Minimum number of jumps required to sort numbers placed on a number line

Given two arrays W[] and L[] consisting of N positive integers, where W[i] is initially located at position i on an infinite number line. In each forward jump, W[i] can jump to the position (j + L[i]) from its current position j to any vacant position. The task is to find the minimum number of forward jumps required to sort the array.

Examples:

Input: W[] = {2, 1, 4, 3}, L[] = {4, 1, 2, 4}
Output: 5
Explanation:
Initially, 2 is at position 0, 1 is at position 1, 4 is at position 2, 3 is at position 3 on the number line as: 2 1 4 3
Push number 2 to jump from its current position 0 to position 4, arrangement on the number line: _ 1 4 3 2
Push number 3 to jump from its current position 3 to position 7, arrangement on the number line: _ 1 4 _ 2 _ _ 3
Push number 4 to jump from its current position 2 to position 8, arrangement on the number line: _ 1 _ _ 2 _ _ 3 4

Therefore, the total number of jumps required is 1 + 1 + 3 = 5.

Input: W[] = {3, 1, 2}, L[] = {1, 4, 5}
Output: 3

Approach: The given problem can be solved by using the Greedy Approach by minimizing the number of jumps required for the smallest element in the array which is not correctly positioned in every operation and update the number of jumps. Follow the steps below to solve the problem:

• Initialize a variable, say ans as 0 to store the minimum number of jumps required.
• Create a copy of the array W[] and store the elements in sorted order in array arr[].
• Initialize a HashMap pos that stores the current position of the element W[i].
• Traverse the array W[] and update the position of W[i] in pos.
• Traverse the array arr[] in the range [1, N – 1] and perform the following steps:
  • Store the position of arr[i] and the position of arr[i – 1] in the array W[] in the variables, say curr and prev respectively.
  • If the value of curr is greater than prev, then continue to the next iteration.
  • Otherwise, iterate until curr ? prev or curr is already occupied and increment the value of curr by the jump and increment the value of ans by 1.
  • Update the position of arr[i] in the HashMap pos to curr.
• After completing the above steps, print the value of ans as the result.

Below is the implementation of the above approach:

C++

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

// Function to find the minimum number
// of jumps required to sort the array
void minJumps(int w[], int l[], int n)
{
    // Base Case
    if (n == 1) {
        cout << 0;
        return;
    }

    // Store the required result
    int ans = 0;

    // Stores the current position of
    // elements and their respective
    // maximum jump
    unordered_map<int, int> pos, jump;

    // Used to check if a position is
    // already taken by another element
    unordered_map<int, bool> filled;

    // Stores the sorted array a[]
    int a[n];

    // Traverse the array w[] & update
    // positions jumps array a[]
    for (int i = 0; i < n; i++) {

        pos[w[i]] = i;
        filled[i] = true;
        jump[w[i]] = l[i];
        a[i] = w[i];
    }

    // Sort the array a[]
    sort(a, a + n);

    // Traverse the array a[] over
    // the range [1, N-1]
    for (int curr = 1;
         curr < n; curr++) {

        // Store the index of current
        // element and its just smaller
        // element in array w[]
        int currElementPos
            = pos[a[curr]];
        int prevElementPos
            = pos[a[curr - 1]];

        if (currElementPos
            > prevElementPos)
            continue;

        // Iterate until current element
        // position is at most its just
        // smaller element position
        while (currElementPos
                   <= prevElementPos
               || filled[currElementPos]) {

            currElementPos += jump[a[curr]];
            ans++;
        }

        // Update the position of the
        // current element
        pos[a[curr]] = currElementPos;
        filled[currElementPos] = true;
    }

    // Print the result
    cout << ans;
}

// Driver Code
int main()
{
    int W[] = { 2, 1, 4, 3 };
    int L[] = { 4, 1, 2, 4 };
    int N = sizeof(W) / sizeof(W[0]);
    minJumps(W, L, N);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;
class GFG {

    // Function to find the minimum number
    // of jumps required to sort the array
    static void minJumps(int[] w, int[] l, int n)
    {

        // Base Case
        if (n == 1) {
            System.out.print(0);
            return;
        }

        // Store the required result
        int ans = 0;

        // Stores the current position of
        // elements and their respective
        // maximum jump
        HashMap<Integer, Integer> pos
            = new HashMap<Integer, Integer>();
        HashMap<Integer, Integer> jump
            = new HashMap<Integer, Integer>();

        // Used to check if a position is
        // already taken by another element
        HashMap<Integer, Boolean> filled
            = new HashMap<Integer, Boolean>();

        // Stores the sorted array a[]
        int[] a = new int[n];

        // Traverse the array w[] & update
        // positions jumps array a[]
        for (int i = 0; i < n; i++) {
            if (pos.containsKey(w[i]))
                pos.put(w[i], i);
            else
                pos.put(w[i], i);
            if (filled.containsKey(w[i]))
                filled.put(i, true);
            else
                filled.put(i, true);
            if (jump.containsKey(w[i]))
                jump.put(w[i], l[i]);
            else
                jump.put(w[i], l[i]);

            a[i] = w[i];
        }

        // Sort the array a[]
        Arrays.sort(a);

        // Traverse the array a[] over
        // the range [1, N-1]
        for (int curr = 1; curr < n; curr++) {

            // Store the index of current
            // element and its just smaller
            // element in array w[]
            int currElementPos = pos.get(a[curr]);
            int prevElementPos = pos.get(a[curr - 1]);

            if (currElementPos > prevElementPos)
                continue;

            // Iterate until current element
            // position is at most its just
            // smaller element position
            while (currElementPos <= prevElementPos
                   || filled.containsKey(currElementPos)
                          && filled.containsKey(
                              currElementPos)) {
                currElementPos += jump.get(a[curr]);
                ans++;
            }

            // Update the position of the
            // current element
            if (pos.containsKey(a[curr]))
                pos.put(a[curr], currElementPos);
            else
                pos.put(a[curr], currElementPos);
            if (filled.containsKey(currElementPos))
                filled.put(currElementPos, true);
            else
                filled.put(currElementPos, true);
        }

        // Print the result
        System.out.print(ans);
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[] W = { 2, 1, 4, 3 };
        int[] L = { 4, 1, 2, 4 };
        int N = W.length;

        minJumps(W, L, N);
    }
}

// This code is contributed by ukasp.

Python3

# Python3 program for the above approach

# Function to find the minimum number
# of jumps required to sort the array
def minJumps(w, l, n):

    # Base Case
    if (n == 1):
        print(0)
        return

    # Store the required result
    ans = 0

    # Stores the current position of
    # elements and their respective
    # maximum jump
    pos = {}
    jump = {}

    # Used to check if a position is
    # already taken by another element
    filled = {}

    # Stores the sorted array a[]
    a = [0 for i in range(n)]

    # Traverse the array w[] & update
    # positions jumps array a[]
    for i in range(n):
        pos[w[i]] = i
        filled[i] = True
        jump[w[i]] = l[i]
        a[i] = w[i]

    # Sort the array a[]
    a.sort()

    # Traverse the array a[] over
    # the range [1, N-1]
    for curr in range(1, n, 1):

        # Store the index of current
        # element and its just smaller
        # element in array w[]
        currElementPos = pos[a[curr]]
        prevElementPos = pos[a[curr - 1]]

        if (currElementPos > prevElementPos):
            continue

        # Iterate until current element
        # position is at most its just
        # smaller element position
        while (currElementPos <= prevElementPos or
              (currElementPos in filled and
              filled[currElementPos])):
            currElementPos += jump[a[curr]]
            ans += 1

        # Update the position of the
        # current element
        pos[a[curr]] = currElementPos
        filled[currElementPos] = True

    # Print the result
    print(ans)

# Driver Code
if __name__ == '__main__':

    W = [ 2, 1, 4, 3 ]
    L = [ 4, 1, 2, 4 ]
    N = len(W)

    minJumps(W, L, N)

# This code is contributed by SURENDRA_GANGWAR

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to find the minimum number
// of jumps required to sort the array
static void minJumps(int []w, int []l, int n)
{

    // Base Case
    if (n == 1)
    {
        Console.Write(0);
        return;
    }

    // Store the required result
    int ans = 0;

    // Stores the current position of
    // elements and their respective
    // maximum jump
    Dictionary<int,
               int> pos = new Dictionary<int,
                                         int>();
    Dictionary<int,
               int> jump = new Dictionary<int,
                                          int>();

    // Used to check if a position is
    // already taken by another element
     Dictionary<int,
                bool> filled = new Dictionary<int,
                                              bool>();

    // Stores the sorted array a[]
    int []a = new int[n];

    // Traverse the array w[] & update
    // positions jumps array a[]
    for(int i = 0; i < n; i++)
    {
        if (pos.ContainsKey(w[i]))
           pos[w[i]] = i;
        else
           pos.Add(w[i], i);
        if (filled.ContainsKey(w[i]))
           filled[i] = true;
        else
           filled.Add(i, true);
        if (jump.ContainsKey(w[i]))
           jump[w[i]] = l[i];
        else
           jump.Add(w[i], l[i]);

        a[i] = w[i];
    }

    // Sort the array a[]
    Array.Sort(a);

    // Traverse the array a[] over
    // the range [1, N-1]
    for(int curr = 1; curr < n; curr++)
    {

        // Store the index of current
        // element and its just smaller
        // element in array w[]
        int currElementPos = pos[a[curr]];
        int prevElementPos = pos[a[curr - 1]];

        if (currElementPos > prevElementPos)
            continue;

        // Iterate until current element
        // position is at most its just
        // smaller element position
        while (currElementPos <= prevElementPos ||
               filled.ContainsKey(currElementPos) &&
               filled[currElementPos])
        {
            currElementPos += jump[a[curr]];
            ans++;
        }

        // Update the position of the
        // current element
        if (pos.ContainsKey(a[curr]))
            pos[a[curr]] = currElementPos;
        else
            pos.Add(a[curr], currElementPos);
        if (filled.ContainsKey(currElementPos))
            filled[currElementPos] = true;
        else
            filled.Add(currElementPos, true);
    }

    // Print the result
    Console.Write(ans);
}

// Driver Code
public static void Main()
{
    int []W = { 2, 1, 4, 3 };
    int []L = { 4, 1, 2, 4 };
    int N = W.Length;

    minJumps(W, L, N);
}
}

// This code is contributed by ipg2016107

Javascript

<script>

// Javascript program for the above approach

// Function to find the minimum number
// of jumps required to sort the array
function minJumps(w, l, n)
{

    // Base Case
    if (n == 1)
    {
        document.write(0);
        return;
    }

    // Store the required result
    var ans = 0;
    var i;

    // Stores the current position of
    // elements and their respective
    // maximum jump
    var pos = new Map();
    var jump = new Map();

    // Used to check if a position is
    // already taken by another element
    var filled = new Map();

    // Stores the sorted array a[]
    var a  = new Array(n);

    // Traverse the array w[] & update
    // positions jumps array a[]
    for(i = 0; i < n; i++)
    {
        pos.set(w[i], i);
        filled.set(i, true);
        jump.set(w[i], l[i]);
        a[i] = w[i];
    }

    // Sort the array a[]
    a = a.sort(function(p, q)
    {
        return p - q;
    });

    // Traverse the array a[] over
    // the range [1, N-1]
    for(curr = 1; curr < n; curr++)
    {

        // Store the index of current
        // element and its just smaller
        // element in array w[]
        var currElementPos = pos.get(a[curr]);
        var prevElementPos = pos.get(a[curr - 1]);

        if (currElementPos > prevElementPos)
            continue;

        // Iterate until current element
        // position is at most its just
        // smaller element position
        while (currElementPos <= prevElementPos ||
               filled[currElementPos])
        {
            currElementPos += jump.get(a[curr]);
            ans += 1;
        }

        // Update the position of the
        // current element
        pos.set(a[curr], currElementPos);
        filled.set(currElementPos, true);
    }

    // Print the result
    document.write(ans);
}

// Driver Code
var W = [ 2, 1, 4, 3 ];
var L = [ 4, 1, 2, 4 ];
var N = W.length;

minJumps(W, L, N);

// This code is contributed by ipg2016107

</script>

Output:

5

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