Given a graph and a source vertex in the graph, find shortest paths from source to all vertices in the given graph.

Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree. Like Prim’s MST, we generate a* SPT (shortest path tree)* with given source as root. We maintain two sets, one set contains vertices included in shortest path tree, other set includes vertices not yet included in shortest path tree. At every step of the algorithm, we find a vertex which is in the other set (set of not yet included) and has a minimum distance from the source.

Below are the detailed steps used in Dijkstra’s algorithm to find the shortest path from a single source vertex to all other vertices in the given graph.

Algorithm**1)** Create a set *sptSet* (shortest path tree set) that keeps track of vertices included in shortest path tree, i.e., whose minimum distance from source is calculated and finalized. Initially, this set is empty.**2)** Assign a distance value to all vertices in the input graph. Initialize all distance values as INFINITE. Assign distance value as 0 for the source vertex so that it is picked first.**3)** While *sptSet* doesn’t include all vertices

….**a)** Pick a vertex u which is not there in *sptSet* and has minimum distance value.

….**b)** Include u to *sptSet*.

….**c)** Update distance value of all adjacent vertices of u. To update the distance values, iterate through all adjacent vertices. For every adjacent vertex v, if sum of distance value of u (from source) and weight of edge u-v, is less than the distance value of v, then update the distance value of v.

Let us understand with the following example:

The set *sptSet* is initially empty and distances assigned to vertices are {0, INF, INF, INF, INF, INF, INF, INF} where INF indicates infinite. Now pick the vertex with minimum distance value. The vertex 0 is picked, include it in *sptSet*. So *sptSet *becomes {0}. After including 0 to *sptSet*, update distance values of its adjacent vertices. Adjacent vertices of 0 are 1 and 7. The distance values of 1 and 7 are updated as 4 and 8. Following subgraph shows vertices and their distance values, only the vertices with finite distance values are shown. The vertices included in SPT are shown in green colour.

Pick the vertex with minimum distance value and not already included in SPT (not in sptSET). The vertex 1 is picked and added to sptSet. So sptSet now becomes {0, 1}. Update the distance values of adjacent vertices of 1. The distance value of vertex 2 becomes 12.

Pick the vertex with minimum distance value and not already included in SPT (not in sptSET). Vertex 7 is picked. So sptSet now becomes {0, 1, 7}. Update the distance values of adjacent vertices of 7. The distance value of vertex 6 and 8 becomes finite (15 and 9 respectively).

Pick the vertex with minimum distance value and not already included in SPT (not in sptSET). Vertex 6 is picked. So sptSet now becomes {0, 1, 7, 6}. Update the distance values of adjacent vertices of 6. The distance value of vertex 5 and 8 are updated.

We repeat the above steps until *sptSet * does include all vertices of given graph. Finally, we get the following Shortest Path Tree (SPT).

*How to implement the above algorithm?*

We use a boolean array sptSet[] to represent the set of vertices included in SPT. If a value sptSet[v] is true, then vertex v is included in SPT, otherwise not. Array dist[] is used to store shortest distance values of all vertices.

`// A C++ program for Dijkstra's single source shortest path algorithm. ` `// The program is for adjacency matrix representation of the graph ` ` ` `#include <limits.h> ` `#include <stdio.h> ` ` ` `// Number of vertices in the graph ` `#define V 9 ` ` ` `// A utility function to find the vertex with minimum distance value, from ` `// the set of vertices not yet included in shortest path tree ` `int` `minDistance(` `int` `dist[], ` `bool` `sptSet[]) `
`{ ` ` ` `// Initialize min value `
` ` `int` `min = INT_MAX, min_index; `
` ` ` ` `for` `(` `int` `v = 0; v < V; v++) `
` ` `if` `(sptSet[v] == ` `false` `&& dist[v] <= min) `
` ` `min = dist[v], min_index = v; `
` ` ` ` `return` `min_index; `
`} ` ` ` `// A utility function to print the constructed distance array ` `void` `printSolution(` `int` `dist[]) `
`{ ` ` ` `printf` `(` `"Vertex \t\t Distance from Source\n"` `); `
` ` `for` `(` `int` `i = 0; i < V; i++) `
` ` `printf` `(` `"%d \t\t %d\n"` `, i, dist[i]); `
`} ` ` ` `// Function that implements Dijkstra's single source shortest path algorithm ` `// for a graph represented using adjacency matrix representation ` `void` `dijkstra(` `int` `graph[V][V], ` `int` `src) `
`{ ` ` ` `int` `dist[V]; ` `// The output array. dist[i] will hold the shortest `
` ` `// distance from src to i `
` ` ` ` `bool` `sptSet[V]; ` `// sptSet[i] will be true if vertex i is included in shortest `
` ` `// path tree or shortest distance from src to i is finalized `
` ` ` ` `// Initialize all distances as INFINITE and stpSet[] as false `
` ` `for` `(` `int` `i = 0; i < V; i++) `
` ` `dist[i] = INT_MAX, sptSet[i] = ` `false` `; `
` ` ` ` `// Distance of source vertex from itself is always 0 `
` ` `dist[src] = 0; `
` ` ` ` `// Find shortest path for all vertices `
` ` `for` `(` `int` `count = 0; count < V - 1; count++) { `
` ` `// Pick the minimum distance vertex from the set of vertices not `
` ` `// yet processed. u is always equal to src in the first iteration. `
` ` `int` `u = minDistance(dist, sptSet); `
` ` ` ` `// Mark the picked vertex as processed `
` ` `sptSet[u] = ` `true` `; `
` ` ` ` `// Update dist value of the adjacent vertices of the picked vertex. `
` ` `for` `(` `int` `v = 0; v < V; v++) `
` ` ` ` `// Update dist[v] only if is not in sptSet, there is an edge from `
` ` `// u to v, and total weight of path from src to v through u is `
` ` `// smaller than current value of dist[v] `
` ` `if` `(!sptSet[v] && graph[u][v] && dist[u] != INT_MAX `
` ` `&& dist[u] + graph[u][v] < dist[v]) `
` ` `dist[v] = dist[u] + graph[u][v]; `
` ` `} `
` ` ` ` `// print the constructed distance array `
` ` `printSolution(dist); `
`} ` ` ` `// driver program to test above function ` `int` `main() `
`{ ` ` ` `/* Let us create the example graph discussed above */`
` ` `int` `graph[V][V] = { { 0, 4, 0, 0, 0, 0, 0, 8, 0 }, `
` ` `{ 4, 0, 8, 0, 0, 0, 0, 11, 0 }, `
` ` `{ 0, 8, 0, 7, 0, 4, 0, 0, 2 }, `
` ` `{ 0, 0, 7, 0, 9, 14, 0, 0, 0 }, `
` ` `{ 0, 0, 0, 9, 0, 10, 0, 0, 0 }, `
` ` `{ 0, 0, 4, 14, 10, 0, 2, 0, 0 }, `
` ` `{ 0, 0, 0, 0, 0, 2, 0, 1, 6 }, `
` ` `{ 8, 11, 0, 0, 0, 0, 1, 0, 7 }, `
` ` `{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } }; `
` ` ` ` `dijkstra(graph, 0); `
` ` ` ` `return` `0; `
`} ` |

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`// A Java program for Dijkstra's single source shortest path algorithm. ` `// The program is for adjacency matrix representation of the graph ` `import` `java.util.*; `
`import` `java.lang.*; `
`import` `java.io.*; `
` ` `class` `ShortestPath { `
` ` `// A utility function to find the vertex with minimum distance value, `
` ` `// from the set of vertices not yet included in shortest path tree `
` ` `static` `final` `int` `V = ` `9` `; `
` ` `int` `minDistance(` `int` `dist[], Boolean sptSet[]) `
` ` `{ `
` ` `// Initialize min value `
` ` `int` `min = Integer.MAX_VALUE, min_index = -` `1` `; `
` ` ` ` `for` `(` `int` `v = ` `0` `; v < V; v++) `
` ` `if` `(sptSet[v] == ` `false` `&& dist[v] <= min) { `
` ` `min = dist[v]; `
` ` `min_index = v; `
` ` `} `
` ` ` ` `return` `min_index; `
` ` `} `
` ` ` ` `// A utility function to print the constructed distance array `
` ` `void` `printSolution(` `int` `dist[]) `
` ` `{ `
` ` `System.out.println(` `"Vertex \t\t Distance from Source"` `); `
` ` `for` `(` `int` `i = ` `0` `; i < V; i++) `
` ` `System.out.println(i + ` `" \t\t "` `+ dist[i]); `
` ` `} `
` ` ` ` `// Function that implements Dijkstra's single source shortest path `
` ` `// algorithm for a graph represented using adjacency matrix `
` ` `// representation `
` ` `void` `dijkstra(` `int` `graph[][], ` `int` `src) `
` ` `{ `
` ` `int` `dist[] = ` `new` `int` `[V]; ` `// The output array. dist[i] will hold `
` ` `// the shortest distance from src to i `
` ` ` ` `// sptSet[i] will true if vertex i is included in shortest `
` ` `// path tree or shortest distance from src to i is finalized `
` ` `Boolean sptSet[] = ` `new` `Boolean[V]; `
` ` ` ` `// Initialize all distances as INFINITE and stpSet[] as false `
` ` `for` `(` `int` `i = ` `0` `; i < V; i++) { `
` ` `dist[i] = Integer.MAX_VALUE; `
` ` `sptSet[i] = ` `false` `; `
` ` `} `
` ` ` ` `// Distance of source vertex from itself is always 0 `
` ` `dist[src] = ` `0` `; `
` ` ` ` `// Find shortest path for all vertices `
` ` `for` `(` `int` `count = ` `0` `; count < V - ` `1` `; count++) { `
` ` `// Pick the minimum distance vertex from the set of vertices `
` ` `// not yet processed. u is always equal to src in first `
` ` `// iteration. `
` ` `int` `u = minDistance(dist, sptSet); `
` ` ` ` `// Mark the picked vertex as processed `
` ` `sptSet[u] = ` `true` `; `
` ` ` ` `// Update dist value of the adjacent vertices of the `
` ` `// picked vertex. `
` ` `for` `(` `int` `v = ` `0` `; v < V; v++) `
` ` ` ` `// Update dist[v] only if is not in sptSet, there is an `
` ` `// edge from u to v, and total weight of path from src to `
` ` `// v through u is smaller than current value of dist[v] `
` ` `if` `(!sptSet[v] && graph[u][v] != ` `0` `&& dist[u] != Integer.MAX_VALUE && dist[u] + graph[u][v] < dist[v]) `
` ` `dist[v] = dist[u] + graph[u][v]; `
` ` `} `
` ` ` ` `// print the constructed distance array `
` ` `printSolution(dist); `
` ` `} `
` ` ` ` `// Driver method `
` ` `public` `static` `void` `main(String[] args) `
` ` `{ `
` ` `/* Let us create the example graph discussed above */`
` ` `int` `graph[][] = ` `new` `int` `[][] { { ` `0` `, ` `4` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `8` `, ` `0` `}, `
` ` `{ ` `4` `, ` `0` `, ` `8` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `11` `, ` `0` `}, `
` ` `{ ` `0` `, ` `8` `, ` `0` `, ` `7` `, ` `0` `, ` `4` `, ` `0` `, ` `0` `, ` `2` `}, `
` ` `{ ` `0` `, ` `0` `, ` `7` `, ` `0` `, ` `9` `, ` `14` `, ` `0` `, ` `0` `, ` `0` `}, `
` ` `{ ` `0` `, ` `0` `, ` `0` `, ` `9` `, ` `0` `, ` `10` `, ` `0` `, ` `0` `, ` `0` `}, `
` ` `{ ` `0` `, ` `0` `, ` `4` `, ` `14` `, ` `10` `, ` `0` `, ` `2` `, ` `0` `, ` `0` `}, `
` ` `{ ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `2` `, ` `0` `, ` `1` `, ` `6` `}, `
` ` `{ ` `8` `, ` `11` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `1` `, ` `0` `, ` `7` `}, `
` ` `{ ` `0` `, ` `0` `, ` `2` `, ` `0` `, ` `0` `, ` `0` `, ` `6` `, ` `7` `, ` `0` `} }; `
` ` `ShortestPath t = ` `new` `ShortestPath(); `
` ` `t.dijkstra(graph, ` `0` `); `
` ` `} `
`} ` `// This code is contributed by Aakash Hasija ` |

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`# Python program for Dijkstra's single ` `# source shortest path algorithm. The program is ` `# for adjacency matrix representation of the graph ` ` ` `# Library for INT_MAX ` `import` `sys `
` ` `class` `Graph(): `
` ` ` ` `def` `__init__(` `self` `, vertices): `
` ` `self` `.V ` `=` `vertices `
` ` `self` `.graph ` `=` `[[` `0` `for` `column ` `in` `range` `(vertices)] `
` ` `for` `row ` `in` `range` `(vertices)] `
` ` ` ` `def` `printSolution(` `self` `, dist): `
` ` `print` `"Vertex \tDistance from Source"`
` ` `for` `node ` `in` `range` `(` `self` `.V): `
` ` `print` `node, ` `"\t"` `, dist[node] `
` ` ` ` `# A utility function to find the vertex with `
` ` `# minimum distance value, from the set of vertices `
` ` `# not yet included in shortest path tree `
` ` `def` `minDistance(` `self` `, dist, sptSet): `
` ` ` ` `# Initilaize minimum distance for next node `
` ` `min` `=` `sys.maxint `
` ` ` ` `# Search not nearest vertex not in the `
` ` `# shortest path tree `
` ` `for` `v ` `in` `range` `(` `self` `.V): `
` ` `if` `dist[v] < ` `min` `and` `sptSet[v] ` `=` `=` `False` `: `
` ` `min` `=` `dist[v] `
` ` `min_index ` `=` `v `
` ` ` ` `return` `min_index `
` ` ` ` `# Funtion that implements Dijkstra's single source `
` ` `# shortest path algorithm for a graph represented `
` ` `# using adjacency matrix representation `
` ` `def` `dijkstra(` `self` `, src): `
` ` ` ` `dist ` `=` `[sys.maxint] ` `*` `self` `.V `
` ` `dist[src] ` `=` `0`
` ` `sptSet ` `=` `[` `False` `] ` `*` `self` `.V `
` ` ` ` `for` `cout ` `in` `range` `(` `self` `.V): `
` ` ` ` `# Pick the minimum distance vertex from `
` ` `# the set of vertices not yet processed. `
` ` `# u is always equal to src in first iteration `
` ` `u ` `=` `self` `.minDistance(dist, sptSet) `
` ` ` ` `# Put the minimum distance vertex in the `
` ` `# shotest path tree `
` ` `sptSet[u] ` `=` `True`
` ` ` ` `# Update dist value of the adjacent vertices `
` ` `# of the picked vertex only if the current `
` ` `# distance is greater than new distance and `
` ` `# the vertex in not in the shotest path tree `
` ` `for` `v ` `in` `range` `(` `self` `.V): `
` ` `if` `self` `.graph[u][v] > ` `0` `and` `sptSet[v] ` `=` `=` `False` `and` `\ `
` ` `dist[v] > dist[u] ` `+` `self` `.graph[u][v]: `
` ` `dist[v] ` `=` `dist[u] ` `+` `self` `.graph[u][v] `
` ` ` ` `self` `.printSolution(dist) `
` ` `# Driver program ` `g ` `=` `Graph(` `9` `) `
`g.graph ` `=` `[[` `0` `, ` `4` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `8` `, ` `0` `], `
` ` `[` `4` `, ` `0` `, ` `8` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `11` `, ` `0` `], `
` ` `[` `0` `, ` `8` `, ` `0` `, ` `7` `, ` `0` `, ` `4` `, ` `0` `, ` `0` `, ` `2` `], `
` ` `[` `0` `, ` `0` `, ` `7` `, ` `0` `, ` `9` `, ` `14` `, ` `0` `, ` `0` `, ` `0` `], `
` ` `[` `0` `, ` `0` `, ` `0` `, ` `9` `, ` `0` `, ` `10` `, ` `0` `, ` `0` `, ` `0` `], `
` ` `[` `0` `, ` `0` `, ` `4` `, ` `14` `, ` `10` `, ` `0` `, ` `2` `, ` `0` `, ` `0` `], `
` ` `[` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `2` `, ` `0` `, ` `1` `, ` `6` `], `
` ` `[` `8` `, ` `11` `, ` `0` `, ` `0` `, ` `0` `, ` `0` `, ` `1` `, ` `0` `, ` `7` `], `
` ` `[` `0` `, ` `0` `, ` `2` `, ` `0` `, ` `0` `, ` `0` `, ` `6` `, ` `7` `, ` `0` `] `
` ` `]; `
` ` `g.dijkstra(` `0` `); `
` ` `# This code is contributed by Divyanshu Mehta ` |

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`// A C# program for Dijkstra's single ` `// source shortest path algorithm. ` `// The program is for adjacency matrix ` `// representation of the graph ` `using` `System; `
` ` `class` `GFG { `
` ` `// A utility function to find the `
` ` `// vertex with minimum distance `
` ` `// value, from the set of vertices `
` ` `// not yet included in shortest `
` ` `// path tree `
` ` `static` `int` `V = 9; `
` ` `int` `minDistance(` `int` `[] dist, `
` ` `bool` `[] sptSet) `
` ` `{ `
` ` `// Initialize min value `
` ` `int` `min = ` `int` `.MaxValue, min_index = -1; `
` ` ` ` `for` `(` `int` `v = 0; v < V; v++) `
` ` `if` `(sptSet[v] == ` `false` `&& dist[v] <= min) { `
` ` `min = dist[v]; `
` ` `min_index = v; `
` ` `} `
` ` ` ` `return` `min_index; `
` ` `} `
` ` ` ` `// A utility function to print `
` ` `// the constructed distance array `
` ` `void` `printSolution(` `int` `[] dist) `
` ` `{ `
` ` `Console.Write(` `"Vertex \t\t Distance "`
` ` `+ ` `"from Source\n"` `); `
` ` `for` `(` `int` `i = 0; i < V; i++) `
` ` `Console.Write(i + ` `" \t\t "` `+ dist[i] + ` `"\n"` `); `
` ` `} `
` ` ` ` `// Funtion that implements Dijkstra's `
` ` `// single source shortest path algorithm `
` ` `// for a graph represented using adjacency `
` ` `// matrix representation `
` ` `void` `dijkstra(` `int` `[, ] graph, ` `int` `src) `
` ` `{ `
` ` `int` `[] dist = ` `new` `int` `[V]; ` `// The output array. dist[i] `
` ` `// will hold the shortest `
` ` `// distance from src to i `
` ` ` ` `// sptSet[i] will true if vertex `
` ` `// i is included in shortest path `
` ` `// tree or shortest distance from `
` ` `// src to i is finalized `
` ` `bool` `[] sptSet = ` `new` `bool` `[V]; `
` ` ` ` `// Initialize all distances as `
` ` `// INFINITE and stpSet[] as false `
` ` `for` `(` `int` `i = 0; i < V; i++) { `
` ` `dist[i] = ` `int` `.MaxValue; `
` ` `sptSet[i] = ` `false` `; `
` ` `} `
` ` ` ` `// Distance of source vertex `
` ` `// from itself is always 0 `
` ` `dist[src] = 0; `
` ` ` ` `// Find shortest path for all vertices `
` ` `for` `(` `int` `count = 0; count < V - 1; count++) { `
` ` `// Pick the minimum distance vertex `
` ` `// from the set of vertices not yet `
` ` `// processed. u is always equal to `
` ` `// src in first iteration. `
` ` `int` `u = minDistance(dist, sptSet); `
` ` ` ` `// Mark the picked vertex as processed `
` ` `sptSet[u] = ` `true` `; `
` ` ` ` `// Update dist value of the adjacent `
` ` `// vertices of the picked vertex. `
` ` `for` `(` `int` `v = 0; v < V; v++) `
` ` ` ` `// Update dist[v] only if is not in `
` ` `// sptSet, there is an edge from u `
` ` `// to v, and total weight of path `
` ` `// from src to v through u is smaller `
` ` `// than current value of dist[v] `
` ` `if` `(!sptSet[v] && graph[u, v] != 0 && dist[u] != ` `int` `.MaxValue && dist[u] + graph[u, v] < dist[v]) `
` ` `dist[v] = dist[u] + graph[u, v]; `
` ` `} `
` ` ` ` `// print the constructed distance array `
` ` `printSolution(dist); `
` ` `} `
` ` ` ` `// Driver Code `
` ` `public` `static` `void` `Main() `
` ` `{ `
` ` `/* Let us create the example `
`graph discussed above */` ` ` `int` `[, ] graph = ` `new` `int` `[, ] { { 0, 4, 0, 0, 0, 0, 0, 8, 0 }, `
` ` `{ 4, 0, 8, 0, 0, 0, 0, 11, 0 }, `
` ` `{ 0, 8, 0, 7, 0, 4, 0, 0, 2 }, `
` ` `{ 0, 0, 7, 0, 9, 14, 0, 0, 0 }, `
` ` `{ 0, 0, 0, 9, 0, 10, 0, 0, 0 }, `
` ` `{ 0, 0, 4, 14, 10, 0, 2, 0, 0 }, `
` ` `{ 0, 0, 0, 0, 0, 2, 0, 1, 6 }, `
` ` `{ 8, 11, 0, 0, 0, 0, 1, 0, 7 }, `
` ` `{ 0, 0, 2, 0, 0, 0, 6, 7, 0 } }; `
` ` `GFG t = ` `new` `GFG(); `
` ` `t.dijkstra(graph, 0); `
` ` `} `
`} ` ` ` `// This code is contributed by ChitraNayal ` |

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**Output:**

Vertex Distance from Source 0 0 1 4 2 12 3 19 4 21 5 11 6 9 7 8 8 14

**Notes:****1)** The code calculates shortest distance, but doesn’t calculate the path information. We can create a parent array, update the parent array when distance is updated (like prim’s implementation) and use it show the shortest path from source to different vertices.

**2)** The code is for undirected graph, same dijkstra function can be used for directed graphs also.

**3)** The code finds shortest distances from source to all vertices. If we are interested only in shortest distance from the source to a single target, we can break the for the loop when the picked minimum distance vertex is equal to target (Step 3.a of the algorithm).

**4)** Time Complexity of the implementation is O(V^2). If the input graph is represented using adjacency list, it can be reduced to O(E log V) with the help of binary heap. Please see

Dijkstra’s Algorithm for Adjacency List Representation for more details.

**5)** Dijkstra’s algorithm doesn’t work for graphs with negative weight cycles, it may give correct results for a graph with negative edges. For graphs with negative weight edges and cycles, Bellman–Ford algorithm can be used, we will soon be discussing it as a separate post.

Dijkstra’s Algorithm for Adjacency List Representation

Printing Paths in Dijkstra’s Shortest Path Algorithm

Dijkstra’s shortest path algorithm using set in STL

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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