{INF, INF, <span class="hljs-number">2</span>, <span class="hljs-number">0</span>}};

floydWarshall(graph);

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs"><span class="hljs-comment">// Floyd-Warshall Algorithm in C++</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><iostream></span></span>

<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> <span class="hljs-built_in">std</span>;

<span class="hljs-comment">// defining the number of vertices</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> nV 4</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> INF 999</span>

<span class="hljs-comment">// Implementing floyd warshall algorithm</span>

<span class="hljs-keyword">int</span> matrix[nV][nV], i, j, k;

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < nV; i++)

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < nV; j++)

matrix[i][j] = graph[i][j];

<span class="hljs-comment">// Adding vertices individually</span>

<span class="hljs-keyword">for</span> (k = <span class="hljs-number">0</span>; k < nV; k++) {

<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < nV; i++) {

<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < nV; j++) {

<span class="hljs-keyword">if</span> (matrix[i][k] + matrix[k][j] < matrix[i][j])

matrix[i][j] = matrix[i][k] + matrix[k][j];

}

printMatrix(matrix);

}

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < nV; i++) {

<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < nV; j++) {

<span class="hljs-keyword">if</span> (matrix[i][j] == INF)

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%4s"</span>, <span class="hljs-string">"INF"</span>);

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%4d"</span>, matrix[i][j]);

}

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"\n"</span>);

}

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-keyword">int</span> graph[nV][nV] = {{<span class="hljs-number">0</span>, <span class="hljs-number">3</span>, INF, <span class="hljs-number">5</span>},

{<span class="hljs-number">2</span>, <span class="hljs-number">0</span>, INF, <span class="hljs-number">4</span>},

{INF, <span class="hljs-number">1</span>, <span class="hljs-number">0</span>, INF},

{INF, INF, <span class="hljs-number">2</span>, <span class="hljs-number">0</span>}};

floydWarshall(graph);

}</code></pre></div>

</div>

<hr><h2 id="complexity">Floyd Warshall Algorithm Complexity</h2>

<h3>Time Complexity</h3>

<p>There are three loops. Each loop has constant complexities. So, the time complexity of the Floyd-Warshall algorithm is <code>O(n<sup>3</sup>)</code>.</p>

<h3>Space Complexity</h3>

<p>The space complexity of the Floyd-Warshall algorithm is <code>O(n<sup>2</sup>)</code>.</p>

<hr><h2 id="application">Floyd Warshall Algorithm Applications</h2>

<ul><li>To find the shortest path is a directed graph</li>

<li>To find the transitive closure of directed graphs</li>

<li>To find the Inversion of real matrices</li>

<li>For testing whether an undirected graph is bipartite</li>

</ul></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#working">How Floyd-Warshall Algorithm Works?</a></li>

<li><a href="#algortihm">Floyd-Warshall Algorithm</a></li>

<li><a href="#code">Python, Java and C/C++ Examples</a></li>

<li><a href="#complexity">Floyd Warshall Algorithm Complexity</a></li>

<li><a href="#application">Floyd Warshall Algorithm Applications</a></li>

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