Shortest distance between two nodes in Graph by reducing weight of an edge by half

Given a weighted undirected graph consisting of N nodes and M edges, the task is to find the shortest distance between two nodes A and B by reducing the weight of one edge by half.

Examples:

Input: A = 0, B = 2, Below is the graph

Output: 8
Explanation:
After reducing the weight of the edge connecting 1 and 2 by half modifies its new weight to 4. Now, the shortest distance to reach 2 from 0 through the path 0 -> 1 -> 2 is (4 + 4) = 8.
Therefore, print 8.

Approach: The given problem can be solved by maintaining two arrays, the shortest distance array taking source node as A which will store the shortest distance of all nodes from A, and similarly the shortest distance array taking source node as B. These arrays can be calculated using Dijkstra’s algorithm. Follow the below steps to solve the above problem:

• Use Dijkstra’s algorithm to store the shortest distance of each node from A into an array disA[].
• Use Dijkstra’s algorithm to store the shortest distance of each node from B into an array disB[].
• Suppose edge_i = {u_i, v_i, wt_i} i.e., edge_i connects node u_i to v_i and has a weight of wt_i.
• Now, iterate over all edges and for every edge keep track of the function as:

f(edge_i) = min( disA[u_i] + disB[v_i], disA[v_i] + disB[u_i]) + (wt_i/2).

• The above relation gives the minimum value of f(edge), which is the resultant shortest distance.

Below is the implementation of the above approach:

C++14

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Stores the input Graph
vector<pair<int, int> > graph[100001];

// Stores edges of input Graph
vector<vector<int> > edges;

// Function to input Edges
void add_edge(int u, int v, int w)
{
    graph[u].push_back({ v, w });
    graph[v].push_back({ u, w });
    edges.push_back({ u, v, w });
}

// Function to find the shortest distance
// to each node from the src node using
// Dijkstra’s Algorithm
vector<int> dijsktras(int src, int N)
{
    // Stores the shortest distance of
    // each node form src node
    vector<int> dis(N, INT_MAX);

    vector<bool> vis(N, false);

    // Stores the node and current
    // minimum distance in a heap
    priority_queue<pair<int, int>,
                   vector<pair<int, int> >,
                   greater<pair<int, int> > >
        pq;
    pq.push({ 0, src });
    dis[src] = 0;

    // BFS for single source shortest
    // path algorithm
    while (!pq.empty()) {

        // Top of the PQ
        auto cur = pq.top();
        pq.pop();

        // Store the node and weight
        int node = cur.second;
        int weight = cur.first;

        // If node is already visited
        if (vis[node])
            continue;
        vis[node] = true;

        // Traverse the adjacency list
        // of the node
        for (auto child : graph[node]) {

            // If the distance obtained
            // from parent is less than
            // the current minimum
            // distance stored for child
            if (dis[child.first]
                > child.second + weight) {
                dis[child.first] = weight
                                   + child.second;

                // Push the next pair
                // in the PQ
                pq.push({ dis[child.first],
                          child.first });
            }
        }
    }

    // Return the maximum distance
    return dis;
}

// Function to find shortest distance
// between two nodes by reducing any
// one weight of an edge by half
int shortestDistance(int N, int M,
                     int A, int B)
{
    // Stores the shortest distance
    // of each node from A
    vector<int> disA = dijsktras(A, N);

    // Stores the shortest distance
    // of each node from B
    vector<int> disB = dijsktras(B, N);

    int ans = disA[B];
    for (auto edge : edges) {
        int u = edge[0], v = edge[1];
        int weight = edge[2];

        // Calculate the value of f(edge)
        // for the current edge
        int cur = min(disA[u] + disB[v],
                      disA[v] + disB[u])
                  + (weight / 2);

        // Keep track of the minimum of
        // f(edge) for all edges
        ans = min(ans, cur);
    }

    // Return Answer
    return ans;
}

// Driver Code
int main()
{
    int N = 9, M = 14, A = 0, B = 2;

    // Create a Graph
    add_edge(0, 1, 4);
    add_edge(1, 2, 8);
    add_edge(2, 3, 7);
    add_edge(3, 4, 9);
    add_edge(4, 5, 10);
    add_edge(5, 6, 2);
    add_edge(6, 7, 1);
    add_edge(7, 0, 8);
    add_edge(1, 7, 11);
    add_edge(7, 8, 7);
    add_edge(2, 8, 2);
    add_edge(6, 8, 6);
    add_edge(2, 5, 4);
    add_edge(3, 5, 14);

    // Function Call
    cout << shortestDistance(N, M, A, B);

    return 0;
}

Java

// jav  program for the above approach
import java.util.*;
// Stores the input Graph
public class ShortestPathWithHalfEdgeWeight {
// Stores edges of input Graph
    static List<List<Pair>> graph = new ArrayList<>();
    static List<List<Integer>> edges = new ArrayList<>();

    static void addEdge(int u, int v, int w) {
        graph.get(u).add(new Pair(v, w));
        graph.get(v).add(new Pair(u, w));
        edges.add(Arrays.asList(u, v, w));
      // Function to find the shortest distance
// to each node from the src node using
// Dijkstra’s Algorithm
    }
 // Stores the shortest distance of
    // each node form src node
    static class Pair implements Comparable<Pair> {
        int first, second;
// Stores the node and current
    // minimum distance in a heap
        Pair(int f, int s) {
            first = f;
            second = s;
        }
// If the distance obtained
            // from parent is less than
            // the current minimum
            // distance stored for child
        public int compareTo(Pair o) {
            return Integer.compare(first, o.first);
        }
    }
  // Function to find shortest distance
// between two nodes by reducing any
// one weight of an edge by half

    static List<Integer> dijkstras(int src, int N) {
        List<Integer> dis = new ArrayList<>(Collections.nCopies(N, Integer.MAX_VALUE));
        List<Boolean> vis = new ArrayList<>(Collections.nCopies(N, false));
        PriorityQueue<Pair> pq = new PriorityQueue<>();

        pq.add(new Pair(0, src));
        dis.set(src, 0);
 // Stores the shortest distance
    // of each node from B
        while (!pq.isEmpty()) {
            Pair cur = pq.poll();
            int node = cur.second, weight = cur.first;
            if (vis.get(node))
                continue;
            vis.set(node, true);

            for (Pair child : graph.get(node)) {
                if (dis.get(child.first) > child.second + weight) {
                    dis.set(child.first, weight + child.second);
                    pq.add(new Pair(dis.get(child.first), child.first));
                }
            }
        }

        return dis;
    }

    static int shortestDistance(int N, int M, int A, int B) {
        List<Integer> disA = dijkstras(A, N);
        List<Integer> disB = dijkstras(B, N);

        int ans = disA.get(B);
        for (List<Integer> edge : edges) {
            int u = edge.get(0), v = edge.get(1), weight = edge.get(2);
            int cur = Math.min(disA.get(u) + disB.get(v), disA.get(v) + disB.get(u)) + (weight / 2);
            ans = Math.min(ans, cur);
        }

        return ans;
    }
// Driver Code
    public static void main(String[] args) {
        int N = 9, M = 14, A = 0, B = 2;

        for (int i = 0; i < N; i++) {
            graph.add(new ArrayList<>());
        }
// Create a Graph
        addEdge(0, 1, 4);
        addEdge(1, 2, 8);
        addEdge(2, 3, 7);
        addEdge(3, 4, 9);
        addEdge(4, 5, 10);
        addEdge(5, 6, 2);
        addEdge(6, 7, 1);
        addEdge(7, 0, 8);
        addEdge(1, 7, 11);
        addEdge(7, 8, 7);
        addEdge(2, 8, 2);
        addEdge(6, 8, 6);
        addEdge(2, 5, 4);
        addEdge(3, 5, 14);
 // Function Call
        System.out.println(shortestDistance(N, M, A, B));
    }
}

Python

import heapq
import sys

# Stores the input Graph
graph = [[] for i in range(100001)]

# Stores edges of input Graph
edges = []

# Function to input Edges


def add_edge(u, v, w):
    graph[u].append((v, w))
    graph[v].append((u, w))
    edges.append((u, v, w))

# Function to find the shortest distance
# to each node from the src node using
# Dijkistra Algorithm



def dijsktras(src, N):
    # Stores the shortest distance of
    # each node form src node
    dis = [sys.maxsize] * N

    vis = [False] * N

    # Stores the node and current
    # minimum distance in a heap
    pq = [(0, src)]
    dis[src] = 0

    # BFS for single source shortest
    # path algorithm
    while pq:

        # Top of the PQ
        weight, node = heapq.heappop(pq)

        # If node is already visited
        if vis[node]:
            continue
        vis[node] = True

        # Traverse the adjacency list
        # of the node
        for child, child_weight in graph[node]:

            # If the distance obtained
            # from parent is less than
            # the current minimum
            # distance stored for child
            if dis[child] > child_weight + weight:
                dis[child] = weight + child_weight

                # Push the next pair
                # in the PQ
                heapq.heappush(pq, (dis[child], child))

    # Return the maximum distance
    return dis

# Function to find shortest distance
# between two nodes by reducing any
# one weight of an edge by half


def shortest_distance(N, M, A, B):
    # Stores the shortest distance
    # of each node from A
    disA = dijsktras(A, N)

    # Stores the shortest distance
    # of each node from B
    disB = dijsktras(B, N)

    ans = disA[B]
    for u, v, weight in edges:

        # Calculate the value of f(edge)
        # for the current edge
        cur = min(disA[u] + disB[v], disA[v] + disB[u]) + (weight // 2)

        # Keep track of the minimum of
        # f(edge) for all edges
        ans = min(ans, cur)

    # Return Answer
    return ans


# Driver Code
if __name__ == "__main__":
    N, M, A, B = 9, 14, 0, 2

    # Create a Graph
    add_edge(0, 1, 4)
    add_edge(1, 2, 8)
    add_edge(2, 3, 7)
    add_edge(3, 4, 9)
    add_edge(4, 5, 10)
    add_edge(5, 6, 2)
    add_edge(6, 7, 1)
    add_edge(7, 0, 8)
    add_edge(1, 7, 11)
    add_edge(7, 8, 7)
    add_edge(2, 8, 2)
    add_edge(6, 8, 6)
    add_edge(2, 5, 4)
    add_edge(3, 5, 14)

    # Function Call
    print(shortest_distance(N, M, A, B))

C#

using System;
using System.Collections.Generic;

namespace ConsoleApp1
{
    class Program
    {
        static void Main(string[] args)
        {
            int N = 9, M = 14, A = 0, B = 2;
            var graph = new List<Tuple<int, int, int>>();
            graph.Add(new Tuple<int, int, int>(0, 1, 4));
            graph.Add(new Tuple<int, int, int>(1, 2, 8));
            graph.Add(new Tuple<int, int, int>(2, 3, 7));
            graph.Add(new Tuple<int, int, int>(3, 4, 9));
            graph.Add(new Tuple<int, int, int>(4, 5, 10));
            graph.Add(new Tuple<int, int, int>(5, 6, 2));
            graph.Add(new Tuple<int, int, int>(6, 7, 1));
            graph.Add(new Tuple<int, int, int>(7, 0, 8));
            graph.Add(new Tuple<int, int, int>(1, 7, 11));
            graph.Add(new Tuple<int, int, int>(7, 8, 7));
            graph.Add(new Tuple<int, int, int>(2, 8, 2));
            graph.Add(new Tuple<int, int, int>(6, 8, 6));
            graph.Add(new Tuple<int, int, int>(2, 5, 4));
            graph.Add(new Tuple<int, int, int>(3, 5, 14));

            Console.WriteLine(ShortestDistance(graph, N, M, A, B));
            Console.ReadLine();
        }

        private static int ShortestDistance(List<Tuple<int, int, int>> graph, int N, int M, int A, int B)
        {
            var distA = Dijkstras(graph, N, A);
            var distB = Dijkstras(graph, N, B);

            int ans = distA[B];
            foreach (var edge in graph)
            {
                int u = edge.Item1;
                int v = edge.Item2;
                int weight = edge.Item3;

                int cur = Math.Min(distA[u] + distB[v], distA[v] + distB[u]) + (weight / 2);
                ans = Math.Min(ans, cur);
            }

            return ans;
        }

        private static int[] Dijkstras(List<Tuple<int, int, int>> graph, int N, int src)
        {
            var dis = new int[N];
            for(int i=0; i<N; i++)
            {
                dis[i] = Int32.MaxValue;
            }

            var vis = new bool[N];

            var pq = new SortedSet<Tuple<int, int>>(Comparer<Tuple<int, int>>.Create((x, y) => x.Item1.CompareTo(y.Item1)));
            pq.Add(new Tuple<int, int>(0, src));
            dis[src] = 0;

            while(pq.Count > 0)
            {
                var cur = pq.Min;
                pq.Remove(cur);
                int node = cur.Item2;
                int weight = cur.Item1;

                if (vis[node])
                    continue;
                vis[node] = true;

                foreach (var child in graph)
                {
                    if(child.Item1 == node || child.Item2 == node)
                    {
                        int adjNode = child.Item1 == node ? child.Item2 : child.Item1;

                        if (dis[adjNode] > child.Item3 + weight)
                        {
                            dis[adjNode] = weight + child.Item3;

                            pq.Add(new Tuple<int, int>(dis[adjNode], adjNode));
                        }
                    }
                }
            }

            return dis;
        }
    }
}

Javascript

// Stores the input Graph
const graph = [];
const edges = [];

function addEdge(u, v, w) {
  graph[u] = graph[u] || [];
  graph[v] = graph[v] || [];
  graph[u].push([v, w]);
  graph[v].push([u, w]);
  edges.push([u, v, w]);
}

// Function to find the shortest distance
// to each node from the src node using
// Dijkstra’s Algorithm
class Pair {
  constructor(first, second) {
    this.first = first;
    this.second = second;
  }

  // If the distance obtained from parent is less than
  // the current minimum distance stored for child
  compareTo(o) {
    return this.first - o.first;
  }
}

// Function to find shortest distance between two nodes by reducing any
// one weight of an edge by half
function dijkstras(src, N) {
  const dis = new Array(N).fill(Number.MAX_SAFE_INTEGER);
  const vis = new Array(N).fill(false);
  const pq = [];

  pq.push(new Pair(0, src));
  dis[src] = 0;

  // Stores the shortest distance of each node from B
  while (pq.length > 0) {
    const cur = pq.shift();
    const node = cur.second;
    const weight = cur.first;
    if (vis[node]) continue;
    vis[node] = true;

    for (const child of graph[node]) {
      const [v, w] = child;
      if (dis[v] > w + weight) {
        dis[v] = w + weight;
        pq.push(new Pair(dis[v], v));
      }
    }
  }

  return dis;
}

function shortestDistance(N, M, A, B) {
  const disA = dijkstras(A, N);
  const disB = dijkstras(B, N);

  let ans = disA[B];
  for (const edge of edges) {
    const [u, v, weight] = edge;
    const cur = Math.min(disA[u] + disB[v], disA[v] + disB[u]) + Math.floor(weight / 2);
    ans = Math.min(ans, cur);
  }

  return ans;
}

// Create a Graph
addEdge(0, 1, 4);
addEdge(1, 2, 8);
addEdge(2, 3, 7);
addEdge(3, 4, 9);
addEdge(4, 5, 10);
addEdge(5, 6, 2);
addEdge(6, 7, 1);
addEdge(7, 0, 8);
addEdge(1, 7, 11);
addEdge(7, 8, 7);
addEdge(2, 8, 2);
addEdge(6, 8, 6);
addEdge(2, 5, 4);
addEdge(3, 5, 14);

// Function Call
const N = 9,
  M = 14,
  A = 0,
  B = 2;
console.log(shortestDistance(N, M, A, B));

Output:

8

Time Complexity: O(M*log N)
Auxiliary Space:O(N + M)