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# Dial’s Algorithm (Optimized Dijkstra for small range weights)

• Difficulty Level : Expert
• Last Updated : 25 Mar, 2023

Dijkstra’s shortest path algorithm runs in O(Elog V) time when implemented with adjacency list representation (See C implementation and STL based C++ implementations for details).

Input : Source = 0, Maximum Weight W = 14
Output :
Vertex       Distance from Source
0                   0
1                   4
2                  12
3                  19
4                  21
5                  11
6                   9
7                   8
8                 14

Consider the below Graph: The shortest path from source 0

We have learned about how to find the shortest path from a given source vertex to all other vertex using Dijkstra’s shortest path algorithm with the Time Complexity of O(E log V) in this article.

Can we optimize Dijkstra’s shortest path algorithm to work better than O(E log V) if the maximum weight is small (or the range of edge weights is small)? For example, in the above diagram, the maximum weight is 14. Many times the range of weights on edges is in a small range (i.e. all edge weights can be mapped to 0, 1, 2.. w where w is a small number). In that case, Dijkstra’s algorithm can be modified by using different data structures, and buckets, which is called dial implementation of Dijkstra’s algorithm. time complexity is O(E + WV) where W is the maximum weight on any edge of the graph, so we can see that, if W is small then this implementation runs much faster than the traditional algorithm. The following are important observations.

• The maximum distance between any two nodes can be at max w(V – 1) (w is maximum edge weight and we can have at max V-1 edges between two vertices).
• In the Dijkstra algorithm, distances are finalized in non-decreasing, i.e., the distance of the closer (to given source) vertices is finalized before the distant vertices.

Algorithm Below is the complete algorithm:

1. Maintains some buckets, numbered 0, 1, 2,…,wV.
2. Bucket k contains all temporarily labeled nodes with a distance equal to k.
3. Nodes in each bucket are represented by a list of vertices.
4. Buckets 0, 1, 2,..wV are checked sequentially until the first non-empty bucket is found. Each node contained in the first non-empty bucket has the minimum distance label by definition.
5. One by one, these nodes with minimum distance labels are permanently labeled and deleted from the bucket during the scanning process.
6. Thus operations involving vertex include:
• Checking if a bucket is empty
• Adding a vertex to a bucket
• Deleting a vertex from a bucket.
7. The position of a temporarily labeled vertex in the buckets is updated accordingly when the distance label of a vertex changes.
8. The process is repeated until all vertices are permanently labeled (or the distances of all vertices are finalized).

Implementation Since the maximum distance can be w(V – 1), we create wV buckets (more for simplicity of code) for implementation of the algorithm which can be large if w is big.

## C++

 `// C++ Program for Dijkstra's dial implementation` `#include` `using` `namespace` `std;` `# define INF 0x3f3f3f3f`   `// This class represents a directed graph using` `// adjacency list representation` `class` `Graph` `{` `    ``int` `V; ``// No. of vertices` `    ``// In a weighted graph, we need to store vertex` `    ``// and weight pair for every edge` `    ``list< pair<``int``, ``int``> > *adj;`   `public``:` `    ``Graph(``int` `V); ``// Constructor`   `    ``// function to add an edge to graph` `    ``void` `addEdge(``int` `u, ``int` `v, ``int` `w);`   `    ``// prints shortest path from s` `    ``void` `shortestPath(``int` `s, ``int` `W);` `};`   `// Allocates memory for adjacency list` `Graph::Graph(``int` `V)` `{` `    ``this``->V = V;` `    ``adj = ``new` `list< pair<``int``, ``int``> >[V];` `}`   `// adds edge between u and v of weight w` `void` `Graph::addEdge(``int` `u, ``int` `v, ``int` `w)` `{` `    ``adj[u].push_back(make_pair(v, w));` `    ``adj[v].push_back(make_pair(u, w));` `}`   `// Prints shortest paths from src to all other vertices.` `// W is the maximum weight of an edge` `void` `Graph::shortestPath(``int` `src, ``int` `W)` `{` `    ``/* With each distance, iterator to that vertex in` `    ``its bucket is stored so that vertex can be deleted` `    ``in O(1) at time of updation. So` `    ``dist[i].first = distance of ith vertex from src vertex` `    ``dits[i].second = iterator to vertex i in bucket number */` `    ``vector::iterator> > dist(V);`   `    ``// Initialize all distances as infinite (INF)` `    ``for` `(``int` `i = 0; i < V; i++)` `        ``dist[i].first = INF;`   `    ``// Create buckets B[].` `    ``// B[i] keep vertex of distance label i` `    ``list<``int``> B[W * V + 1];`   `    ``B.push_back(src);` `    ``dist[src].first = 0;`   `    ``//` `    ``int` `idx = 0;` `    ``while` `(1)` `    ``{` `        ``// Go sequentially through buckets till one non-empty` `        ``// bucket is found` `        ``while` `(B[idx].size() == 0 && idx < W*V)` `            ``idx++;`   `        ``// If all buckets are empty, we are done.` `        ``if` `(idx == W * V)` `            ``break``;`   `        ``// Take top vertex from bucket and pop it` `        ``int` `u = B[idx].front();` `        ``B[idx].pop_front();`   `        ``// Process all adjacents of extracted vertex 'u' and` `        ``// update their distanced if required.` `        ``for` `(``auto` `i = adj[u].begin(); i != adj[u].end(); ++i)` `        ``{` `            ``int` `v = (*i).first;` `            ``int` `weight = (*i).second;`   `            ``int` `du = dist[u].first;` `            ``int` `dv = dist[v].first;`   `            ``// If there is shorted path to v through u.` `            ``if` `(dv > du + weight)` `            ``{` `                ``// If dv is not INF then it must be in B[dv]` `                ``// bucket, so erase its entry using iterator` `                ``// in O(1)` `                ``if` `(dv != INF)` `                    ``B[dv].erase(dist[v].second);`   `                ``// updating the distance` `                ``dist[v].first = du + weight;` `                ``dv = dist[v].first;`   `                ``// pushing vertex v into updated distance's bucket` `                ``B[dv].push_front(v);`   `                ``// storing updated iterator in dist[v].second` `                ``dist[v].second = B[dv].begin();` `            ``}` `        ``}` `    ``}`   `    ``// Print shortest distances stored in dist[]` `    ``printf``(``"Vertex Distance from Source\n"``);` `    ``for` `(``int` `i = 0; i < V; ++i)` `        ``printf``(``"%d     %d\n"``, i, dist[i].first);` `}`   `// Driver program to test methods of graph class` `int` `main()` `{` `    ``// create the graph given in above figure` `    ``int` `V = 9;` `    ``Graph g(V);`   `    ``// making above shown graph` `    ``g.addEdge(0, 1, 4);` `    ``g.addEdge(0, 7, 8);` `    ``g.addEdge(1, 2, 8);` `    ``g.addEdge(1, 7, 11);` `    ``g.addEdge(2, 3, 7);` `    ``g.addEdge(2, 8, 2);` `    ``g.addEdge(2, 5, 4);` `    ``g.addEdge(3, 4, 9);` `    ``g.addEdge(3, 5, 14);` `    ``g.addEdge(4, 5, 10);` `    ``g.addEdge(5, 6, 2);` `    ``g.addEdge(6, 7, 1);` `    ``g.addEdge(6, 8, 6);` `    ``g.addEdge(7, 8, 7);`   `    ``// maximum weighted edge - 14` `    ``g.shortestPath(0, 14);`   `    ``return` `0;` `}`   `// This code is contributed by Aditya Kumar (adityakumar129)`

## Python3

 `from` `typing ``import` `List``, ``Tuple` `INF ``=` `0x3f3f3f3f`   `# This class represents a directed graph using adjacency list representation` `class` `Graph:` `    ``def` `__init__(``self``, V: ``int``):` `        ``# No. of vertices` `        ``self``.V ``=` `V` `        `  `        ``# In a weighted graph, we need to store vertex and weight pair for every edge` `        ``self``.adj ``=` `[[] ``for` `_ ``in` `range``(V)]`   `    ``# function to add an edge to graph` `    ``def` `addEdge(``self``, u: ``int``, v: ``int``, w: ``int``):` `        ``self``.adj[u].append((v, w))` `        ``self``.adj[v].append((u, w))`   `    ``# Prints shortest paths from src to all other vertices.` `    ``# W is the maximum weight of an edge` `    ``def` `shortestPath(``self``, src: ``int``, W: ``int``):` `        ``# With each distance, iterator to that vertex in its bucket is stored so that vertex can be deleted` `        ``# in O(1) at time of updation. So dist[i] = distance of ith vertex from src vertex` `        ``# dist[i] = iterator to vertex i in bucket number` `        ``dist ``=` `[[INF, ``None``] ``for` `_ ``in` `range``(``self``.V)]`   `        ``# Initialize distance of source vertex` `        ``dist[src][``0``] ``=` `0`   `        ``# Create buckets B[].` `        ``# B[i] keep vertex of distance label i` `        ``B ``=` `[[] ``for` `_ ``in` `range``(W ``*` `self``.V ``+` `1``)]` `        ``B[``0``].append(src)`   `        ``idx ``=` `0` `        ``while` `True``:` `            ``# Go sequentially through buckets till one non-empty bucket is found` `            ``while` `len``(B[idx]) ``=``=` `0` `and` `idx < W ``*` `self``.V:` `                ``idx ``+``=` `1`   `            ``# If all buckets are empty, we are done.` `            ``if` `idx ``=``=` `W ``*` `self``.V:` `                ``break`   `            ``# Take top vertex from bucket and pop it` `            ``u ``=` `B[idx][``0``]` `            ``B[idx].pop(``0``)`   `            ``# Process all adjacents of extracted vertex ` `            ``# 'u' and update their distances if required.` `            ``for` `v, weight ``in` `self``.adj[u]:` `                ``du ``=` `dist[u][``0``]` `                ``dv ``=` `dist[v][``0``]`   `                ``# If there is shorted path to v through u.` `                ``if` `dv > du ``+` `weight:` `                    ``# If dv is not INF then it must be in ` `                    ``# B[dv] bucket, so erase its entry using iterator` `                    ``# in O(1)` `                    ``if` `dv !``=` `INF:` `                        ``B[dv].remove(v)`   `                    ``# updating the distance` `                    ``dist[v][``0``] ``=` `du ``+` `weight` `                    ``dv ``=` `dist[v][``0``]`   `                    ``# pushing vertex v into updated distance's bucket` `                    ``B[dv].append(v)`   `                    ``# storing updated iterator in dist[v]` `                    ``dist[v][``1``] ``=` `len``(B[dv]) ``-` `1`   `        ``# Print shortest distances stored in dist[]` `        ``print``(``"Vertex Distance from Source"``)` `        ``for` `i ``in` `range``(``self``.V):` `            ``print``(f``"{i}     {dist[i]}"``)`   `# Driver program to test methods of graph class` `def` `main():` `    ``# create the graph given in above figure` `    ``V ``=` `9` `    ``W ``=` `14` `    ``g ``=` `Graph(V)`   `    ``g.addEdge(``0``, ``1``, ``4``)` `    ``g.addEdge(``0``, ``7``, ``8``)` `    ``g.addEdge(``1``, ``2``, ``8``)` `    ``g.addEdge(``1``, ``7``, ``11``)` `    ``g.addEdge(``2``, ``3``, ``7``)` `    ``g.addEdge(``2``, ``8``, ``2``)` `    ``g.addEdge(``2``, ``5``, ``4``)` `    ``g.addEdge(``3``, ``4``, ``9``)` `    ``g.addEdge(``3``, ``5``, ``14``)` `    ``g.addEdge(``4``, ``5``, ``10``)` `    ``g.addEdge(``5``, ``6``, ``2``)` `    ``g.addEdge(``6``, ``7``, ``1``)` `    ``g.addEdge(``6``, ``8``, ``6``)` `    ``g.addEdge(``7``, ``8``, ``7``)`   `    ``g.shortestPath(``0``, W)`   `if` `__name__ ``=``=` `"__main__"``:` `    ``main()` `    `  `# This code is contributed by sanjanasikarwar24`

## C#

 `// C# Program for Dijkstra's dial implementation` `using` `System;` `using` `System.Collections.Generic;`   `// This class represents a directed graph using` `// adjacency list representation` `public` `class` `Graph ` `{` `  ``static` `readonly` `int` `INF = Int32.MaxValue;` `  ``private` `int` `V; ``// No. of vertices` `  ``// In a weighted graph, we need to store vertex` `  ``// and weight pair for every edge` `  ``private` `List >[] adj;`   `  ``public` `Graph(``int` `v) ``// Constructor` `  ``{` `    ``this``.V = v;` `    ``this``.adj = ``new` `List >[ v ];` `    ``for` `(``int` `i = 0; i < v; i++)` `      ``this``.adj[i] = ``new` `List >();` `  ``}`   `  ``// function to Add an edge to graph` `  ``// Adds edge between u and v of weight w` `  ``public` `void` `AddEdge(``int` `u, ``int` `v, ``int` `w)` `  ``{` `    ``adj[u].Add(Tuple.Create(v, w));` `    ``adj[v].Add(Tuple.Create(u, w));` `  ``}`   `  ``// Prints shortest paths from src to all other vertices.` `  ``// W is the maximum weight of an edge` `  ``public` `void` `shortestPath(``int` `src, ``int` `W)` `  ``{` `    ``/* With each distance, iterator to that vertex in` `        ``its bucket is stored so that vertex can be deleted` `        ``in O(1) at time of updation. So` `        ``dist[i].first = distance of ith vertex from src` `        ``vertex dits[i].second = iterator to vertex i in` `        ``bucket number */` `    ``int``[] dist = ``new` `int``[V];`   `    ``// Initialize all distances as infinite (INF)` `    ``for` `(``int` `i = 0; i < V; i++)` `      ``dist[i] = INF;`   `    ``// Create buckets B[].` `    ``// B[i] keep vertex of distance label i` `    ``List<``int``>[] B = ``new` `List<``int``>[ W * V + 1 ];` `    ``for` `(``int` `i = 0; i < W * V + 1; i++)` `      ``B[i] = ``new` `List<``int``>();`   `    ``B.Add(src);` `    ``dist[src] = 0;`   `    ``int` `idx = 0;` `    ``while` `(``true``) {` `      ``// Go sequentially through buckets till one` `      ``// non-empty bucket is found` `      ``while` `(B[idx].Count == 0 && idx < W * V)` `        ``idx++;`   `      ``// If all buckets are empty, we are done.` `      ``if` `(idx == W * V)` `        ``break``;`   `      ``// Take top vertex from bucket and pop it` `      ``int` `u = B[idx];` `      ``B[idx].Remove(u);`   `      ``// Process all adjacents of extracted vertex 'u'` `      ``// and update their distances if required.` `      ``foreach``(Tuple<``int``, ``int``> i ``in` `adj[u])` `      ``{` `        ``int` `v = i.Item1;` `        ``int` `weight = i.Item2;`   `        ``int` `du = dist[u];` `        ``int` `dv = dist[v];`   `        ``// If there is shorted path to v through u.` `        ``if` `(dv > du + weight) {` `          ``// updating the distance` `          ``dist[v] = du + weight;` `          ``dv = dist[v];`   `          ``// pushing vertex v into updated` `          ``// distance's bucket` `          ``B[dv].Insert(0, v);` `        ``}` `      ``}` `    ``}`   `    ``// Print shortest distances stored in dist[]` `    ``Console.WriteLine(``"Vertex Distance from Source"``);` `    ``for` `(``int` `i = 0; i < V; ++i)` `      ``Console.WriteLine(``"{0}     {1}"``, i, dist[i]);` `  ``}` `}`   `class` `GFG {` `  ``// Driver program to test methods of graph class` `  ``static` `void` `Main(``string``[] args)` `  ``{` `    ``// create the graph given in above figure` `    ``int` `V = 9;` `    ``Graph g = ``new` `Graph(V);`   `    ``// making above shown graph` `    ``g.AddEdge(0, 1, 4);` `    ``g.AddEdge(0, 7, 8);` `    ``g.AddEdge(1, 2, 8);` `    ``g.AddEdge(1, 7, 11);` `    ``g.AddEdge(2, 3, 7);` `    ``g.AddEdge(2, 8, 2);` `    ``g.AddEdge(2, 5, 4);` `    ``g.AddEdge(3, 4, 9);` `    ``g.AddEdge(3, 5, 14);` `    ``g.AddEdge(4, 5, 10);` `    ``g.AddEdge(5, 6, 2);` `    ``g.AddEdge(6, 7, 1);` `    ``g.AddEdge(6, 8, 6);` `    ``g.AddEdge(7, 8, 7);`   `    ``// maximum weighted edge - 14` `    ``g.shortestPath(0, 14);` `  ``}` `}`   `// This code is contributed by cavi4762`

## Javascript

 `// JS Program for Dijkstra's dial implementation` `const INF = 0x3f3f3f3f;`   `// This class represents a directed graph using` `// adjacency list representation` `class Graph {` `  ``constructor(V)` `  ``{` `    ``this``.V = V;``// No. of vertices` `    `  `    ``// In a weighted graph, we need to store vertex` `    ``// and weight pair for every edge` `    ``this``.adj = Array.from({ length: V }, () => []);` `  ``}`   `// adds edge between u and v of weight w` `  ``addEdge(u, v, w) {` `    ``this``.adj[u].push([v, w]);` `    ``this``.adj[v].push([u, w]);` `  ``}` `  `  `// Prints shortest paths from src to all other vertices.` `// W is the maximum weight of an edge` `  ``shortestPath(src, W)` `  ``{` `  `  `       ``/* With each distance, iterator to that vertex in` `    ``its bucket is stored so that vertex can be deleted` `    ``in O(1) at time of updation. So` `    ``dist[i].first = distance of ith vertex from src vertex` `    ``dits[i].second = iterator to vertex i in bucket number */` `    ``const dist = Array.from({ length: ``this``.V }, () => [INF, ``null``]);` `    ``dist[src] = 0;` ` ``// Create buckets B[].` `    ``// B[i] keep vertex of distance label i` `    ``const B = Array.from({ length: W * ``this``.V + 1 }, () => []);` `    ``B.push(src);`   `    ``let idx = 0;` `    ``while` `(``true``)` `    ``{` `    `  `        ``// Go sequentially through buckets till one non-empty` `        ``// bucket is found` `      ``while` `(B[idx].length === 0 && idx < W * ``this``.V) {` `        ``idx += 1;` `      ``}` `      ``if` `(idx === W * ``this``.V) {` `        ``break``;` `      ``}` `      ``const u = B[idx].shift();` `      `  `    ``// Process all adjacents of extracted vertex 'u' and` `        ``// update their distanced if required.` `      ``for` `(const [v, weight] of ``this``.adj[u]) {` `        ``const du = dist[u];` `        ``let dv = dist[v];` `        `  `       ``// If there is shorted path to v through u.` `        ``if` `(dv > du + weight)` `        ``{` `        `  `               ``// If dv is not INF then it must be in B[dv]` `                ``// bucket, so erase its entry using iterator` `                ``// in O(1)` `          ``if` `(dv !== INF) {` `            ``B[dv].splice(dist[v], 1);` `          ``}` `          `  `           ``// updating the distance` `          ``dist[v] = du + weight;` `          ``dv = dist[v];` `          `  `                ``// pushing vertex v into updated distance's bucket` `          ``B[dv].push(v);` `          `  `          ``// storing updated iterator in dist[v].second` `          ``dist[v] = B[dv].length - 1;` `        ``}` `      ``}` `    ``}` `    `  ` ``// Print shortest distances stored in dist[]` `    ``console.log(``"Vertex Distance from Source"``);` `    ``for` `(let i = 0; i < ``this``.V; i++) {` `      ``console.log(`\${i}     \${dist[i]}`);` `    ``}` `  ``}` `}`   `// Driver code` `  ``const V = 9;` `  ``const W = 14;` `  ``const g = ``new` `Graph(V);`   `  ``g.addEdge(0, 1, 4);` `  ``g.addEdge(0, 7, 8);` `  ``g.addEdge(1, 2, 8);` `  ``g.addEdge(1, 7, 11);` `  ``g.addEdge(2, 3, 7);` `  ``g.addEdge(2, 8, 2);` `  ``g.addEdge(2, 5, 4);` `  ``g.addEdge(3, 4, 9);` `  ``g.addEdge(3, 5, 14);` `  ``g.addEdge(4, 5, 10);` `  ``g.addEdge(5, 6, 2);` `  ``g.addEdge(6, 7, 1);` `  ``g.addEdge(6, 8, 6);` `  ``g.addEdge(7, 8, 7);`   `  ``g.shortestPath(0, W);`   `// This code is contributed by lokeshpotta20.`

## Java

 `import` `java.util.*;`   `public` `class` `Graph {` `    ``static` `final` `int` `INF = Integer.MAX_VALUE;` `    ``private` `int` `V; ``// No. of vertices` `    ``// In a weighted graph, we need to store vertex` `    ``// and weight pair for every edge` `    ``private` `ArrayList > adj;`   `    ``public` `Graph(``int` `v) ``// Constructor` `    ``{` `        ``this``.V = v;` `        ``this``.adj = ``new` `ArrayList >();` `        ``for` `(``int` `i = ``0``; i < v; i++)` `            ``this``.adj.add(``new` `ArrayList());` `    ``}`   `    ``// function to Add an edge to graph` `    ``// Adds edge between u and v of weight w` `    ``public` `void` `AddEdge(``int` `u, ``int` `v, ``int` `w)` `    ``{` `        ``adj.get(u).add(``new` `Tuple(v, w));` `        ``adj.get(v).add(``new` `Tuple(u, w));` `    ``}`   `    ``// Prints shortest paths from src to all other vertices.` `    ``// W is the maximum weight of an edge` `    ``public` `void` `shortestPath(``int` `src, ``int` `W)` `    ``{` `        ``/* With each distance, iterator to that vertex in` `            ``its bucket is stored so that vertex can be` `           ``deleted in O(1) at time of updation. So` `            ``dist[i].first = distance of ith vertex from src` `            ``vertex dits[i].second = iterator to vertex i in` `            ``bucket number */` `        ``int``[] dist = ``new` `int``[V];`   `        ``// Initialize all distances as infinite (INF)` `        ``Arrays.fill(dist, INF);`   `        ``// Create buckets B[].` `        ``// B[i] keep vertex of distance label i` `        ``ArrayList[] B = ``new` `ArrayList[W * V + ``1``];` `        ``for` `(``int` `i = ``0``; i < W * V + ``1``; i++)` `            ``B[i] = ``new` `ArrayList();`   `        ``B[``0``].add(src);` `        ``dist[src] = ``0``;`   `        ``int` `idx = ``0``;` `        ``while` `(``true``) {` `            ``// Go sequentially through buckets till one` `            ``// non-empty bucket is found` `            ``while` `(B[idx].size() == ``0` `&& idx < W * V)` `                ``idx++;`   `            ``// If all buckets are empty, we are done.` `            ``if` `(idx == W * V)` `                ``break``;`   `            ``// Take top vertex from bucket and pop it` `            ``int` `u = B[idx].get(``0``);` `            ``B[idx].remove(``0``);`   `            ``// Process all adjacents of extracted vertex 'u'` `            ``// and update their distances if required.` `            ``for` `(Tuple i : adj.get(u)) {` `                ``int` `v = i.v;` `                ``int` `weight = i.w;`   `                ``int` `du = dist[u];` `                ``int` `dv = dist[v];`   `                ``// If there is shorted path to v through u.` `                ``if` `(dv > du + weight) {` `                    ``// updating the distance` `                    ``dist[v] = du + weight;` `                    ``dv = dist[v];`   `                    ``// pushing vertex v into updated` `                    ``// distance's bucket` `                    ``B[dv].add(``0``, v);` `                ``}` `            ``}` `        ``}`   `        ``// Print shortest distances stored in dist[]` `        ``System.out.println(``"Vertex Distance from Source"``);` `        ``for` `(``int` `i = ``0``; i < V; ++i)` `            ``System.out.println(i + ``"\t\t"` `+ dist[i]);` `    ``}`   `    ``static` `class` `Tuple {` `        ``int` `v, w;` `        ``Tuple(``int` `v, ``int` `w)` `        ``{` `            ``this``.v = v;` `            ``this``.w = w;` `        ``}` `    ``}` `    ``public` `static` `void` `main(String[] args)` `    ``{` `        ``// create the graph given in above figure` `        ``int` `V = ``9``;` `        ``Graph g = ``new` `Graph(V);`   `        ``// making above shown graph` `        ``g.AddEdge(``0``, ``1``, ``4``);` `        ``g.AddEdge(``0``, ``7``, ``8``);` `        ``g.AddEdge(``1``, ``2``, ``8``);` `        ``g.AddEdge(``1``, ``7``, ``11``);` `        ``g.AddEdge(``2``, ``3``, ``7``);` `        ``g.AddEdge(``2``, ``8``, ``2``);` `        ``g.AddEdge(``2``, ``5``, ``4``);` `        ``g.AddEdge(``3``, ``4``, ``9``);` `        ``g.AddEdge(``3``, ``5``, ``14``);` `        ``g.AddEdge(``4``, ``5``, ``10``);` `        ``g.AddEdge(``5``, ``6``, ``2``);` `        ``g.AddEdge(``6``, ``7``, ``1``);` `        ``g.AddEdge(``6``, ``8``, ``6``);` `        ``g.AddEdge(``7``, ``8``, ``7``);`   `        ``// maximum weighted edge - 14` `        ``g.shortestPath(``0``, ``14``);` `    ``}` `}`

Output

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

Time Complexity: The time complexity of the above implementation is O(V*W). Here V is the number of vertices in the graph and W is the maximum weight of the edge.

Space Complexity: The space complexity of the above implementation is O(V*W). In the above implementation, we have used a 2D vector to store the distances of each vertex from the source vertex.

Dial’s algorithm is an optimized version of Dijkstra’s algorithm for finding the shortest paths in a graph with non-negative edge weights. It is particularly useful when the range of the edge weights is small, meaning that the weights only take on a small number of distinct values.

### Here are the steps of Dial’s algorithm:

1. Initialize a set of buckets, where each bucket i contains all nodes with a current shortest path estimate of i.
Start with the source node s and set its shortest path estimate to 0.
2. For each node v adjacent to s, add it to the bucket corresponding to its distance from s.
While there are non-empty buckets:
a. Find the non-empty bucket with the smallest index i.
b. Remove the node u with the smallest shortest path estimate from the bucket.
c. For each node v adjacent to u:
3. If the shortest path estimate for v is greater than the new estimate via u, update v’s estimate and move it to the corresponding bucket.
Return the shortest path estimates for all nodes.
4. The idea behind Dial’s algorithm is to avoid scanning all nodes in each iteration of Dijkstra’s algorithm. Instead, we group the nodes into buckets based on their shortest path estimate, and only process the nodes with the smallest estimate in each iteration. This reduces the number of nodes that need to be processed and speeds up the algorithm, especially when the range of the edge weights is small.

1. Dial’s algorithm can be much faster than Dijkstra’s algorithm for graphs with small range weights.
2. It has a time complexity of O(n + m), where n is the number of nodes and m is the number of edges.
3. It is easy to implement and modify from Dijkstra’s algorithm.