Minimum cost to traverse from one index to another in the String

Given a string S of length N consisting of lower case character, the task is to find the minimum cost to reach from index i to index j. 
At any index k, the cost to jump to the index k+1 and k-1(without going out of bounds) is 1. 
Additionally, the cost to jump to any index m such that S[m] = S[k] is 0. 

Examples:

Input : S = “abcde”, i = 0, j = 4 
Output : 4 
Explanation:
The shortest path will be: 
0->1->2->3->4 
Thus, the answer will be 4.

Input : S = “abcdefb”, i = 0, j = 5 
Output : 2 
Explanation:
0->1->6->5 
0->1 edge weight is 1, 1->6 edge weight is 0, and 6->5 edge weight is 1. 
Thus, the answer will be 2  

Approach:

1. One approach to solve this problem is 0-1 BFS.
2. The setup can be visualized as a graph with N nodes.
3. All the nodes will be connected to adjacent nodes with an edge of the weight of ‘1’ and nodes with the same characters with an edge with weight ‘0’.
4. In this setup, 0-1 BFS can be run to find the shortest path from index ‘i’ to index ‘j’.

Time complexity: O(N^2) – As the number of vertices would be of O(N^2)

Efficient Approach: 

1. For each character X, all the characters are found for which it is adjacent.
2. A graph is created with the number of nodes as the number of distinct characters in the string, each representing a character.
3. Each node X will have an edge of weight 1 with all the nodes representing characters adjacent to character X.
4. Then BFS can be run from nodes representing S[i] to nodes representing S[j] in this new graph

Time complexity: O(N)

Below is the implementation of the above approach:  

C++

// C++ implementation of the above approach.
#include <bits/stdc++.h>
using namespace std;

// function to find the minimum cost
int findMinCost(string s, int i, int j)
{
    // graph
    vector<vector<int> > gr(26);

    // adjacency matrix
    bool edge[26][26];

    // initialising adjacency matrix
    for (int k = 0; k < 26; k++)
        for (int l = 0; l < 26; l++)
            edge[k][l] = 0;

    // creating adjacency list
    for (int k = 0; k < s.size(); k++) {
        // pushing left adjacent element for index 'k'
        if (k - 1 >= 0
            and !edge[s[k] - 97][s[k - 1] - 97])
            gr[s[k] - 97].push_back(s[k - 1] - 97),
                edge[s[k] - 97][s[k - 1] - 97] = 1;
        // pushing right adjacent element for index 'k'
        if (k + 1 <= s.size() - 1
            and !edge[s[k] - 97][s[k + 1] - 97])
            gr[s[k] - 97].push_back(s[k + 1] - 97),
                edge[s[k] - 97][s[k + 1] - 97] = 1;
    }

    // queue to perform BFS
    queue<int> q;
    q.push(s[i] - 97);

    // visited array
    bool v[26] = { 0 };

    // variable to store depth of BFS
    int d = 0;

    // BFS
    while (q.size()) {

        // number of elements in the current level
        int cnt = q.size();

        // inner loop
        while (cnt--) {

            // current element
            int curr = q.front();

            // popping queue
            q.pop();

            // base case
            if (v[curr])
                continue;
            v[curr] = 1;

            // checking if the current node is required node
            if (curr == s[j] - 97)
                return d;

            // iterating through the current node
            for (auto it : gr[curr])
                q.push(it);
        }

        // updating depth
        d++;
    }

    return -1;
}

// Driver Code
int main()
{
    // input variables
    string s = "abcde";
    int i = 0;
    int j = 4;

    // function to find the minimum cost
    cout << findMinCost(s, i, j);
}

Java

// Java implementation of the above approach.
import java.util.*;

class GFG
{

    // function to find the minimum cost
    static int findMinCost(char[] s, int i, int j)
    {
        // graph
        Vector<Integer>[] gr = new Vector[26];
        for (int iN = 0; iN < 26; iN++)
            gr[iN] = new Vector<Integer>();

        // adjacency matrix
        boolean[][] edge = new boolean[26][26];

        // initialising adjacency matrix
        for (int k = 0; k < 26; k++)
            for (int l = 0; l < 26; l++)
                edge[k][l] = false;

        // creating adjacency list
        for (int k = 0; k < s.length; k++)
        {
            // pushing left adjacent element for index 'k'
            if (k - 1 >= 0 && !edge[s[k] - 97][s[k - 1] - 97])
            {
                gr[s[k] - 97].add(s[k - 1] - 97);
                edge[s[k] - 97][s[k - 1] - 97] = true;
            }
            // pushing right adjacent element for index 'k'
            if (k + 1 <= s.length - 1 && !edge[s[k] - 97][s[k + 1] - 97])
            {
                gr[s[k] - 97].add(s[k + 1] - 97);
                edge[s[k] - 97][s[k + 1] - 97] = true;
            }
        }

        // queue to perform BFS
        Queue<Integer> q = new LinkedList<Integer>();
        q.add(s[i] - 97);

        // visited array
        boolean[] v = new boolean[26];

        // variable to store depth of BFS
        int d = 0;

        // BFS
        while (q.size() > 0)
        {

            // number of elements in the current level
            int cnt = q.size();

            // inner loop
            while (cnt-- > 0)
            {

                // current element
                int curr = q.peek();

                // popping queue
                q.remove();

                // base case
                if (v[curr])
                    continue;
                v[curr] = true;

                // checking if the current node is required node
                if (curr == s[j] - 97)
                    return d;

                // iterating through the current node
                for (Integer it : gr[curr])
                    q.add(it);
            }

            // updating depth
            d++;
        }

        return -1;
    }

    // Driver Code
    public static void main(String[] args)
    {
        // input variables
        String s = "abcde";
        int i = 0;
        int j = 4;

        // function to find the minimum cost
        System.out.print(findMinCost(s.toCharArray(), i, j));
    }
}

// This code is contributed by 29AjayKumar

Python3

# Python3 implementation of the above approach.
from collections import deque as a queue

# function to find minimum cost
def findMinCost(s, i, j):

    # graph
    gr = [[] for i in range(26)]

    # adjacency matrix
    edge = [[ 0 for i in range(26)] for i in range(26)]

    # initialising adjacency matrix
    for k in range(26):
        for l in range(26):
            edge[k][l] = 0

    # creating adjacency list
    for k in range(len(s)):

        # pushing left adjacent element for index 'k'
        if (k - 1 >= 0 and edge[ord(s[k]) - 97][ord(s[k - 1]) - 97] == 0):
            gr[ord(s[k]) - 97].append(ord(s[k - 1]) - 97)
            edge[ord(s[k]) - 97][ord(s[k - 1]) - 97] = 1

        # pushing right adjacent element for index 'k'
        if (k + 1 <= len(s) - 1 and edge[ord(s[k]) - 97][ord(s[k + 1]) - 97] == 0):
            gr[ord(s[k]) - 97].append(ord(s[k + 1]) - 97)
            edge[ord(s[k]) - 97][ord(s[k + 1]) - 97] = 1

    # queue to perform BFS
    q = queue()
    q.append(ord(s[i]) - 97)

    # visited array
    v = [0] * (26)

    # variable to store depth of BFS
    d = 0

    # BFS
    while (len(q)):

        # number of elements in the current level
        cnt = len(q)

        # inner loop
        while (cnt > 0):

            # current element
            curr = q.popleft()


            # base case
            if (v[curr] == 1):
                continue
            v[curr] = 1

            # checking if the current node is required node
            if (curr == ord(s[j]) - 97):
                return curr

            # iterating through the current node
            for it in gr[curr]:
                q.append(it)
            print()
            cnt -= 1

        # updating depth
        d = d + 1

    return -1

# Driver Code

# input variables
s = "abcde"
i = 0
j = 4

# function to find the minimum cost
print(findMinCost(s, i, j))

# This code is contributed by mohit kumar 29

C#

// C# implementation of the above approach.
using System;
using System.Collections.Generic;

class GFG
{

    // function to find the minimum cost
    static int findMinCost(char[] s, int i, int j)
    {
        // graph
        List<int>[] gr = new List<int>[26];
        for (int iN = 0; iN < 26; iN++)
            gr[iN] = new List<int>();

        // adjacency matrix
        bool[,] edge = new bool[26, 26];

        // initialising adjacency matrix
        for (int k = 0; k < 26; k++)
            for (int l = 0; l < 26; l++)
                edge[k, l] = false;

        // creating adjacency list
        for (int k = 0; k < s.Length; k++)
        {
            // pushing left adjacent element for index 'k'
            if (k - 1 >= 0 && !edge[s[k] - 97, s[k - 1] - 97])
            {
                gr[s[k] - 97].Add(s[k - 1] - 97);
                edge[s[k] - 97, s[k - 1] - 97] = true;
            }

            // pushing right adjacent element for index 'k'
            if (k + 1 <= s.Length - 1 &&
                !edge[s[k] - 97, s[k + 1] - 97])
            {
                gr[s[k] - 97].Add(s[k + 1] - 97);
                edge[s[k] - 97, s[k + 1] - 97] = true;
            }
        }

        // queue to perform BFS
        Queue<int> q = new Queue<int>();
        q.Enqueue(s[i] - 97);

        // visited array
        bool[] v = new bool[26];

        // variable to store depth of BFS
        int d = 0;

        // BFS
        while (q.Count > 0)
        {

            // number of elements in the current level
            int cnt = q.Count;

            // inner loop
            while (cnt-- > 0)
            {

                // current element
                int curr = q.Peek();

                // popping queue
                q.Dequeue();

                // base case
                if (v[curr])
                    continue;
                v[curr] = true;

                // checking if the current node is required node
                if (curr == s[j] - 97)
                    return d;

                // iterating through the current node
                foreach (int it in gr[curr])
                    q.Enqueue(it);
            }

            // updating depth
            d++;
        }

        return -1;
    }

    // Driver Code
    public static void Main(String[] args)
    {
        // input variables
        String s = "abcde";
        int i = 0;
        int j = 4;

        // function to find the minimum cost
        Console.Write(findMinCost(s.ToCharArray(), i, j));
    }
}

// This code is contributed by 29AjayKumar

Javascript

<script>

// Javascript implementation of the above approach.

// function to find the minimum cost
function findMinCost(s,i, j)
{
    // graph
    var gr = Array.from(Array(26), ()=> new Array());

    // adjacency matrix
    var edge = Array.from(Array(26), ()=> Array(26));

    // initialising adjacency matrix
    for (var k = 0; k < 26; k++)
        for (var l = 0; l < 26; l++)
            edge[k][l] = 0;

    // creating adjacency list
    for (var k = 0; k < s.length; k++) {
        // pushing left adjacent element for index 'k'
        if (k - 1 >= 0
            && !edge[s[k].charCodeAt(0) - 97][s[k - 1].charCodeAt(0) - 97])
            gr[s[k].charCodeAt(0) - 97].push(s[k - 1].charCodeAt(0) - 97),
                edge[s[k].charCodeAt(0) - 97][s[k - 1].charCodeAt(0) - 97] = 1;
        // pushing right adjacent element for index 'k'
        if (k + 1 <= s.length - 1
            && !edge[s[k].charCodeAt(0) - 97][s[k + 1].charCodeAt(0) - 97])
            gr[s[k].charCodeAt(0) - 97].push(s[k + 1].charCodeAt(0) - 97),
                edge[s[k].charCodeAt(0) - 97][s[k + 1].charCodeAt(0) - 97] = 1;
    }

    // queue to perform BFS
    var q = [];
    q.push(s[i].charCodeAt(0) - 97);

    // visited array
    var v = Array(26).fill(0);

    // variable to store depth of BFS
    var d = 0;

    // BFS
    while (q.length>0) {

        // number of elements in the current level
        var cnt = q.length;

        // inner loop
        while (cnt-->0) {

            // current element
            var curr = q[0];

            // popping queue
            q.shift();

            // base case
            if (v[curr])
                continue;
            v[curr] = 1;

            // checking if the current node is required node
            if (curr == s[j].charCodeAt(0) - 97)
                return d;

            // iterating through the current node
            for(var it =0 ;it< gr[curr].length; it++)
            {
                q.push(gr[curr][it]);
            }
        }

        // updating depth
        d++;
    }

    return -1;
}

// Driver Code
// input variables
var s = "abcde";
var i = 0;
var j = 4;
// function to find the minimum cost
document.write( findMinCost(s, i, j));


</script>

Output:

4