Multistage Graph (Shortest Path)

A Multistage graph is a directed graph in which the nodes can be divided into a set of stages such that all edges are from a stage to next stage only (In other words there is no edge between vertices of same stage and from a vertex of current stage to previous stage).

We are give a multistage graph, a source and a destination, we need to find shortest path from source to destination. By convention, we consider source at stage 1 and destination as last stage.

Following is an example graph we will consider in this article :-

Now there are various strategies we can apply :-

  • The Brute force method of finding all possible paths between Source and Destination and then finding the minimum. That’s the WORST possible strategy.
  • Dijkstra’s Algorithm of Single Source shortest paths. This method will find shortest paths from source to all other nodes which is not required in this case. So it will take a lot of time and it doesn’t even use the SPECIAL feature that this MULTI-STAGE graph has.
  • Simple Greedy Method – At each node, choose the shortest outgoing path. If we apply this approach to the example graph give above we get the solution as 1 + 4 + 18 = 23. But a quick look at the graph will show much shorter paths available than 23. So the greedy method fails !
  • The best option is Dynamic Programming. So we need to find Optimal Sub-structure, Recursive Equations and Overlapping Sub-problems.

Optimal Substructure and Recursive Equation :-

We define the notation :- M(x, y) as the minimum cost to T(target node) from Stage x, Node y.



Shortest distance from stage 1, node 0 to 
destination, i.e., 7 is M(1, 0).

// From 0, we can go to 1 or 2 or 3 to
// reach 7.              
M(1, 0) = min(1 + M(2, 1),
              2 + M(2, 2),
              5 + M(2, 3))

This means that our problem of 0 —> 7 is now sub-divided into 3 sub-problems :-

So if we have total 'n' stages and target
as T, then the stopping condition  will be :-
M(n-1, i) = i ---> T + M(n, T) = i ---> T

Recursion Tree and Overlapping Sub-Problems:-
So, the hierarchy of M(x, y) evaluations will look something like this :-

In M(i, j), i is stage number and
j is node number

                   M(1, 0)
           /          |         \                             
          /           |          \                            
       M(2, 1)      M(2, 2)        M(2, 3)
    /      \        /     \         /    \
M(3, 4)  M(3, 5)  M(3, 4)  M(3, 5) M(3, 6)  M(3, 6)
 .         .       .       .          .        .
 .         .       .       .          .        .
 .         .       .       .          .        .

So, here we have drawn a very small part of the Recursion Tree and we can already see Overlapping Sub-Problems. We can largely reduce the number of M(x, y) evaluations using Dynamic Programming.

Implementation details:
The below implementation assumes that nodes are numbered from 0 to N-1 from first stage (source) to last stage (destination). We also assume that the input graph is multistage.

C++

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// CPP program to find shortest distance
// in a multistage graph.
#include<bits/stdc++.h>
using namespace std;
  
#define N 8
#define INF INT_MAX
  
// Returns shortest distance from 0 to
// N-1.
int shortestDist(int graph[N][N]) {
  
    // dist[i] is going to store shortest
    // distance from node i to node N-1.
    int dist[N];
  
    dist[N-1] = 0;
  
    // Calculating shortest path for
    // rest of the nodes
    for (int i = N-2 ; i >= 0 ; i--)
    {
  
        // Initialize distance from i to
        // destination (N-1)
        dist[i] = INF;
  
        // Check all nodes of next stages
        // to find shortest distance from
        // i to N-1.
        for (int j = i ; j < N ; j++)
        {
            // Reject if no edge exists
            if (graph[i][j] == INF)
                continue;
  
            // We apply recursive equation to
            // distance to target through j.
            // and compare with minimum distance 
            // so far.
            dist[i] = min(dist[i], graph[i][j] +
                                        dist[j]);
        }
    }
  
    return dist[0];
}
  
// Driver code
int main()
{
    // Graph stored in the form of an
    // adjacency Matrix
    int graph[N][N] =
      {{INF, 1, 2, 5, INF, INF, INF, INF},
       {INF, INF, INF, INF, 4, 11, INF, INF},
       {INF, INF, INF, INF, 9, 5, 16, INF},
       {INF, INF, INF, INF, INF, INF, 2, INF},
       {INF, INF, INF, INF, INF, INF, INF, 18},
       {INF, INF, INF, INF, INF, INF, INF, 13},
       {INF, INF, INF, INF, INF, INF, INF, 2}};
  
    cout << shortestDist(graph);
    return 0;
}

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Java

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// Java program to find shortest distance 
// in a multistage graph.
class GFG 
{
  
    static int N = 8;
    static int INF = Integer.MAX_VALUE;
  
    // Returns shortest distance from 0 to 
    // N-1. 
    public static int shortestDist(int[][] graph) 
    {
  
        // dist[i] is going to store shortest 
        // distance from node i to node N-1. 
        int[] dist = new int[N];
  
        dist[N - 1] = 0;
  
        // Calculating shortest path for 
        // rest of the nodes 
        for (int i = N - 2; i >= 0; i--) 
        {
  
            // Initialize distance from i to 
            // destination (N-1) 
            dist[i] = INF;
  
            // Check all nodes of next stages 
            // to find shortest distance from 
            // i to N-1. 
            for (int j = i; j < N; j++) 
            {
                // Reject if no edge exists 
                if (graph[i][j] == INF) 
                {
                    continue;
                }
  
                // We apply recursive equation to 
                // distance to target through j. 
                // and compare with minimum distance 
                // so far. 
                dist[i] = Math.min(dist[i], graph[i][j]
                        + dist[j]);
            }
        }
  
        return dist[0];
    }
  
    // Driver code 
    public static void main(String[] args) 
    {
        // Graph stored in the form of an 
        // adjacency Matrix 
        int[][] graph = new int[][]{{INF, 1, 2, 5, INF, INF, INF, INF},
        {INF, INF, INF, INF, 4, 11, INF, INF},
        {INF, INF, INF, INF, 9, 5, 16, INF},
        {INF, INF, INF, INF, INF, INF, 2, INF},
        {INF, INF, INF, INF, INF, INF, INF, 18},
        {INF, INF, INF, INF, INF, INF, INF, 13},
        {INF, INF, INF, INF, INF, INF, INF, 2}};
  
        System.out.println(shortestDist(graph));
    }
}
  
// This code has been contributed by 29AjayKumar

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Python3

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# Python3 program to find shortest 
# distance in a multistage graph. 
  
# Returns shortest distance from 
# 0 to N-1. 
def shortestDist(graph):
    global INF
  
    # dist[i] is going to store shortest 
    # distance from node i to node N-1. 
    dist = [0] *
  
    dist[N - 1] = 0
  
    # Calculating shortest path 
    # for rest of the nodes 
    for i in range(N - 2, -1, -1):
  
        # Initialize distance from  
        # i to destination (N-1) 
        dist[i] = INF 
  
        # Check all nodes of next stages 
        # to find shortest distance from 
        # i to N-1.
        for j in range(N):
              
            # Reject if no edge exists 
            if graph[i][j] == INF:
                continue
  
            # We apply recursive equation to 
            # distance to target through j. 
            # and compare with minimum 
            # distance so far. 
            dist[i] = min(dist[i], 
                          graph[i][j] + dist[j])
  
    return dist[0]
  
# Driver code 
N = 8
INF = 999999999999
  
# Graph stored in the form of an 
# adjacency Matrix 
graph = [[INF, 1, 2, 5, INF, INF, INF, INF], 
         [INF, INF, INF, INF, 4, 11, INF, INF], 
         [INF, INF, INF, INF, 9, 5, 16, INF], 
         [INF, INF, INF, INF, INF, INF, 2, INF], 
         [INF, INF, INF, INF, INF, INF, INF, 18],
         [INF, INF, INF, INF, INF, INF, INF, 13], 
         [INF, INF, INF, INF, INF, INF, INF, 2]] 
  
print(shortestDist(graph))
  
# This code is contributed by PranchalK

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C#

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// C# program to find shortest distance 
// in a multistage graph.
using System; 
    
class GFG 
    static int N = 8; 
    static int INF = int.MaxValue; 
        
    // Returns shortest distance from 0 to 
    // N-1. 
    public static int shortestDist(int[,] graph) { 
        
        // dist[i] is going to store shortest 
        // distance from node i to node N-1. 
        int[] dist = new int[N]; 
        
        dist[N-1] = 0; 
        
        // Calculating shortest path for 
        // rest of the nodes 
        for (int i = N-2 ; i >= 0 ; i--) 
        
        
            // Initialize distance from i to 
            // destination (N-1) 
            dist[i] = INF; 
        
            // Check all nodes of next stages 
            // to find shortest distance from 
            // i to N-1. 
            for (int j = i ; j < N ; j++) 
            
                // Reject if no edge exists 
                if (graph[i,j] == INF) 
                    continue
        
                // We apply recursive equation to 
                // distance to target through j. 
                // and compare with minimum distance  
                // so far. 
                dist[i] = Math.Min(dist[i], graph[i,j] + 
                                            dist[j]); 
            
        
        
        return dist[0]; 
    
        
    // Driver code 
    static void Main() 
    
        // Graph stored in the form of an 
        // adjacency Matrix 
        int[,] graph = new int[,] 
          {{INF, 1, 2, 5, INF, INF, INF, INF}, 
           {INF, INF, INF, INF, 4, 11, INF, INF}, 
           {INF, INF, INF, INF, 9, 5, 16, INF}, 
           {INF, INF, INF, INF, INF, INF, 2, INF}, 
           {INF, INF, INF, INF, INF, INF, INF, 18}, 
           {INF, INF, INF, INF, INF, INF, INF, 13}, 
           {INF, INF, INF, INF, INF, INF, INF, 2}}; 
        
        Console.Write(shortestDist(graph));
    }
}
  
// This code is contributed by DrRoot_

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Output:

9

Time Complexity : O(n2)

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