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# Sum of all pair shortest paths in a Tree

• Difficulty Level : Easy
• Last Updated : 22 Jun, 2021

Given a weighted undirected graph T consisting of nodes valued [0, N – 1] and an array Edges[] of type {u, v, w} that denotes an edge between vertices u and v having weight w. The task is to find the sum of all pair shortest paths in the given tree.

Examples:

Input: N = 3, Edges[][] = {{0, 2, 15}, {1, 0, 90}}
Output: 210
Explanation:
Sum of weights of path between nodes 0 and 1 = 90
Sum of weights of path between nodes 0 and 2 = 15
Sum of weights of path between nodes 1 and 2 = 105
Hence, sum = 90 + 15 + 105

Input: N = 4, Edges[][] = {{0, 1, 1}, {1, 2, 2}, {2, 3, 3}}
Output: 20
Explanation:
Sum of weights of path between nodes 0 and 1 = 1
Sum of weights of path between nodes 0 and 2 = 3
Sum of weights of path between nodes 0 and 3 = 6
Sum of weights of path between nodes 1 and 2 = 2
Sum of weights of path between nodes 1 and 3 = 5
Sum of weights of path between nodes 2 and 3 = 3
Hence, sum = 1 + 3 + 6 + 2 + 5 + 3 = 20.

Naive Approach: The simplest approach is to find the shortest path between every pair of vertices using the Floyd Warshall Algorithm. After precomputing the cost of the shortest path between every pair of nodes, print the sum of all the shortest paths.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;``#define INF 99999` `// Function that performs the Floyd``// Warshall to find all shortest paths``int` `floyd_warshall(``int``* graph, ``int` `V)``{` `    ``int` `dist[V][V], i, j, k;` `    ``// Initialize the distance matrix``    ``for` `(i = 0; i < V; i++) {``        ``for` `(j = 0; j < V; j++) {``            ``dist[i][j] = *((graph + i * V) + j);``        ``}``    ``}` `    ``for` `(k = 0; k < V; k++) {` `        ``// Pick all vertices as``        ``// source one by one``        ``for` `(i = 0; i < V; i++) {` `            ``// Pick all vertices as``            ``// destination for the``            ``// above picked source``            ``for` `(j = 0; j < V; j++) {` `                ``// If vertex k is on the``                ``// shortest path from i to``                ``// j then update dist[i][j]``                ``if` `(dist[i][k]``                        ``+ dist[k][j]``                    ``< dist[i][j]) {``                    ``dist[i][j]``                        ``= dist[i][k]``                          ``+ dist[k][j];``                ``}``            ``}``        ``}``    ``}` `    ``// Sum the upper diagonal of the``    ``// shortest distance matrix``    ``int` `sum = 0;` `    ``// Traverse the given dist[][]``    ``for` `(i = 0; i < V; i++) {` `        ``for` `(j = i + 1; j < V; j++) {` `            ``// Add the distance``            ``sum += dist[i][j];``        ``}``    ``}` `    ``// Return the final sum``    ``return` `sum;``}` `// Function to generate the tree``int` `sumOfshortestPath(``int` `N, ``int` `E,``                      ``int` `edges[])``{``    ``int` `g[N][N];``    ``for` `(``int` `i = 0; i < N; i++) {``        ``for` `(``int` `j = 0; j < N; j++) {``            ``g[i][j] = INF;``        ``}``    ``}` `    ``// Add edges``    ``for` `(``int` `i = 0; i < E; i++) {` `        ``// Get source and destination``        ``// with weight``        ``int` `u = edges[i];``        ``int` `v = edges[i];``        ``int` `w = edges[i];` `        ``// Add the edges``        ``g[u][v] = w;``        ``g[v][u] = w;``    ``}` `    ``// Perform Floyd Warshal Algorithm``    ``return` `floyd_warshall((``int``*)g, N);``}` `// Driver code``int` `main()``{``    ``// Number of Vertices``    ``int` `N = 4;` `    ``// Number of Edges``    ``int` `E = 3;` `    ``// Given Edges with weight``    ``int` `Edges[]``        ``= { { 0, 1, 1 }, { 1, 2, 2 },``            ``{ 2, 3, 3 } };` `    ``// Function Call``    ``cout << sumOfshortestPath(N, E, Edges);` `    ``return` `0;``}`

## Java

 `// Java program for``// the above approach``class` `GFG{``  ` `static` `final` `int` `INF = ``99999``;` `// Function that performs the Floyd``// Warshall to find all shortest paths``static` `int` `floyd_warshall(``int``[][] graph,``                          ``int` `V)``{``  ``int` `[][]dist = ``new` `int``[V][V];``  ``int` `i, j, k;` `  ``// Initialize the distance matrix``  ``for` `(i = ``0``; i < V; i++)``  ``{``    ``for` `(j = ``0``; j < V; j++)``    ``{``      ``dist[i][j] = graph[i][j];``    ``}``  ``}` `  ``for` `(k = ``0``; k < V; k++)``  ``{``    ``// Pick all vertices as``    ``// source one by one``    ``for` `(i = ``0``; i < V; i++)``    ``{``      ``// Pick all vertices as``      ``// destination for the``      ``// above picked source``      ``for` `(j = ``0``; j < V; j++)``      ``{``        ``// If vertex k is on the``        ``// shortest path from i to``        ``// j then update dist[i][j]``        ``if` `(dist[i][k] + dist[k][j] <``            ``dist[i][j])``        ``{``          ``dist[i][j] = dist[i][k] +``                       ``dist[k][j];``        ``}``      ``}``    ``}``  ``}` `  ``// Sum the upper diagonal of the``  ``// shortest distance matrix``  ``int` `sum = ``0``;` `  ``// Traverse the given dist[][]``  ``for` `(i = ``0``; i < V; i++)``  ``{``    ``for` `(j = i + ``1``; j < V; j++)``    ``{``      ``// Add the distance``      ``sum += dist[i][j];``    ``}``  ``}` `  ``// Return the final sum``  ``return` `sum;``}` `// Function to generate the tree``static` `int` `sumOfshortestPath(``int` `N, ``int` `E,``                             ``int` `edges[][])``{``  ``int` `[][]g = ``new` `int``[N][N];``  ``for` `(``int` `i = ``0``; i < N; i++)``  ``{``    ``for` `(``int` `j = ``0``; j < N; j++)``    ``{``      ``g[i][j] = INF;``    ``}``  ``}` `  ``// Add edges``  ``for` `(``int` `i = ``0``; i < E; i++)``  ``{``    ``// Get source and destination``    ``// with weight``    ``int` `u = edges[i][``0``];``    ``int` `v = edges[i][``1``];``    ``int` `w = edges[i][``2``];` `    ``// Add the edges``    ``g[u][v] = w;``    ``g[v][u] = w;``  ``}` `  ``// Perform Floyd Warshal Algorithm``  ``return` `floyd_warshall(g, N);``}` `// Driver code``public` `static` `void` `main(String[] args)``{``  ``// Number of Vertices``  ``int` `N = ``4``;` `  ``// Number of Edges``  ``int` `E = ``3``;` `  ``// Given Edges with weight``  ``int` `Edges[][] = {{``0``, ``1``, ``1``}, {``1``, ``2``, ``2``},``                   ``{``2``, ``3``, ``3``}};` `  ``// Function Call``  ``System.out.print(``         ``sumOfshortestPath(N, E, Edges));``}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 program for the above approach``INF ``=` `99999` `# Function that performs the Floyd``# Warshall to find all shortest paths``def` `floyd_warshall(graph, V):` `    ``dist ``=` `[[``0` `for` `i ``in` `range``(V)]``               ``for` `i ``in` `range``(V)]` `    ``# Initialize the distance matrix``    ``for` `i ``in` `range``(V):``        ``for` `j ``in` `range``(V):``            ``dist[i][j] ``=` `graph[i][j]` `    ``for` `k ``in` `range``(V):``        ` `        ``# Pick all vertices as``        ``# source one by one``        ``for` `i ``in` `range``(V):``            ` `            ``# Pick all vertices as``            ``# destination for the``            ``# above picked source``            ``for` `j ``in` `range``(V):``                ` `                ``# If vertex k is on the``                ``# shortest path from i to``                ``# j then update dist[i][j]``                ``if` `(dist[i][k] ``+` `dist[k][j] < dist[i][j]):``                    ``dist[i][j] ``=` `dist[i][k] ``+` `dist[k][j]` `    ``# Sum the upper diagonal of the``    ``# shortest distance matrix``    ``sum` `=` `0` `    ``# Traverse the given dist[][]``    ``for` `i ``in` `range``(V):``        ``for` `j ``in` `range``(i ``+` `1``, V):``            ` `            ``# Add the distance``            ``sum` `+``=` `dist[i][j]` `    ``# Return the final sum``    ``return` `sum` `# Function to generate the tree``def` `sumOfshortestPath(N, E,edges):``    ` `    ``g ``=` `[[INF ``for` `i ``in` `range``(N)]``              ``for` `i ``in` `range``(N)]``              ` `    ``# Add edges``    ``for` `i ``in` `range``(E):``        ` `        ``# Get source and destination``        ``# with weight``        ``u ``=` `edges[i][``0``]``        ``v ``=` `edges[i][``1``]``        ``w ``=` `edges[i][``2``]` `        ``# Add the edges``        ``g[u][v] ``=` `w``        ``g[v][u] ``=` `w` `    ``# Perform Floyd Warshal Algorithm``    ``return` `floyd_warshall(g, N)` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``# Number of Vertices``    ``N ``=` `4` `    ``# Number of Edges``    ``E ``=` `3` `    ``# Given Edges with weight``    ``Edges ``=` `[ [ ``0``, ``1``, ``1` `],``              ``[ ``1``, ``2``, ``2` `],``              ``[ ``2``, ``3``, ``3` `] ]` `    ``# Function Call``    ``print``(sumOfshortestPath(N, E, Edges))` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program for``// the above approach``using` `System;``class` `GFG{``  ` `static` `readonly` `int` `INF = 99999;` `// Function that performs the Floyd``// Warshall to find all shortest paths``static` `int` `floyd_warshall(``int``[,] graph,``                          ``int` `V)``{``  ``int` `[,]dist = ``new` `int``[V, V];``  ``int` `i, j, k;` `  ``// Initialize the distance matrix``  ``for` `(i = 0; i < V; i++)``  ``{``    ``for` `(j = 0; j < V; j++)``    ``{``      ``dist[i, j] = graph[i, j];``    ``}``  ``}` `  ``for` `(k = 0; k < V; k++)``  ``{``    ``// Pick all vertices as``    ``// source one by one``    ``for` `(i = 0; i < V; i++)``    ``{``      ``// Pick all vertices as``      ``// destination for the``      ``// above picked source``      ``for` `(j = 0; j < V; j++)``      ``{``        ``// If vertex k is on the``        ``// shortest path from i to``        ``// j then update dist[i,j]``        ``if` `(dist[i, k] + dist[k, j] <``            ``dist[i, j])``        ``{``          ``dist[i, j] = dist[i, k] +``                       ``dist[k, j];``        ``}``      ``}``    ``}``  ``}` `  ``// Sum the upper diagonal of the``  ``// shortest distance matrix``  ``int` `sum = 0;` `  ``// Traverse the given dist[,]``  ``for` `(i = 0; i < V; i++)``  ``{``    ``for` `(j = i + 1; j < V; j++)``    ``{``      ``// Add the distance``      ``sum += dist[i, j];``    ``}``  ``}` `  ``// Return the readonly sum``  ``return` `sum;``}` `// Function to generate the tree``static` `int` `sumOfshortestPath(``int` `N, ``int` `E,``                             ``int` `[,]edges)``{``  ``int` `[,]g = ``new` `int``[N, N];``  ` `  ``for` `(``int` `i = 0; i < N; i++)``  ``{``    ``for` `(``int` `j = 0; j < N; j++)``    ``{``      ``g[i, j] = INF;``    ``}``  ``}` `  ``// Add edges``  ``for` `(``int` `i = 0; i < E; i++)``  ``{``    ``// Get source and destination``    ``// with weight``    ``int` `u = edges[i, 0];``    ``int` `v = edges[i, 1];``    ``int` `w = edges[i, 2];` `    ``// Add the edges``    ``g[u, v] = w;``    ``g[v, u] = w;``  ``}` `  ``// Perform Floyd Warshal Algorithm``  ``return` `floyd_warshall(g, N);``}` `// Driver code``public` `static` `void` `Main(String[] args)``{``  ``// Number of Vertices``  ``int` `N = 4;` `  ``// Number of Edges``  ``int` `E = 3;` `  ``// Given Edges with weight``  ``int` `[,]Edges = {{0, 1, 1},``                  ``{1, 2, 2},``                  ``{2, 3, 3}};` `  ``// Function Call``  ``Console.Write(sumOfshortestPath(N,``                                  ``E, Edges));``}``}` `// This code is contributed by 29AjayKumar`

## Javascript

 ``
Output:
`20`

Time Complexity:O(N3), where N is the number of vertices.
Auxiliary Space: O(N)

Efficient Approach: The idea is to use the DFS algorithm, using the DFS, for each vertex, the cost to visit every other vertex from this vertex can be found in linear time. Follow the below steps to solve the problem:

1. Traverse the nodes 0 to N – 1.
2. For each node i, find the sum of the cost to visit every other vertex using DFS where the source will be node i, and let’s denote this sum by Si.
3. Now, calculate S = S0 + S1 + … + SN-1. and divide S by 2 because every path is calculated twice.
4. After completing the above steps, print the value of sum S obtained.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include``using` `namespace` `std;` `// Function that performs the DFS``// traversal to find cost to reach``// from vertex v to other vertexes``void` `dfs(``int` `v, ``int` `p,``         ``vector> t[],``         ``int` `h, ``int` `ans[])``{``    ` `    ``// Traverse the Adjacency list``    ``// of u``    ``for``(pair<``int``, ``int``> u : t[v])``    ``{``        ``if` `(u.first == p)``            ``continue``;``            ` `        ``// Recursive Call``        ``dfs(u.first, v, t, h + u.second, ans);``    ``}``    ` `    ``// Update ans[v]``    ``ans[v] = h;``}` `// Function to find the sum of``// weights of all paths``int` `solve(``int` `n, ``int` `edges[])``{``    ` `    ``// Stores the Adjacency List``    ``vector> t[n];` `    ``// Store the edges``    ``for``(``int` `i = 0; i < n - 1; i++)``    ``{``        ``t[edges[i]].push_back({edges[i],``                                  ``edges[i]});` `        ``t[edges[i]].push_back({edges[i],``                                  ``edges[i]});``    ``}` `    ``// sum is the answer``    ``int` `sum = 0;` `    ``// Calculate sum for each vertex``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``int` `ans[n];``        ` `        ``// Perform the DFS Traversal``        ``dfs(i, -1, t, 0, ans);` `        ``// Sum of distance``        ``for``(``int` `j = 0; j < n; j++)``            ``sum += ans[j];``    ``}``    ` `    ``// Return the final sum``    ``return` `sum / 2;``}` `// Driver Code``int` `main()``{``    ` `    ``// No of vertices``    ``int` `N = 4;` `    ``// Given Edges``    ``int` `edges[] = { { 0, 1, 1 },``                       ``{ 1, 2, 2 },``                       ``{ 2, 3, 3 } };` `    ``// Function Call``    ``cout << solve(N, edges) << endl;``    ` `    ``return` `0;``}` `// This code is contributed by pratham76`

## Java

 `// Java program for the above approach` `import` `java.io.*;``import` `java.awt.*;``import` `java.io.*;``import` `java.util.*;` `@SuppressWarnings``(``"unchecked"``)``class` `GFG {` `    ``// Function that performs the DFS``    ``// traversal to find cost to reach``    ``// from vertex v to other vertexes``    ``static` `void` `dfs(``int` `v, ``int` `p,``                    ``ArrayList t[],``                    ``int` `h, ``int` `ans[])``    ``{` `        ``// Traverse the Adjacency list``        ``// of u``        ``for` `(Point u : t[v]) {``            ``if` `(u.x == p)``                ``continue``;` `            ``// Recursive Call``            ``dfs(u.x, v, t, h + u.y, ans);``        ``}` `        ``// Update ans[v]``        ``ans[v] = h;``    ``}` `    ``// Function to find the sum of``    ``// weights of all paths``    ``static` `int` `solve(``int` `n, ``int` `edges[][])``    ``{` `        ``// Stores the Adjacency List``        ``ArrayList t[]``            ``= ``new` `ArrayList[n];` `        ``for` `(``int` `i = ``0``; i < n; i++)``            ``t[i] = ``new` `ArrayList<>();` `        ``// Store the edges``        ``for` `(``int` `i = ``0``; i < n - ``1``; i++) {` `            ``t[edges[i][``0``]].add(``                ``new` `Point(edges[i][``1``],``                          ``edges[i][``2``]));` `            ``t[edges[i][``1``]].add(``                ``new` `Point(edges[i][``0``],``                          ``edges[i][``2``]));``        ``}` `        ``// sum is the answer``        ``int` `sum = ``0``;` `        ``// Calculate sum for each vertex``        ``for` `(``int` `i = ``0``; i < n; i++) {` `            ``int` `ans[] = ``new` `int``[n];` `            ``// Perform the DFS Traversal``            ``dfs(i, -``1``, t, ``0``, ans);` `            ``// Sum of distance``            ``for` `(``int` `j = ``0``; j < n; j++)``                ``sum += ans[j];``        ``}` `        ``// Return the final sum``        ``return` `sum / ``2``;``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``// No of vertices``        ``int` `N = ``4``;` `        ``// Given Edges``        ``int` `edges[][]``            ``= ``new` `int``[][] { { ``0``, ``1``, ``1` `},``                            ``{ ``1``, ``2``, ``2` `},``                            ``{ ``2``, ``3``, ``3` `} };` `        ``// Function Call``        ``System.out.println(solve(N, edges));``    ``}``}`

## Python3

 `# Python3 program for the above approach` `# Function that performs the DFS``# traversal to find cost to reach``# from vertex v to other vertexes``def` `dfs(v, p, t, h, ans):` `    ``# Traverse the Adjacency list``    ``# of u``    ``for` `u ``in` `t[v]:``        ``if` `(u[``0``] ``=``=` `p):``            ``continue``        ` `        ``# Recursive Call``        ``dfs(u[``0``], v, t, h ``+` `u[``1``], ans)` `    ``# Update ans[v]``    ``ans[v] ``=` `h` `# Function to find the sum of``# weights of all paths``def` `solve(n, edges):`` ` `    ``# Stores the Adjacency List``    ``t ``=` `[[] ``for` `i ``in` `range``(n)]``    ` `    ``# Store the edges``    ``for` `i ``in` `range``(n ``-` `1``):``        ``t[edges[i][``0``]].append([edges[i][``1``],``                               ``edges[i][``2``]])``        ``t[edges[i][``1``]].append([edges[i][``0``],``                               ``edges[i][``2``]])` `    ``# sum is the answer``    ``sum` `=` `0`` ` `    ``# Calculate sum for each vertex``    ``for` `i ``in` `range``(n):``        ``ans ``=` `[``0` `for` `i ``in` `range``(n)]`` ` `        ``# Perform the DFS Traversal``        ``dfs(i, ``-``1``, t, ``0``, ans)`` ` `        ``# Sum of distance``        ``for` `j ``in` `range``(n):``            ``sum` `+``=` `ans[j]`` ` `    ``# Return the final sum``    ``return` `sum` `/``/` `2``    ` `# Driver Code``if` `__name__ ``=``=` `"__main__"``:``   ` `    ``# No of vertices``    ``N ``=` `4`` ` `    ``# Given Edges``    ``edges ``=` `[ [ ``0``, ``1``, ``1` `],``              ``[ ``1``, ``2``, ``2` `],``              ``[ ``2``, ``3``, ``3` `] ]`` ` `    ``# Function Call``    ``print``(solve(N, edges))` `# This code is contributed by rutvik_56`

## C#

 `// C# program for the above approach``using` `System;``using` `System.Collections.Generic;` `class` `GFG{` `// Function that performs the DFS``// traversal to find cost to reach``// from vertex v to other vertexes``static` `void` `dfs(``int` `v, ``int` `p,``                ``List> []t,``                ``int` `h, ``int` `[]ans)``{``    ` `    ``// Traverse the Adjacency list``    ``// of u``    ``foreach``(Tuple<``int``, ``int``> u ``in` `t[v])``    ``{``        ``if` `(u.Item1 == p)``            ``continue``;` `        ``// Recursive call``        ``dfs(u.Item1, v, t,``        ``h + u.Item2, ans);``    ``}` `    ``// Update ans[v]``    ``ans[v] = h;``}` `// Function to find the sum of``// weights of all paths``static` `int` `solve(``int` `n, ``int` `[,]edges)``{``    ` `    ``// Stores the Adjacency List``    ``List> []t = ``new` `List>[n];` `    ``for``(``int` `i = 0; i < n; i++)``        ``t[i] = ``new` `List>();` `    ``// Store the edges``    ``for``(``int` `i = 0; i < n - 1; i++)``    ``{``        ``t[edges[i, 0]].Add(``            ``new` `Tuple<``int``, ``int``>(edges[i, 1],``                                ``edges[i, 2]));` `        ``t[edges[i, 1]].Add(``            ``new` `Tuple<``int``, ``int``>(edges[i, 0],``                                ``edges[i, 2]));``    ``}` `    ``// sum is the answer``    ``int` `sum = 0;` `    ``// Calculate sum for each vertex``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ``int` `[]ans = ``new` `int``[n];` `        ``// Perform the DFS Traversal``        ``dfs(i, -1, t, 0, ans);` `        ``// Sum of distance``        ``for``(``int` `j = 0; j < n; j++)``            ``sum += ans[j];``    ``}` `    ``// Return the readonly sum``    ``return` `sum / 2;``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{``    ` `    ``// No of vertices``    ``int` `N = 4;` `    ``// Given Edges``    ``int` `[,]edges = ``new` `int``[,] { { 0, 1, 1 },``                                ``{ 1, 2, 2 },``                                ``{ 2, 3, 3 } };` `    ``// Function call``    ``Console.WriteLine(solve(N, edges));``}``}` `// This code is contributed by Amit Katiyar`

## Javascript

 ``
Output:
`20`

Time Complexity: O(N2), where N is the number of vertices.
Auxiliary Space: O(N)

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