Maximum possible balanced binary substring splits with at most cost k

The array in the C programming language has a fixed size, which means that once the size is specified, it cannot be changed; you can neither shrink it or extend it.

As we know, an array is a group of identically data-typed elements kept in consecutive memory regions.

Given an array of values v[] and a binary array a[]. The objective is to use as many k coins to divide the binary array as much as is possible while ensuring that each segment has an equal amount of 0s and 1s. i and j are the neighboring indices of the split segment, and the cost of each split is (v[i] - v[j])2.

Problem Statement

Implement a program to find maximum possible balanced binary substring splits with at most cost k.

Sample Example 1

Let the Input array be: 
a: [1,0,0, 1, 0, 0, 1, 1]
The given values be: 
v: [7, 8, 9, 10, 11, 12,13,14]
K: 1

Output obtained is: 1

Explanation

We are able to make a single cut between the first and second indexes since K has a value of 1.

In this case [0, 1] and [0, 0, 1, 1] are the balanced binary substrings that are the end result.

Making this cut will cost (8 - 9)^ 2 = 1 as well as 1 = 1.

Sample Example 2

Let the Input array be: 
a: [1,0 1, 0, 1, 1, 0,0]
The given values be: 
v: [2, 4, 7, 10, 11, 12, 13, 14]
K: 14

Output obtained is: 2

Explanation

The first cut will be made between the first and second index that is 4 and 7, costing us (4 - 7)^2 = 9 and the second cut will be made between the third and fourth index that is 7 and 10, costing us (7 - 10)^ 2 = 9. No more cuts are possible at this time. The balanced binary substrings in this case that would arise are [1, 0], [1, 0], and [1, 1, 0, 0].

Approach

In order to find maximum possible balanced binary substring splits with at most cost k, we take the following methodology.

Here we take a top-down approach to solve this problem and to find maximum possible balanced binary substring splits with at most cost k.

Coming to top-down approach or widely-known as dynamic programming approach. The main benefit of dynamic programming is an improvement over simple recursion. Dynamic Programming can be used to optimize any recursive solution that contains repeated calls for the same inputs. To avoid having to recompute the outcomes of subproblems later, the idea is to just store them. With this straightforward optimization, time complexity is reduced from polynomial to exponential.

Algorithm

The algorithm to find maximum possible balanced binary substring splits with at most cost K is given below.

• Step 1 − Start

• Step 2 − Define a two-dimensional matrix m

• Step 3 − Define a function to find maximum possible balanced binary substring splits.

• Step 4 − Define integer variables zeroCount to count the number of zeros and oneCount to count the number of ones respectively

• Step 5 − Define an integer variable cntSplits to count the number of splits

• Step 6 − Iterate through the given array a

• Step 7 − check whether the number of zeros is equal to the number of ones, then store the maximum feasible one

• Step 8 − Suppose the index is at the position 0, then find whether it is a one or a zero and then increment the count

• Step 9 − set the cntSplits to zero, if count of one and count of zero is unequal.

• Step 10 − store the resultant value in the matrix

• Step 11 − Print the desired result obtained

• Step 12 − Stop

Example: C Program

Here is the C program implementation of the above written algorithm to find maximum possible balanced binary substring splits with at most cost k.

#include <stdio.h>
#include <limits.h>
#include <string.h>
//Define a two-dimensional matrix m
int m[1001][1001];

//Define a function to find maximum possible //balanced binary substring splits
int maxSplits(int a[], int v[], int k, int s) {
   if (k < 0) {
      return INT_MIN;
   }
   if (m[k][s] != -1) {
      return m[k][s];
   }
   
   //Define integer variables to count the number of zeros and ones 
   // Define an integer variable to count the //number of splits
   int zeroCount = 0, oneCount = 0;
   int cntSplits = 0;
   int i;
   
   //Iterating through the given array a
   for (i = s - 1; i > 0; i--) {
      a[i] == 0 ? zeroCount++ : oneCount++;
      
   // check whether the number of zeros is equal to the number of ones, then store the maximum feasible one
      if (zeroCount == oneCount) {
         cntSplits = cntSplits > (1 + maxSplits(a, v, k - (v[i] - v[i - 1]) * (v[i] - v[i - 1]), i)) ? cntSplits : (1 + maxSplits(a, v, k - (v[i] - v[i - 1]) * (v[i] - v[i - 1]), i));
      }
   }
   
   //Suppose the index is at the position 0, then find whether it is a one or a zero. then increment the count
   if (i == 0) {
      a[0] == 0 ? zeroCount++ : oneCount++;
      
   // set the cntSplits to zero , if count of one and count of zero is unequal.
      if (zeroCount != oneCount) {
         cntSplits = 0;
      }
   }
   
   // store the resultant value in the matrix
   return m[k][s] = cntSplits;
}
int main() {
   int a[] = { 1, 0, 0, 1, 0, 0, 1, 1 };
   int v[] = { 7, 8, 9, 10, 11, 12, 13, 14 };
   int k = 1;
   int s = sizeof(a) / sizeof(a[0]);
   
   //To assign a specific value to a block of memory, we use the memset() function.
   memset(m, -1, sizeof(m));
   printf("%d\n", maxSplits(a, v, k, s));
   return 0;
}

Output

1

Conclusion

Likewise, we can find the maximum possible balanced binary substring splits with at most cost K.

The challenge of obtaining the program to find maximum possible balanced binary substring splits with at most cost K is resolved in this article.

Here C programming code as well as the algorithm to find maximum possible balanced binary substring splits with at most cost K are provided.