The key points of Dijkstra’s single source shortest path algorithm is as below :
Algorithm : Dijkstra’s Shortest Path C++
1. Initialize the distance from the source node S to all other nodes as infinite (999999999999) and to itself as 0.
2. Insert the pair of < distance , node > for source i.e < 0, S > in a priority-based SET [C++]
where the priority of the elements in the set is based on the length of the distance.
3. While the SET is not empty do
4. pair_at_top = SET . top( );
Remove the element from the top of the SET.
current_source_node = pair_at_top . second.
5. For every adjacent_node to current_source_node do
6. If ( distance [ adjacent_node ] > length_of_path_to_adjacent_node_from_current_source + distance [ current_source_node ] ) then
7. distance [ adjacent_node ] = length_of_path_to_adjacent_node_from_current_source + distance [ current_source_node ]
8. Update the SET with the new pair < distance [ adjacent_node ], adjacent_node >
Below data structures are used for storing the graph before running Dijkstra’s algorithm Adjacency List : List of pairs of adjacent nodes and their corresponding weights in the graph.
Dijkstra’s algorithm step-by-step
This example of Dijkstra’s algorithm finds the shortest distance of all the nodes in the graph from the single / original source node 0.
Step 1 : Initialize the distance of the source node to itself as 0 and to all other nodes as ∞.
Insert the pair < distance_from_original_source, node > in the set.
i.e Insert < 0, 0 > in the set as the distance from the original source (0) to itself is 0.
Step 2 : Remove the topmost pair < 0, 0 > from the set and relax the edge going towards every adjacent node(s) from the original source node ( 0 ).
Currentsource node | Adjacentnode | Distance to the adjacent nodefrom the original source ( 0 ) | Edge relaxation |
---|---|---|---|
0 | 1 | distance [ 1 ] = ∞ | distance [ 1 ] > distance_between [ 0 - 1 ] + distance [ 0 ]i.e Since ∞ > 5 + 0Update distance, i.e distance [ 1 ] = 5 and insert pair < 5, 1 > in the set. |
0 | 2 | distance [ 2 ] = ∞ | distance [ 2 ] > distance_between [ 0 - 2 ] + distance [ 0 ]i.e Since ∞ > 1 + 0Update distance, i.e distance [ 2 ] = 1 and insert pair < 1, 2 > in the set. |
0 | 3 | distance [ 3 ] = ∞ | distance [ 3 ] > distance_between [ 0 - 3 ] + distance [ 0 ]i.e Since ∞ > 4 + 0Update distance, i.e distance [ 3 ] = 4 and insert pair < 4, 3 > in the set. |
Step 3 : Remove the topmost pair < 1, 2 > from the set and relax the edge going towards every adjacent node(s) from the current source ( 2 ).
Currentsource node | Adjacentnode | Distance to the adjacent nodefrom the original source ( 0 ) | Edge relaxation |
---|---|---|---|
2 | 0 | distance [ 0 ] = 0 | distance [ 0 ] < distance_between [ 2 - 0 ] + distance [ 2 ]i.e Since 0 < 1 + 1 No edge relaxation needed. |
2 | 1 | distance [ 1 ] = 5 | distance [ 1 ] > distance_between [ 2 - 1 ] + distance [ 2 ]i.e Since 5 > 3 + 1Update distance, i.e distance [ 1 ] = 4 and insert pair < 4, 1 > in the set. |
2 | 3 | distance [ 3 ] = 4 | distance [ 3 ] > distance_between [ 2 - 3 ] + distance [ 2 ]i.e Since 4 > 2 + 1Update distance, i.e distance [ 3 ] = 3 and insert pair < 3, 3 > in the set. |
2 | 4 | distance [ 4 ] = ∞ | distance [ 4 ] > distance_between [ 2 - 4 ] + distance [ 2 ]i.e Since ∞ > 1 + 1Update distance, i.e distance [ 4 ] = 2 and insert pair < 2, 4 > in the set. |
Step 4 : Remove the topmost pair < 2, 4 > from the set and relax the edge going towards every adjacent node(s) from the current source ( 4 ).
Currentsource node | Adjacentnode | Distance to the adjacent nodefrom the original source ( 0 ) | Edge relaxation |
---|---|---|---|
4 | 1 | distance [ 1 ] = 4 | distance [ 1 ] < distance_between [ 4 - 1 ] + distance [ 4 ]i.e Since 4 < 8 + 2 No edge relaxation needed. |
4 | 2 | distance [ 2 ] = 1 | distance [ 2 ] < distance_between [ 4 - 2 ] + distance [ 4 ]i.e Since 1 < 1 + 2 No edge relaxation needed. |
4 | 3 | distance [ 3 ] = 3 | distance [ 3 ] < distance_between [ 4 - 3 ] + distance [ 4 ]i.e Since 3 < 2 + 2 No edge relaxation needed. |
4 | 5 | distance [ 5 ] = ∞ | distance [ 5 ] > distance_between [ 4 - 5 ] + distance [ 4 ]i.e Since ∞ > 3 + 2Update distance, i.e distance [ 5 ] = 5 and insert pair < 5, 5 > in the set. |
Step 5 : Remove the topmost pair < 3, 3 > from the set and relax the edge going towards every adjacent node(s) from the current source ( 3 ).
Currentsource node | Adjacentnode | Distance to the adjacent nodefrom the original source ( 0 ) | Edge relaxation |
---|---|---|---|
3 | 0 | distance [ 0 ] = 0 | distance [ 0 ] < distance_between [ 3 - 0 ] + distance [ 3 ]i.e Since 0 < 4 + 3 No edge relaxation needed. |
3 | 2 | distance [ 2 ] = 1 | distance [ 2 ] < distance_between [ 3 - 2 ] + distance [ 3 ]i.e Since 1 < 2 + 3 No edge relaxation needed. |
3 | 4 | distance [ 4 ] = 2 | distance [ 4 ] < distance_between [ 3 - 4 ] + distance [ 3 ]i.e Since 2 < 2 + 3 No edge relaxation needed. |
3 | 5 | distance [ 5 ] = 5 | distance [ 5 ] > distance_between [ 3 - 5 ] + distance [ 3 ]i.e Since 5 > 1 + 3Update distance, i.e distance [ 5 ] = 4 and insert pair < 4, 5 > in the set. |
Step 6 : Remove the topmost pair < 4, 1 > from the set and relax the edge going towards every adjacent node(s) from the current source ( 1 ).
Currentsource node | Adjacentnode | Distance to the adjacent nodefrom the original source ( 0 ) | Edge relaxation |
---|---|---|---|
1 | 0 | distance [ 0 ] = 0 | distance [ 0 ] < distance_between [ 1 - 0 ] + distance [ 1 ]i.e Since 0 < 5 + 4 No edge relaxation needed. |
1 | 2 | distance [ 2 ] = 1 | distance [ 2 ] < distance_between [ 1 - 2 ] + distance [ 1 ]i.e Since 1 < 3 + 4 No edge relaxation needed. |
1 | 4 | distance [ 4 ] = 2 | distance [ 4 ] < distance_between [ 1 - 4 ] + distance [ 1 ]i.e Since 2 < 8 + 4 No edge relaxation needed. |
Step 7 : Remove the topmost pair < 4, 5 > from the set and relax the edge going towards every adjacent node(s) from the current source ( 5 ).
Currentsource node | Adjacentnode | Distance to the adjacent nodefrom the original source ( 0 ) | Edge relaxation |
---|---|---|---|
5 | 3 | distance [ 3 ] = 3 | distance [ 3 ] < distance_between [ 5 - 3 ] + distance [ 5 ]i.e Since 3 < 1 + 4 No edge relaxation needed. |
5 | 4 | distance [ 4 ] = 2 | distance [ 4 ] < distance_between [ 5 - 4 ] + distance [ 5 ]i.e Since 2 < 3 + 4 No edge relaxation needed. |
The algorithm terminates here as the set is empty and we have calculated the shortest path to all the nodes from the original source 0 as shown below.
Data structure used for running Dijkstra’s shortest path : Distance based priority set for choosing the vertex nearest to the source.
Graph type: Designed for weighted (directed / un-directed) graph containing positve edge weights.
Time complexity of Dijkstra’s algorithm : O ( (E+V) Log(V) ) for an adjacency list implementation of a graph. V is the
number of vertices and E is the number of edges in a graph.
#include<iostream>
#include<vector>
#include<set>
using namespace std;
typedef pair<int,unsigned long long> PII;
typedef vector<PII> VPII;
typedef vector<VPII> VVPII;
void DijkstrasShortestPath (int source_node, int node_count, VVPII& graph) {
// Assume that the distance from source_node to other nodes is infinite
// in the beginnging, i.e initialize the distance vector to a max value
const long long INF = 999999999999;
vector<unsigned long long> dist(node_count, INF);
set<PII> set_length_node;
// Distance from starting vertex to itself is 0
dist[source_node] = 0;
set_length_node.insert(PII(0,source_node));
while (!set_length_node.empty()) {
PII top = *set_length_node.begin();
set_length_node.erase(set_length_node.begin());
int current_source_node = top.second;
for (auto& it : graph[current_source_node]) {
int adj_node = it.first;
int length_to_adjnode = it.second;
// Edge relaxation
if (dist[adj_node] > length_to_adjnode + dist[current_source_node]) {
// If the distance to the adjacent node is not INF, means the pair <dist, node> is in the set
// Remove the pair before updating it in the set.
if (dist[adj_node] != INF) {
set_length_node.erase(set_length_node.find(PII(dist[adj_node],adj_node)));
}
dist[adj_node] = length_to_adjnode + dist[current_source_node];
set_length_node.insert(PII(dist[adj_node], adj_node));
}
}
}
for (int i=0; i<node_count; i++)
cout << "Source Node(" << source_node << ") -> Destination Node(" << i << ") : " << dist[i] << endl;
}
int main(){
vector<VPII> graph;
// Node 0: <1,5> <2,1> <3,4>
VPII a = {{1,5}, {2,1}, {3,4}};
graph.push_back(a);
// Node 1: <0,5> <2,3> <4,8>
VPII b = {{0,5}, {2,3}, {4,8}};
graph.push_back(b);
// Node 2: <0,1> <1,3> <3,2> <4,1>
VPII c = {{0,1}, {1,3}, {3,2}, {4,1}};
graph.push_back(c);
// Node 3: <0,4> <2,2> <4,2> <5,1>
VPII d = {{0,4}, {2,2}, {4,2}, {5,1}};
graph.push_back(d);
// Node 4: <1,8> <2,1> <3,2> <5,3>
VPII e = {{1,8}, {2,1}, {3,2}, {5,3}};
graph.push_back(e);
// Node 5: <3,1> <4,3>
VPII f = {{3,1}, {4,3}};
graph.push_back(f);
int node_count = 6;
int source_node = 0;
DijkstrasShortestPath(source_node, node_count, graph);
cout << endl;
source_node = 5;
DijkstrasShortestPath(source_node, node_count, graph);
return 0;
}
Output
Source Node(0) -> Destination Node(0) : 0
Source Node(0) -> Destination Node(1) : 4
Source Node(0) -> Destination Node(2) : 1
Source Node(0) -> Destination Node(3) : 3
Source Node(0) -> Destination Node(4) : 2
Source Node(0) -> Destination Node(5) : 4
Source Node(5) -> Destination Node(0) : 4
Source Node(5) -> Destination Node(1) : 6
Source Node(5) -> Destination Node(2) : 3
Source Node(5) -> Destination Node(3) : 1
Source Node(5) -> Destination Node(4) : 3
Source Node(5) -> Destination Node(5) : 0