Minimize the cost of partitioning an array into K groups

Given an array arr[] and an integer K, the task is to partition the array into K non-empty groups where each group is a subarray of the given array and each element of the array is part of only one group. All the elements in a given group must have the same value. You can perform the following operation any number of times:
Choose an element from the array and change it’s value to any value. Print the minimum number of such operations required to partition the array.

Examples:

Input: arr[] = {3, 1, 3, 3, 2, 1, 8, 5}, K = 3
Output: 3
The array can be partitioned in {3, 3, 3, 3}, {2, 2} and {8, 8}
by changing the 2nd element to 3, the 6th
element to 2 and the last element to 8.



Input:arr[] = {3, 3, 9, 10}, K = 3
Output: 0
Divide the array in groups {3, 3}, {9} and {10}
without performing any operations.

Observations:

  • If K = 1 then the group is the complete array itself. To minimize the number of operations needed the most intuitive thing to do is to change all the elements of the array and make them equal to the mode of the array (element with the highest frequency).
  • For K groups, the last element of the array will always belong to the Kth group while the 1st element will belong to the 1st group.
  • If Kth group has been found correctly then the problem will reduce to partitioning the remaining array into K – 1 groups using minimum operations.

Approach: This problem can be solved using dynamic programming.

  1. Let DP(i, j) represent the minimum operations needed to partition the array[1..i] into j groups.
  2. Now, the task is to find DP(N, K) which is the minimum operations needed to partition the array[1..N] into K groups.
  3. The base cases DP(i, j) where j = 1 can be easily answered. Since the complete array array[1..i] needs to be partitioned into a single group only. From the observations, find the mode of the array[1..i] and change all the elements in array[1..i] to the mode. If the mode occurred x times then i – x elements will have to be changed i.e. i – x operations.
  4. Since, the Kth group ends at the last element. However it may start at various possible positions. Suppose that the Kth group starts at some index it then array[it..N] needs to be partitioned into a single group and array[1..(it – 1)] needs to be partitioned into K – 1 groups. The cost of partitioning array[1..(it – 1)] into K – 1 groups is DP(it – 1, K – 1) and the cost of partitioning array[it..N] in a single group can be calculated using the mode and it’s frequency observation.
  5. To find the frequency of the most occurring element in a range [it..i] we can use a hashmap and an integer variable. The integer variable represents the current highest frequency. The map stores all elements seen till now along with their frequencies. Whenever an element is seen it’s frequency is incremented in the map, if now the frequency of this element is higher than the current highest frequency we update the current highest frequency to the frequency of the just seen element. Refer this for the approach.
  6. Therefore DP(i, j) is the minimum of DP(it – 1, j – 1) + cost of partitioning array[it..i] into 1 group for all possible values of it.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the minimum number
// of operations needed to partition
// the array in k contiguous groups
// such that all elements of a
// given group are identical
int getMinimumOps(vector<int> ar, int k)
{
    // n is the size of the array
    int n = ar.size();
  
    // dp(i, j) represents the minimum cost for
    // partitioning the array[0..i] into j groups
    int dp[n][k + 1];
  
    // Base case, cost is 0 for parititoning the
    // array[0..0] into 1 group
    dp[0][1] = 0;
  
    // Fill dp(i, j) and the answer will
    // be stored at dp(n-1, k)
    for (int i = 1; i < n; i++) {
  
        // The maximum groups that the segment 0..i can
        // be divided in is represented by maxGroups
        int maxGroups = min(k, i + 1);
  
        for (int j = 1; j <= maxGroups; j++) {
  
            // Initialize dp(i, j) to infinity
            dp[i][j] = INT_MAX;
  
            // Divide segment 0..i in 1 group
            if (j == 1) {
  
                // map and freqOfMode are together used to
                // keep track of the frequency of the most
                // occuring element in [0..i]
                unordered_map<int, int> freq;
                int freqOfMode = 0;
                for (int it = 0; it <= i; it++) {
                    freq[ar[it]]++;
                    int newElementFreq = freq[ar[it]];
                    if (newElementFreq > freqOfMode)
                        freqOfMode = newElementFreq;
                }
  
                // Change all the elements in the range
                // 0..i to the most frequent element
                // in this range
                dp[i][1] = (i + 1) - freqOfMode;
            }
            else {
                unordered_map<int, int> freq;
                int freqOfMode = 0;
  
                // If the jth group is the segment from
                // it..i, we change all the elements in this
                // range to this range's most occurring element
                for (int it = i; it >= j - 1; it--) {
                    freq[ar[it]]++;
                    int newElementFreq = freq[ar[it]];
                    if (newElementFreq > freqOfMode)
                        freqOfMode = newElementFreq;
  
                    // Number of elements we need to change
                    // in the jth group i.e. the range it..i
                    int elementsToChange = i - it + 1;
                    elementsToChange -= freqOfMode;
  
                    // For all the possible sizes of the jth
                    // group that end at the ith element
                    // we pick the size that gives us the minimum
                    // cost for dp(i, j)
                    // elementsToChange is the cost of making
                    // all the elements in the jth group identical
                    // and we make use of dp(it - 1, j - 1) to
                    // find the overall minimal cost
                    dp[i][j] = min(dp[it - 1][j - 1]
                                       + elementsToChange,
                                   dp[i][j]);
                }
            }
        }
    }
  
    // Return the minimum cost for
    // partitioning array[0..n-1]
    // into k groups which is
    // stored at dp(n-1, k)
    return dp[n - 1][k];
}
  
// Driver code
int main()
{
    int k = 3;
    vector<int> ar = { 3, 1, 3, 3, 2, 1, 8, 5 };
  
    cout << getMinimumOps(ar, k);
  
    return 0;
}

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Java

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// Java implementation of above approach
class GFG
{
      
// Function to return the minimum number
// of operations needed to partition
// the array in k contiguous groups
// such that all elements of a
// given group are identical
static int getMinimumOps(int ar[], int k)
{
    // n is the size of the array
    int n = ar.length;
  
    // dp(i, j) represents the minimum cost for
    // partitioning the array[0..i] into j groups
    int dp[][] = new int[n][k + 1];
  
    // Base case, cost is 0 for parititoning the
    // array[0..0] into 1 group
    dp[0][1] = 0;
  
    // Fill dp(i, j) and the answer will
    // be stored at dp(n-1, k)
    for (int i = 1; i < n; i++)
    {
  
        // The maximum groups that the segment 0..i can
        // be divided in is represented by maxGroups
        int maxGroups = Math.min(k, i + 1);
  
        for (int j = 1; j <= maxGroups; j++) 
        {
  
            // Initialize dp(i, j) to infinity
            dp[i][j] = Integer.MAX_VALUE;
  
            // Divide segment 0..i in 1 group
            if (j == 1
            {
  
                // map and freqOfMode are together used to
                // keep track of the frequency of the most
                // occuring element in [0..i]
                int freq[] = new int[100000];
                int freqOfMode = 0;
                for (int it = 0; it <= i; it++) 
                {
                    freq[ar[it]]++;
                    int newElementFreq = freq[ar[it]];
                    if (newElementFreq > freqOfMode)
                        freqOfMode = newElementFreq;
                }
  
                // Change all the elements in the range
                // 0..i to the most frequent element
                // in this range
                dp[i][1] = (i + 1) - freqOfMode;
            }
            else
            {
                int freq[] = new int[100000];
                int freqOfMode = 0;
  
                // If the jth group is the segment from
                // it..i, we change all the elements in this
                // range to this range's most occurring element
                for (int it = i; it >= j - 1; it--) 
                {
                    freq[ar[it]]++;
                    int newElementFreq = freq[ar[it]];
                    if (newElementFreq > freqOfMode)
                        freqOfMode = newElementFreq;
  
                    // Number of elements we need to change
                    // in the jth group i.e. the range it..i
                    int elementsToChange = i - it + 1;
                    elementsToChange -= freqOfMode;
  
                    // For all the possible sizes of the jth
                    // group that end at the ith element
                    // we pick the size that gives us the minimum
                    // cost for dp(i, j)
                    // elementsToChange is the cost of making
                    // all the elements in the jth group identical
                    // and we make use of dp(it - 1, j - 1) to
                    // find the overall minimal cost
                    dp[i][j] = Math.min(dp[it - 1][j - 1] +
                                        elementsToChange, dp[i][j]);
                }
            }
        }
    }
  
    // Return the minimum cost for
    // partitioning array[0..n-1]
    // into k groups which is
    // stored at dp(n-1, k)
    return dp[n - 1][k];
}
  
// Driver code
public static void main(String args[])
{
    int k = 3;
    int ar[] = { 3, 1, 3, 3, 2, 1, 8, 5 };
  
    System.out.println(getMinimumOps(ar, k));
}
}
  
// This code is contributed by Arnab Kundu

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Python3

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# Python3 implementation of the approach
  
# Function to return the minimum number
# of operations needed to partition
# the array in k contiguous groups
# such that all elements of a
# given group are identical
def getMinimumOps(ar, k):
      
    # n is the size of the array
    n = len(ar)
  
    # dp(i, j) represents the minimum cost for
    # partitioning the array[0..i] into j groups
    dp = [[ 0 for i in range(k + 1)] 
              for i in range(n)]
  
    # Base case, cost is 0 for parititoning the
    # array[0..0] into 1 group
    dp[0][1] = 0
  
    # Fill dp(i, j) and the answer will
    # be stored at dp(n-1, k)
    for i in range(1, n):
  
        # The maximum groups that the segment 0..i can
        # be divided in is represented by maxGroups
        maxGroups = min(k, i + 1)
  
        for j in range(1, maxGroups + 1):
  
            # Initialize dp(i, j) to infinity
            dp[i][j] = 10**9
  
            # Divide segment 0..i in 1 group
            if (j == 1):
  
                # map and freqOfMode are together used to
                # keep track of the frequency of the most
                # occuring element in [0..i]
                freq1 = dict()
                freqOfMode = 0
                for it in range(0, i + 1):
  
                    freq1[ar[it]] = freq1.get(ar[it], 0) + 1
                    newElementFreq = freq1[ar[it]]
                    if (newElementFreq > freqOfMode):
                        freqOfMode = newElementFreq
  
                # Change all the elements in the range
                # 0..i to the most frequent element
                # in this range
                dp[i][1] = (i + 1) - freqOfMode
  
            else:
                freq = dict()
                freqOfMode = 0
  
                # If the jth group is the segment from
                # it..i, we change all the elements in this
                # range to this range's most occurring element
                for it in range(i, j - 2, -1):
                      
                    #print(i,j,it)
                    freq[ar[it]] = freq.get(ar[it], 0) + 1
                    newElementFreq = freq[ar[it]]
                    if (newElementFreq > freqOfMode):
                        freqOfMode = newElementFreq
  
                    # Number of elements we need to change
                    # in the jth group i.e. the range it..i
                    elementsToChange = i - it + 1
                    elementsToChange -= freqOfMode
  
                    # For all the possible sizes of the jth
                    # group that end at the ith element
                    # we pick the size that gives us the minimum
                    # cost for dp(i, j)
                    # elementsToChange is the cost of making
                    # all the elements in the jth group identical
                    # and we make use of dp(it - 1, j - 1) to
                    # find the overall minimal cost
                    dp[i][j] = min(dp[it - 1][j - 1] +  
                                   elementsToChange, dp[i][j])
  
    # Return the minimum cost for
    # partitioning array[0..n-1]
    # into k groups which is
    # stored at dp(n-1, k)
    return dp[n - 1][k]
  
# Driver code
k = 3
ar =[3, 1, 3, 3, 2, 1, 8, 5]
  
print(getMinimumOps(ar, k))
  
# This code is contributed by Mohit Kumar

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Output:

3

Time Complexity: O(N * N * K) where N is the size of the array and K is the number of groups the array should be partitioned into.
Space Complexity: O(N * K)



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Final year BTech IT student at DTU, Upcoming Technology Analyst at Morgan Stanley

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