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<title>Bellman Ford Algorithm</title>
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<h1>Bellman Ford's Algorithm</h1>
<p class="editor-contents__short-description">Bellman Ford algorithm helps us find the shortest path from a vertex to all other vertices of a weighted graph.</p>
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<span property="dc:title" content="Bellman Ford's Algorithm" class="rdf-meta element-hidden"></span>
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<p id="introduction">It is similar to <a href="/dsa/dijkstra-algorithm.html">Dijkstra's algorithm</a> but it can work with graphs in which edges can have negative weights.</p>
<hr><h2 id="why">Why would one ever have edges with negative weights in real life?</h2>
<p>Negative weight edges might seem useless at first but they can explain a lot of phenomena like cashflow, the heat released/absorbed in a chemical reaction, etc.</p>
<p>For instance, if there are different ways to reach from one chemical A to another chemical B, each method will have sub-reactions involving both heat dissipation and absorption.</p>
<p>If we want to find the set of reactions where minimum energy is required, then we will need to be able to factor in the heat absorption as negative weights and heat dissipation as positive weights.</p>
<hr><h2 id="negative-weights">Why do we need to be careful with negative weights?</h2>
<p>Negative weight edges can create negative weight cycles i.e. a cycle that will reduce the total path distance by coming back to the same point.</p>
<figure><img alt="negative weight cycles can give an incorrect result when trying to find out the shortest path" src="//cdn.programiz.com/sites/tutorial2program/files/negative-weight-cycle_1.png" title="Negative Weights" width="730" height="223"><figcaption>Negative weight cycles can give an incorrect result when trying to find out the shortest path</figcaption></figure><p>Shortest path algorithms like Dijkstra's Algorithm that aren't able to detect such a cycle can give an incorrect result because they can go through a negative weight cycle and reduce the path length.</p>
<hr><h2 id="working">How Bellman Ford's algorithm works</h2>
<p>Bellman Ford algorithm works by overestimating the length of the path from the starting vertex to all other vertices. Then it iteratively relaxes those estimates by finding new paths that are shorter than the previously overestimated paths.</p>
<p></p><div class="clearfix"></div><p>By doing this repeatedly for all vertices, we can guarantee that the result is optimized.</p>
<figure><img alt="steps for bellman ford algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/Bellman-Ford-Algorithm-1.png" title="Bellman Ford's algorithm steps" width="730" height="334"><figcaption>Step-1 for Bellman Ford's algorithm</figcaption></figure><figure><img alt="steps for bellman ford algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/Bellman-Ford-Algorithm-2.png" title="Bellman Ford's algorithm steps" width="730" height="364"><figcaption>Step-2 for Bellman Ford's algorithm</figcaption></figure><figure><img alt="steps for bellman ford algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/Bellman-Ford-Algorithm-3.png" title="Bellman Ford's algorithm steps" width="730" height="364"><figcaption>Step-3 for Bellman Ford's algorithm</figcaption></figure><figure><img alt="steps for bellman ford algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/Bellman-Ford-Algorithm-4.png" title="Bellman Ford's algorithm steps" width="730" height="364"><figcaption>Step-4 for Bellman Ford's algorithm</figcaption></figure><figure><img alt="steps for bellman ford algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/Bellman-Ford-Algorithm-5.png" title="Bellman Ford's algorithm steps" width="730" height="364"><figcaption>Step-5 for Bellman Ford's algorithm</figcaption></figure><figure><img alt="steps for bellman ford algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/Bellman-Ford-Algorithm-6.png" title="Bellman Ford's algorithm steps" width="730" height="464"><figcaption>Step-6 for Bellman Ford's algorithm</figcaption></figure><hr><h2 id="pseudocode">Bellman Ford Pseudocode</h2>
<p>We need to maintain the path distance of every vertex. We can store that in an array of size v, where v is the number of vertices.</p>
<p>We also want to be able to get the shortest path, not only know the length of the shortest path. For this, we map each vertex to the vertex that last updated its path length.</p>
<p>Once the algorithm is over, we can backtrack from the destination vertex to the source vertex to find the path.</p>
<pre>function bellmanFord(G, S)
for each vertex V in G
distance[V] <- infinite
previous[V] <- NULL
distance[S] <- 0
for each vertex V in G
for each edge (U,V) in G
tempDistance <- distance[U] + edge_weight(U, V)
if tempDistance < distance[V]
distance[V] <- tempDistance
previous[V] <- U
for each edge (U,V) in G
If distance[U] + edge_weight(U, V) < distance[V}
Error: Negative Cycle Exists
return distance[], previous[]</pre>
<hr><h2 id="vs">Bellman Ford vs Dijkstra</h2>
<p>Bellman Ford's algorithm and Dijkstra's algorithm are very similar in structure. While Dijkstra looks only to the immediate neighbors of a vertex, Bellman goes through each edge in every iteration.</p>
<figure><img alt="Dijkstra's vs Bellman Ford's Algorithm" src="//cdn.programiz.com/sites/tutorial2program/files/bellman-ford-vs-dijkstra.jpg" title="Dijkstra's vs Bellman Ford's Algorithm" width="1200" height="444"><figcaption>Bellman Ford's Algorithm vs Dijkstra's Algorithm</figcaption></figure><hr><h2 id="code">Python, Java and C/C++ Examples</h2>
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<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"># Bellman Ford Algorithm in Python</span>
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Graph</span>:</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">__init__</span><span class="hljs-params">(self, vertices)</span>:</span>
self.V = vertices <span class="hljs-comment"># Total number of vertices in the graph</span>
self.graph = [] <span class="hljs-comment"># Array of edges</span>
<span class="hljs-comment"># Add edges</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">add_edge</span><span class="hljs-params">(self, s, d, w)</span>:</span>
self.graph.append([s, d, w])
<span class="hljs-comment"># Print the solution</span>
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">print_solution</span><span class="hljs-params">(self, dist)</span>:</span>
<span class="hljs-keyword">print</span>(<span class="hljs-string">"Vertex Distance from Source"</span>)
<span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> range(self.V):
<span class="hljs-keyword">print</span>(<span class="hljs-string">"{0}\t\t{1}"</span>.format(i, dist[i]))
<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">bellman_ford</span><span class="hljs-params">(self, src)</span>:</span>
<span class="hljs-comment"># Step 1: fill the distance array and predecessor array</span>
dist = [float(<span class="hljs-string">"Inf"</span>)] * self.V
<span class="hljs-comment"># Mark the source vertex</span>
dist[src] = <span class="hljs-number">0</span>
<span class="hljs-comment"># Step 2: relax edges |V| - 1 times</span>
<span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> range(self.V - <span class="hljs-number">1</span>):
<span class="hljs-keyword">for</span> s, d, w <span class="hljs-keyword">in</span> self.graph:
<span class="hljs-keyword">if</span> dist[s] != float(<span class="hljs-string">"Inf"</span>) <span class="hljs-keyword">and</span> dist[s] + w < dist[d]:
dist[d] = dist[s] + w
<span class="hljs-comment"># Step 3: detect negative cycle</span>
<span class="hljs-comment"># if value changes then we have a negative cycle in the graph</span>
<span class="hljs-comment"># and we cannot find the shortest distances</span>
<span class="hljs-keyword">for</span> s, d, w <span class="hljs-keyword">in</span> self.graph:
<span class="hljs-keyword">if</span> dist[s] != float(<span class="hljs-string">"Inf"</span>) <span class="hljs-keyword">and</span> dist[s] + w < dist[d]:
<span class="hljs-keyword">print</span>(<span class="hljs-string">"Graph contains negative weight cycle"</span>)
<span class="hljs-keyword">return</span>
<span class="hljs-comment"># No negative weight cycle found!</span>
<span class="hljs-comment"># Print the distance and predecessor array</span>
self.print_solution(dist)
g = Graph(<span class="hljs-number">5</span>)
g.add_edge(<span class="hljs-number">0</span>, <span class="hljs-number">1</span>, <span class="hljs-number">5</span>)
g.add_edge(<span class="hljs-number">0</span>, <span class="hljs-number">2</span>, <span class="hljs-number">4</span>)
g.add_edge(<span class="hljs-number">1</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>)
g.add_edge(<span class="hljs-number">2</span>, <span class="hljs-number">1</span>, <span class="hljs-number">6</span>)
g.add_edge(<span class="hljs-number">3</span>, <span class="hljs-number">2</span>, <span class="hljs-number">2</span>)
g.bellman_ford(<span class="hljs-number">0</span>)</code></pre></div>
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<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">// Bellman Ford Algorithm in Java</span>
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">CreateGraph</span> </span>{
<span class="hljs-comment">// CreateGraph - it consists of edges</span>
<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">CreateEdge</span> </span>{
<span class="hljs-keyword">int</span> s, d, w;
CreateEdge() {
s = d = w = <span class="hljs-number">0</span>;
}
};
<span class="hljs-keyword">int</span> V, E;
CreateEdge edge[];
<span class="hljs-comment">// Creates a graph with V vertices and E edges</span>
CreateGraph(<span class="hljs-keyword">int</span> v, <span class="hljs-keyword">int</span> e) {
V = v;
E = e;
edge = <span class="hljs-keyword">new</span> CreateEdge[e];
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < e; ++i)
edge[i] = <span class="hljs-keyword">new</span> CreateEdge();
}
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">BellmanFord</span><span class="hljs-params">(CreateGraph graph, <span class="hljs-keyword">int</span> s)</span> </span>{
<span class="hljs-keyword">int</span> V = graph.V, E = graph.E;
<span class="hljs-keyword">int</span> dist[] = <span class="hljs-keyword">new</span> <span class="hljs-keyword">int</span>[V];
<span class="hljs-comment">// Step 1: fill the distance array and predecessor array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; ++i)
dist[i] = Integer.MAX_VALUE;
<span class="hljs-comment">// Mark the source vertex</span>
dist[s] = <span class="hljs-number">0</span>;
<span class="hljs-comment">// Step 2: relax edges |V| - 1 times</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i < V; ++i) {
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < E; ++j) {
<span class="hljs-comment">// Get the edge data</span>
<span class="hljs-keyword">int</span> u = graph.edge[j].s;
<span class="hljs-keyword">int</span> v = graph.edge[j].d;
<span class="hljs-keyword">int</span> w = graph.edge[j].w;
<span class="hljs-keyword">if</span> (dist[u] != Integer.MAX_VALUE && dist[u] + w < dist[v])
dist[v] = dist[u] + w;
}
}
<span class="hljs-comment">// Step 3: detect negative cycle</span>
<span class="hljs-comment">// if value changes then we have a negative cycle in the graph</span>
<span class="hljs-comment">// and we cannot find the shortest distances</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < E; ++j) {
<span class="hljs-keyword">int</span> u = graph.edge[j].s;
<span class="hljs-keyword">int</span> v = graph.edge[j].d;
<span class="hljs-keyword">int</span> w = graph.edge[j].w;
<span class="hljs-keyword">if</span> (dist[u] != Integer.MAX_VALUE && dist[u] + w < dist[v]) {
System.out.println(<span class="hljs-string">"CreateGraph contains negative w cycle"</span>);
<span class="hljs-keyword">return</span>;
}
}
<span class="hljs-comment">// No negative w cycle found!</span>
<span class="hljs-comment">// Print the distance and predecessor array</span>
printSolution(dist, V);
}
<span class="hljs-comment">// Print the solution</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printSolution</span><span class="hljs-params">(<span class="hljs-keyword">int</span> dist[], <span class="hljs-keyword">int</span> V)</span> </span>{
System.out.println(<span class="hljs-string">"Vertex Distance from Source"</span>);
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; ++i)
System.out.println(i + <span class="hljs-string">"\t\t"</span> + dist[i]);
}
<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> V = <span class="hljs-number">5</span>; <span class="hljs-comment">// Total vertices</span>
<span class="hljs-keyword">int</span> E = <span class="hljs-number">8</span>; <span class="hljs-comment">// Total Edges</span>
CreateGraph graph = <span class="hljs-keyword">new</span> CreateGraph(V, E);
<span class="hljs-comment">// edge 0 --> 1</span>
graph.edge[<span class="hljs-number">0</span>].s = <span class="hljs-number">0</span>;
graph.edge[<span class="hljs-number">0</span>].d = <span class="hljs-number">1</span>;
graph.edge[<span class="hljs-number">0</span>].w = <span class="hljs-number">5</span>;
<span class="hljs-comment">// edge 0 --> 2</span>
graph.edge[<span class="hljs-number">1</span>].s = <span class="hljs-number">0</span>;
graph.edge[<span class="hljs-number">1</span>].d = <span class="hljs-number">2</span>;
graph.edge[<span class="hljs-number">1</span>].w = <span class="hljs-number">4</span>;
<span class="hljs-comment">// edge 1 --> 3</span>
graph.edge[<span class="hljs-number">2</span>].s = <span class="hljs-number">1</span>;
graph.edge[<span class="hljs-number">2</span>].d = <span class="hljs-number">3</span>;
graph.edge[<span class="hljs-number">2</span>].w = <span class="hljs-number">3</span>;
<span class="hljs-comment">// edge 2 --> 1</span>
graph.edge[<span class="hljs-number">3</span>].s = <span class="hljs-number">2</span>;
graph.edge[<span class="hljs-number">3</span>].d = <span class="hljs-number">1</span>;
graph.edge[<span class="hljs-number">3</span>].w = <span class="hljs-number">6</span>;
<span class="hljs-comment">// edge 3 --> 2</span>
graph.edge[<span class="hljs-number">4</span>].s = <span class="hljs-number">3</span>;
graph.edge[<span class="hljs-number">4</span>].d = <span class="hljs-number">2</span>;
graph.edge[<span class="hljs-number">4</span>].w = <span class="hljs-number">2</span>;
graph.BellmanFord(graph, <span class="hljs-number">0</span>); <span class="hljs-comment">// 0 is the source vertex</span>
}
}</code></pre></div>
</div>
<div class="code-editor__area" id="c-code">
<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">// Bellman Ford 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-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdlib.h></span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> INFINITY 99999</span>
<span class="hljs-comment">//struct for the edges of the graph</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Edge</span> {</span>
<span class="hljs-keyword">int</span> u; <span class="hljs-comment">//start vertex of the edge</span>
<span class="hljs-keyword">int</span> v; <span class="hljs-comment">//end vertex of the edge</span>
<span class="hljs-keyword">int</span> w; <span class="hljs-comment">//weight of the edge (u,v)</span>
};
<span class="hljs-comment">//Graph - it consists of edges</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Graph</span> {</span>
<span class="hljs-keyword">int</span> V; <span class="hljs-comment">//total number of vertices in the graph</span>
<span class="hljs-keyword">int</span> E; <span class="hljs-comment">//total number of edges in the graph</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Edge</span> *<span class="hljs-title">edge</span>;</span> <span class="hljs-comment">//array of edges</span>
};
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">bellmanford</span><span class="hljs-params">(struct Graph *g, <span class="hljs-keyword">int</span> source)</span></span>;
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">display</span><span class="hljs-params">(<span class="hljs-keyword">int</span> arr[], <span class="hljs-keyword">int</span> size)</span></span>;
<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">(<span class="hljs-keyword">void</span>)</span> </span>{
<span class="hljs-comment">//create graph</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Graph</span> *<span class="hljs-title">g</span> = (<span class="hljs-title">struct</span> <span class="hljs-title">Graph</span> *)<span class="hljs-title">malloc</span>(<span class="hljs-title">sizeof</span>(<span class="hljs-title">struct</span> <span class="hljs-title">Graph</span>));</span>
g->V = <span class="hljs-number">4</span>; <span class="hljs-comment">//total vertices</span>
g->E = <span class="hljs-number">5</span>; <span class="hljs-comment">//total edges</span>
<span class="hljs-comment">//array of edges for graph</span>
g->edge = (struct Edge *)<span class="hljs-built_in">malloc</span>(g->E * <span class="hljs-keyword">sizeof</span>(struct Edge));
<span class="hljs-comment">//------- adding the edges of the graph</span>
<span class="hljs-comment">/*
edge(u, v)
where u = start vertex of the edge (u,v)
v = end vertex of the edge (u,v)
w is the weight of the edge (u,v)
*/</span>
<span class="hljs-comment">//edge 0 --> 1</span>
g->edge[<span class="hljs-number">0</span>].u = <span class="hljs-number">0</span>;
g->edge[<span class="hljs-number">0</span>].v = <span class="hljs-number">1</span>;
g->edge[<span class="hljs-number">0</span>].w = <span class="hljs-number">5</span>;
<span class="hljs-comment">//edge 0 --> 2</span>
g->edge[<span class="hljs-number">1</span>].u = <span class="hljs-number">0</span>;
g->edge[<span class="hljs-number">1</span>].v = <span class="hljs-number">2</span>;
g->edge[<span class="hljs-number">1</span>].w = <span class="hljs-number">4</span>;
<span class="hljs-comment">//edge 1 --> 3</span>
g->edge[<span class="hljs-number">2</span>].u = <span class="hljs-number">1</span>;
g->edge[<span class="hljs-number">2</span>].v = <span class="hljs-number">3</span>;
g->edge[<span class="hljs-number">2</span>].w = <span class="hljs-number">3</span>;
<span class="hljs-comment">//edge 2 --> 1</span>
g->edge[<span class="hljs-number">3</span>].u = <span class="hljs-number">2</span>;
g->edge[<span class="hljs-number">3</span>].v = <span class="hljs-number">1</span>;
g->edge[<span class="hljs-number">3</span>].w = <span class="hljs-number">6</span>;
<span class="hljs-comment">//edge 3 --> 2</span>
g->edge[<span class="hljs-number">4</span>].u = <span class="hljs-number">3</span>;
g->edge[<span class="hljs-number">4</span>].v = <span class="hljs-number">2</span>;
g->edge[<span class="hljs-number">4</span>].w = <span class="hljs-number">2</span>;
bellmanford(g, <span class="hljs-number">0</span>); <span class="hljs-comment">//0 is the source vertex</span>
<span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
}
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">bellmanford</span><span class="hljs-params">(struct Graph *g, <span class="hljs-keyword">int</span> source)</span> </span>{
<span class="hljs-comment">//variables</span>
<span class="hljs-keyword">int</span> i, j, u, v, w;
<span class="hljs-comment">//total vertex in the graph g</span>
<span class="hljs-keyword">int</span> tV = g->V;
<span class="hljs-comment">//total edge in the graph g</span>
<span class="hljs-keyword">int</span> tE = g->E;
<span class="hljs-comment">//distance array</span>
<span class="hljs-comment">//size equal to the number of vertices of the graph g</span>
<span class="hljs-keyword">int</span> d[tV];
<span class="hljs-comment">//predecessor array</span>
<span class="hljs-comment">//size equal to the number of vertices of the graph g</span>
<span class="hljs-keyword">int</span> p[tV];
<span class="hljs-comment">//step 1: fill the distance array and predecessor array</span>
<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < tV; i++) {
d[i] = INFINITY;
p[i] = <span class="hljs-number">0</span>;
}
<span class="hljs-comment">//mark the source vertex</span>
d[source] = <span class="hljs-number">0</span>;
<span class="hljs-comment">//step 2: relax edges |V| - 1 times</span>
<span class="hljs-keyword">for</span> (i = <span class="hljs-number">1</span>; i <= tV - <span class="hljs-number">1</span>; i++) {
<span class="hljs-keyword">for</span> (j = <span class="hljs-number">0</span>; j < tE; j++) {
<span class="hljs-comment">//get the edge data</span>
u = g->edge[j].u;
v = g->edge[j].v;
w = g->edge[j].w;
<span class="hljs-keyword">if</span> (d[u] != INFINITY && d[v] > d[u] + w) {
d[v] = d[u] + w;
p[v] = u;
}
}
}
<span class="hljs-comment">//step 3: detect negative cycle</span>
<span class="hljs-comment">//if value changes then we have a negative cycle in the graph</span>
<span class="hljs-comment">//and we cannot find the shortest distances</span>
<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < tE; i++) {
u = g->edge[i].u;
v = g->edge[i].v;
w = g->edge[i].w;
<span class="hljs-keyword">if</span> (d[u] != INFINITY && d[v] > d[u] + w) {
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"Negative weight cycle detected!\n"</span>);
<span class="hljs-keyword">return</span>;
}
}
<span class="hljs-comment">//No negative weight cycle found!</span>
<span class="hljs-comment">//print the distance and predecessor array</span>
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"Distance array: "</span>);
display(d, tV);
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"Predecessor array: "</span>);
display(p, tV);
}
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">display</span><span class="hljs-params">(<span class="hljs-keyword">int</span> arr[], <span class="hljs-keyword">int</span> size)</span> </span>{
<span class="hljs-keyword">int</span> i;
<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < size; i++) {
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%d "</span>, arr[i]);
}
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"\n"</span>);
}</code></pre></div>
</div>
<div class="code-editor__area" id="cpp-code">
<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">// Bellman Ford Algorithm in C++</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><bits/stdc++.h></span></span>
<span class="hljs-comment">// Struct for the edges of the graph</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Edge</span> {</span>
<span class="hljs-keyword">int</span> u; <span class="hljs-comment">//start vertex of the edge</span>
<span class="hljs-keyword">int</span> v; <span class="hljs-comment">//end vertex of the edge</span>
<span class="hljs-keyword">int</span> w; <span class="hljs-comment">//w of the edge (u,v)</span>
};
<span class="hljs-comment">// Graph - it consists of edges</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Graph</span> {</span>
<span class="hljs-keyword">int</span> V; <span class="hljs-comment">// Total number of vertices in the graph</span>
<span class="hljs-keyword">int</span> E; <span class="hljs-comment">// Total number of edges in the graph</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Edge</span>* <span class="hljs-title">edge</span>;</span> <span class="hljs-comment">// Array of edges</span>
};
<span class="hljs-comment">// Creates a graph with V vertices and E edges</span>
<span class="hljs-function">struct Graph* <span class="hljs-title">createGraph</span><span class="hljs-params">(<span class="hljs-keyword">int</span> V, <span class="hljs-keyword">int</span> E)</span> </span>{
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Graph</span>* <span class="hljs-title">graph</span> = <span class="hljs-title">new</span> <span class="hljs-title">Graph</span>;</span>
graph->V = V; <span class="hljs-comment">// Total Vertices</span>
graph->E = E; <span class="hljs-comment">// Total edges</span>
<span class="hljs-comment">// Array of edges for graph</span>
graph->edge = <span class="hljs-keyword">new</span> Edge[E];
<span class="hljs-keyword">return</span> graph;
}
<span class="hljs-comment">// Printing the solution</span>
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">printArr</span><span class="hljs-params">(<span class="hljs-keyword">int</span> arr[], <span class="hljs-keyword">int</span> size)</span> </span>{
<span class="hljs-keyword">int</span> i;
<span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i < size; i++) {
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"%d "</span>, arr[i]);
}
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"\n"</span>);
}
<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">BellmanFord</span><span class="hljs-params">(struct Graph* graph, <span class="hljs-keyword">int</span> u)</span> </span>{
<span class="hljs-keyword">int</span> V = graph->V;
<span class="hljs-keyword">int</span> E = graph->E;
<span class="hljs-keyword">int</span> dist[V];
<span class="hljs-comment">// Step 1: fill the distance array and predecessor array</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < V; i++)
dist[i] = INT_MAX;
<span class="hljs-comment">// Mark the source vertex</span>
dist[u] = <span class="hljs-number">0</span>;
<span class="hljs-comment">// Step 2: relax edges |V| - 1 times</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">1</span>; i <= V - <span class="hljs-number">1</span>; i++) {
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> j = <span class="hljs-number">0</span>; j < E; j++) {
<span class="hljs-comment">// Get the edge data</span>
<span class="hljs-keyword">int</span> u = graph->edge[j].u;
<span class="hljs-keyword">int</span> v = graph->edge[j].v;
<span class="hljs-keyword">int</span> w = graph->edge[j].w;
<span class="hljs-keyword">if</span> (dist[u] != INT_MAX && dist[u] + w < dist[v])
dist[v] = dist[u] + w;
}
}
<span class="hljs-comment">// Step 3: detect negative cycle</span>
<span class="hljs-comment">// if value changes then we have a negative cycle in the graph</span>
<span class="hljs-comment">// and we cannot find the shortest distances</span>
<span class="hljs-keyword">for</span> (<span class="hljs-keyword">int</span> i = <span class="hljs-number">0</span>; i < E; i++) {
<span class="hljs-keyword">int</span> u = graph->edge[i].u;
<span class="hljs-keyword">int</span> v = graph->edge[i].v;
<span class="hljs-keyword">int</span> w = graph->edge[i].w;
<span class="hljs-keyword">if</span> (dist[u] != INT_MAX && dist[u] + w < dist[v]) {
<span class="hljs-built_in">printf</span>(<span class="hljs-string">"Graph contains negative w cycle"</span>);
<span class="hljs-keyword">return</span>;
}
}
<span class="hljs-comment">// No negative weight cycle found!</span>
<span class="hljs-comment">// Print the distance and predecessor array</span>
printArr(dist, V);
<span class="hljs-keyword">return</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-comment">// Create a graph</span>
<span class="hljs-keyword">int</span> V = <span class="hljs-number">5</span>; <span class="hljs-comment">// Total vertices</span>
<span class="hljs-keyword">int</span> E = <span class="hljs-number">8</span>; <span class="hljs-comment">// Total edges</span>
<span class="hljs-comment">// Array of edges for graph</span>
<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Graph</span>* <span class="hljs-title">graph</span> = <span class="hljs-title">createGraph</span>(<span class="hljs-title">V</span>, <span class="hljs-title">E</span>);</span>
<span class="hljs-comment">//------- adding the edges of the graph</span>
<span class="hljs-comment">/*
edge(u, v)
where u = start vertex of the edge (u,v)
v = end vertex of the edge (u,v)
w is the weight of the edge (u,v)
*/</span>
<span class="hljs-comment">//edge 0 --> 1</span>
graph->edge[<span class="hljs-number">0</span>].u = <span class="hljs-number">0</span>;
graph->edge[<span class="hljs-number">0</span>].v = <span class="hljs-number">1</span>;
graph->edge[<span class="hljs-number">0</span>].w = <span class="hljs-number">5</span>;
<span class="hljs-comment">//edge 0 --> 2</span>
graph->edge[<span class="hljs-number">1</span>].u = <span class="hljs-number">0</span>;
graph->edge[<span class="hljs-number">1</span>].v = <span class="hljs-number">2</span>;
graph->edge[<span class="hljs-number">1</span>].w = <span class="hljs-number">4</span>;
<span class="hljs-comment">//edge 1 --> 3</span>
graph->edge[<span class="hljs-number">2</span>].u = <span class="hljs-number">1</span>;
graph->edge[<span class="hljs-number">2</span>].v = <span class="hljs-number">3</span>;
graph->edge[<span class="hljs-number">2</span>].w = <span class="hljs-number">3</span>;
<span class="hljs-comment">//edge 2 --> 1</span>
graph->edge[<span class="hljs-number">3</span>].u = <span class="hljs-number">2</span>;
graph->edge[<span class="hljs-number">3</span>].v = <span class="hljs-number">1</span>;
graph->edge[<span class="hljs-number">3</span>].w = <span class="hljs-number">6</span>;
<span class="hljs-comment">//edge 3 --> 2</span>
graph->edge[<span class="hljs-number">4</span>].u = <span class="hljs-number">3</span>;
graph->edge[<span class="hljs-number">4</span>].v = <span class="hljs-number">2</span>;
graph->edge[<span class="hljs-number">4</span>].w = <span class="hljs-number">2</span>;
BellmanFord(graph, <span class="hljs-number">0</span>); <span class="hljs-comment">//0 is the source vertex</span>
<span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
}</code></pre></div>
</div>
</div>
<hr><h2 id="complexity">Bellman Ford's Complexity</h2>
<h3>Time Complexity</h3>
<div class="table-responsive"><table border="0"><tbody><tr><td>Best Case Complexity</td>
<td>O(E)</td>
</tr><tr><td>Average Case Complexity</td>
<td>O(VE)</td>
</tr><tr><td>Worst Case Complexity</td>
<td>O(VE)</td>
</tr></tbody></table></div><h3>Space Complexity</h3>
<p>And, the space complexity is <code>O(V)</code>.</p>
<hr><h2 id="application">Bellman Ford's Algorithm Applications</h2>
<ol><li>For calculating shortest paths in routing algorithms</li>
<li>For finding the shortest path</li>
</ol></div>
</div>
<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents
<button class="btn btn--clear align-items-center">
<svg class="programiz-icon"><use xlink:href="/sites/all/themes/programiz/assets/feather-sprite.svg#x"></use></svg></button></h3><div class="tutorial-toc__links"><ul><li><a href="#introduction">Introduction</a></li>
<li><a href="#why">Why would one ever have edges with negative weights in real life?</a></li>
<li><a href="#negative-weights">Why do we need to be careful with negative weights?</a></li>
<li><a href="#working">How Bellman Ford's algorithm works</a></li>
<li><a href="#pseudocode">Bellman Ford Pseudocode</a></li>
<li><a href="#vs">Bellman Ford vs Dijkstra</a></li>
<li><a href="#code">Bellman Ford's Algorithm Code</a></li>
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Stack</a></button>
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Queue</a></button>
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Types of Queue</a></button>
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Circular Queue</a></button>
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Priority Queue</a></button>
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Deque</a></button>
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Linked List</a></button>
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Linked List Operations</a></button>
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Types of Linked List</a></button>
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Hash Table</a></button>
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Heap Data Structure</a></button>
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Fibonacci Heap</a></button>
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Decrease Key and Delete node from Fibonacci Heap</a></button>
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Tree Data Structure</a></button>
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Tree Traversal</a></button>
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Binary Tree</a></button>
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Full Binary Tree</a></button>
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Perfect Binary Tree</a></button>
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Complete Binary Tree</a></button>
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Balanced Binary Tree</a></button>
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Binary Search Tree</a></button>
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AVL Tree</a></button>
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B Tree</a></button>
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Insertion into B-tree</a></button>
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Deletion from B-tree</a></button>
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B+ Tree</a></button>
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Insertion on a B+ Tree</a></button>
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Deletion from a B+ Tree</a></button>
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Red Black Tree</a></button>
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Insertion in Red Black Tree</a></button>
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Deletion from Red Black Tree</a></button>
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<button style="padding:5px; margin-bottom:2px"><a href="
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Graph Data Structure</a></button>
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Spanning Tree</a></button>
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Strongly Connected Components</a></button>
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Adjacency Matrix</a></button>
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Adjacency List</a></button>
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DFS Algorithm</a></button>
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Breadth-first Search</a></button>
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Bellman Ford's Algorithm</a></button>
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Bubble Sort</a></button>
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Selection Sort</a></button>
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Insertion Sort</a></button>
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Merge Sort</a></button>
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Quick Sort</a></button>
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Counting Sort</a></button>
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Radix Sort</a></button>
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Bucket Sort</a></button>
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Heap Sort</a></button>
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Shell Sort</a></button>
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Linear Search</a></button>
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Binary Search</a></button>
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<button style="padding:5px; margin-bottom:2px"><a href="
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Greedy Algorithm</a></button>
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Ford-Fulkerson Algorithm</a></button>
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Dijkstra's Algorithm</a></button>
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Kruskal's Algorithm</a></button>
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Prim's Algorithm</a></button>
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Huffman Code</a></button>
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