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Sum of special triplets having elements from 3 arrays

  • Difficulty Level : Hard
  • Last Updated : 26 May, 2021

Given three arrays A, B, and C, the task is to find sum of values of all special triplets. A special triplet is defined as a triplet (X, Y, Z) where the condition : 
X ≤ Y and Z ≤ Y always hold true. The value of each triplet (X, Y, Z) is given by: 
 

f(X, Y, Z) = (X + Y) * (Y + Z)

Note: If a triplet is not ‘special’, f(x, y, z) = 0 for that particular triplet.
Examples:  

Input : A = {1, 4, 5}, B = {2, 3}, C = {2, 1, 3}
Output : 81
Explanation
The special triplets and their values are given below
Triplet      f(x, y, z) = (x + y) * (y + z)
(1, 2, 2)         (1 + 2) * (2 + 2)  = 12
(1, 2, 1)         (1 + 2) * (2 + 1)  =  9
(1, 3, 2)         (1 + 3) * (3 + 2)  = 20
(1, 3, 1)         (1 + 3) * (3 + 1)  = 16
(1, 3, 3)         (1 + 3) * (3 + 3)  = 24
-------------------------------------
                           Sum = 81

Method 1 (Brute Force) We generate all triplets and check if a triplet is a special triplet, we calculate the value of the triplet using f(x, y, z) where (x, y, z) is a special triplet, and add it to the final sum of all such special triplets
 

C++




// C++ Program to find sum of values of
// all special triplets
#include <bits/stdc++.h>
using namespace std;
 
/* Finding special triplets (x, y, z) where
   x belongs to A; y belongs to B and z
   belongs to C; p, q and r are size of
   A, B and C respectively */
int findSplTripletsSum(int a[], int b[], int c[],
                             int p, int q, int r)
{
 
    int sum = 0;
    for (int i = 0; i < p; i++) {
        for (int j = 0; j < q; j++) {
            for (int k = 0; k < r; k++) {
 
                // (a[i], b[j], c[k]) is special if
                // a[i] <= b[j] and c[k] <= b[j];
                if (a[i] <= b[j] && c[k] <= b[j]) {
 
                    // calculate the value of this special
                    // triplet and add sum of all values
                    // of such triplets
                    sum +=  (a[i] + b[j]) * (b[j] + c[k]);
                }
            }
        }
    }
    return sum;
}
 
// Driver Code
int main()
{
    int A[] = { 1, 4, 5 };
    int B[] = { 2, 3 };
    int C[] = { 2, 1, 3 };
 
    int p = sizeof(A) / sizeof(A[0]);
    int q = sizeof(B) / sizeof(B[0]);
    int r = sizeof(C) / sizeof(C[0]);
 
    cout << "Sum of values of all special triplets = "
         << findSplTripletsSum(A, B, C, p, q, r) << endl;
}

Java




// Java Program to find sum of values of
// all special triplets
class GFG
{
 
/* Finding special triplets (x, y, z) where
x belongs to A; y belongs to B and z
belongs to C; p, q and r are size of
A, B and C respectively */
static int findSplTripletsSum(int a[], int b[], int c[],
                            int p, int q, int r)
{
 
    int sum = 0;
    for (int i = 0; i < p; i++)
    {
        for (int j = 0; j < q; j++)
        {
            for (int k = 0; k < r; k++)
            {
 
                // (a[i], b[j], c[k]) is special if
                // a[i] <= b[j] and c[k] <= b[j];
                if (a[i] <= b[j] && c[k] <= b[j])
                {
 
                    // calculate the value of this special
                    // triplet and add sum of all values
                    // of such triplets
                    sum += (a[i] + b[j]) * (b[j] + c[k]);
                }
            }
        }
    }
    return sum;
}
 
// Driver Code
public static void main(String[] args)
{
    int A[] = { 1, 4, 5 };
    int B[] = { 2, 3 };
    int C[] = { 2, 1, 3 };
 
    int p = A.length;
    int q = B.length;
    int r = C.length;
 
    System.out.print("Sum of values of all special triplets = "
                    + findSplTripletsSum(A, B, C, p, q, r) +"\n");
}
}
 
// This code is contributed by 29AjayKumar

Python3




# Python3 Program to find sum of values of
# all special triplets
 
# Finding special triplets (x, y, z) where
# x belongs to A y belongs to B and z
# belongs to C p, q and r are size of
# A, B and C respectively
def findSplTripletsSum(a, b, c, p, q, r):
    summ = 0
    for i in range(p):
        for j in range(q):
            for k in range(r):
 
                # (a[i], b[j], c[k]) is special if
                # a[i] <= b[j] and c[k] <= b[j]
                if (a[i] <= b[j] and c[k] <= b[j]):
 
                    # calculate the value of this special
                    # triplet and add sum of all values
                    # of such triplets
                    summ += (a[i] + b[j]) * (b[j] + c[k])
    return summ
 
# Driver Code
A = [1, 4, 5 ]
B = [2, 3 ]
C = [2, 1, 3 ]
 
p = len(A)
q = len(B)
r = len(C)
 
print("Sum of values of all special triplets = ",
            findSplTripletsSum(A, B, C, p, q, r))
 
# This code is contributed by Mohit kumar 29

C#




// C# Program to find sum of values of
// all special triplets
using System;
 
class GFG
{
 
/* Finding special triplets (x, y, z) where
x belongs to A; y belongs to B and z
belongs to C; p, q and r are size of
A, B and C respectively */
static int findSplTripletsSum(int []a, int []b, int []c,
                            int p, int q, int r)
{
 
    int sum = 0;
    for (int i = 0; i < p; i++)
    {
        for (int j = 0; j < q; j++)
        {
            for (int k = 0; k < r; k++)
            {
 
                // (a[i], b[j], c[k]) is special if
                // a[i] <= b[j] and c[k] <= b[j];
                if (a[i] <= b[j] && c[k] <= b[j])
                {
 
                    // calculate the value of this special
                    // triplet and add sum of all values
                    // of such triplets
                    sum += (a[i] + b[j]) * (b[j] + c[k]);
                }
            }
        }
    }
    return sum;
}
 
// Driver Code
public static void Main(String[] args)
{
    int []A = { 1, 4, 5 };
    int []B = { 2, 3 };
    int []C = { 2, 1, 3 };
 
    int p = A.Length;
    int q = B.Length;
    int r = C.Length;
 
    Console.Write("Sum of values of all special triplets = "
                    + findSplTripletsSum(A, B, C, p, q, r) +"\n");
}
}
 
// This code is contributed by PrinciRaj1992

Javascript




<script>
// javascript Program to find sum of values of
// all special triplets   
     /* Finding special triplets (x, y, z) where x belongs to A;
        y belongs to B and z  belongs to C; p, q and r are size
        of A, B and C respectively */
    function findSplTripletsSum(a , b , c , p , q , r) {
 
        var sum = 0;
        for (i = 0; i < p; i++) {
            for (j = 0; j < q; j++) {
                for (k = 0; k < r; k++) {
 
                    // (a[i], b[j], c[k]) is special if
                    // a[i] <= b[j] and c[k] <= b[j];
                    if (a[i] <= b[j] && c[k] <= b[j]) {
 
                        // calculate the value of this special
                        // triplet and add sum of all values
                        // of such triplets
                        sum += (a[i] + b[j]) * (b[j] + c[k]);
                    }
                }
            }
        }
        return sum;
    }
 
    // Driver Code
     
        var A = [ 1, 4, 5 ];
        var B = [ 2, 3 ];
        var C = [ 2, 1, 3 ];
 
        var p = A.length;
        var q = B.length;
        var r = C.length;
 
        document.write("Sum of values of all special triplets = "
        + findSplTripletsSum(A, B, C, p, q, r) + "\n");
 
// This code is contributed by todaysgaurav
</script>
Output: 
Sum of values of all special triplets = 81

 

The Time Complexity of this approach is O(P * Q * R) where P, Q, and R are the sizes of the three arrays A, B, and C respectively.
Method 2 (Efficient) 
Suppose, 
Array A contains elements {a, b, c, d, e}, 
Array B contains elements {f, g, h, i} and 
Array C contains elements {j, k, l, m}.
First, we sort the arrays A and C so that we are able to find the number of elements in arrays A and C that are less than a particular Bi which can be done by applying binary search on each value of Bi.
Let’s suppose that at a particular index i, the element of array B is Bi. Let’s also suppose that after we are done sorting A and C, we have elements {a, b, c} belonging to array A which are less than or equal to Bi and elements {j, k} belonging to array C which is also less than Bi
 

Lets take Bi = Y from here on.

Let, Total Sum of values of all special triplets = S 
We Know S = Σ f(x, y, z) for all possible (x, y, z)

Since elements {a, b, c} of Array A and elements {j, k} of array C are less than Y,
the Special Triplets formed consists of triplets formed only using these elements 
with Y always being the second element of every possible triplet

All the Special Triplets and their corresponding values are shown below:

Triplet   f(x, y, z) = (x + y) * (y + z)
(a, Y, j)         (a + Y)(Y + j)  
(a, Y, k)         (a + Y)(Y + k)  
(b, Y, j)         (b + Y)(Y + j)  
(b, Y, k)         (b + Y)(Y + k)  
(c, Y, j)         (c + Y)(Y + j)  
(c, Y, k)         (c + Y)(Y + k)

The sum of these triplets is
S = (a + Y)(Y + j) + (a + Y)(Y + k) + (b + Y)(Y + j) + (b + Y)(Y + k) 
    + (c + Y)(Y + j) + (c + Y)(Y + k)

Taking (a + X), (b + X) and (c + x) as common terms we have,

S = (a + Y)(Y + j + Y + k) + (b + Y)(Y + j + Y + k) + (c + Y)(Y + j + Y + k)
Taking (2Y + j + k) common from every term,
S = (a + Y + b + Y + c + Y)(2Y + j + k)  

∴ S = (3Y + a + b + c)(2Y + j + k)

Thus,
S = (N * Y + S1) * (M * Y + S2) 
where, 
N = Number of elements in A less than Y,
M = Number of elements in C less than Y,
S1 = Sum of elements in A less than Y and
S2 = Sum of elements in C less than Y

So for every element in B, we can find the number of elements less than it in arrays A and C using Binary Search and the sum of these elements can be found using prefix sums 
 

C++




// C++ Program to find sum of values
// of all special triplets
#include <bits/stdc++.h>
using namespace std;
 
/* Utility function for findSplTripletsSum()
finds total sum of values of all special
triplets */
int findSplTripletsSumUtil(int A[], int B[], int C[],
                   int prefixSumA[], int prefixSumC[],
                                 int p, int q, int r)
{
 
    int totalSum = 0;
 
    // Traverse through whole array B
    for (int i = 0; i < q; i++) {
 
        // store current element Bi
        int currentElement = B[i];
 
        // n = number of elements in A less than current
        // element
        int n = upper_bound(A, A + p, currentElement) - A;
 
        // m = number of elements in C less than current
        // element
        int m = upper_bound(C, C + r, currentElement) - C;
 
        // if there are Elements neither in A nor C which
        // are less than or equal to the current element
        if (n == 0 || m == 0)
            continue;
 
        /* total sum = (n * currentElement + sum of first
           n elements in A) + (m * currentElement + sum of
           first m elements in C) */
        totalSum +=
            ((prefixSumA[n - 1] + (n * currentElement)) *
              (prefixSumC[m - 1] + (m * currentElement)));
    }
 
    return totalSum;
}
 
/* Builds prefix sum array for arr of size n
and returns a pointer to it */
int* buildPrefixSum(int* arr, int n)
{
 
    // Dynamically allocate memory tp Prefix Sum Array
    int* prefixSumArr = new int[n];  
 
    // building the prefix sum
    prefixSumArr[0] = arr[0];
    for (int i = 1; i < n; i++)
        prefixSumArr[i] = prefixSumArr[i - 1] + arr[i];   
 
    return prefixSumArr;
}
 
/* Wrapper for Finding special triplets (x, y, z)
   where x belongs to A; y belongs to B and z
   belongs to C; p, q and r are size of
   A, B and C respectively */
int findSplTripletsSum(int A[], int B[], int C[],
                             int p, int q, int r)
{
 
    int specialTripletSum = 0;
 
    // sort arrays A and C
    sort(A, A + p);
    sort(C, C + r);
 
    // build prefix arrays for A and C
    int* prefixSumA = buildPrefixSum(A, p);
    int* prefixSumC = buildPrefixSum(C, r);
 
    return findSplTripletsSumUtil(A, B, C,
           prefixSumA, prefixSumC, p, q, r);
}
 
// Driver Code
int main()
{
    int A[] = { 1, 4, 5 };
    int B[] = { 2, 3 };
    int C[] = { 2, 1, 3 };
    int p = sizeof(A) / sizeof(A[0]);
    int q = sizeof(B) / sizeof(B[0]);
    int r = sizeof(C) / sizeof(C[0]);
 
    cout << "Sum of values of all special triplets = "
         << findSplTripletsSum(A, B, C, p, q, r);
}

Java




// Java Program to find sum of values of
// all special triplets
import java.io.*;
import java.util.*;
 
public class GFG {
     
    /* Finding special triplets (x, y, z)
    where x belongs to A; y belongs to B
    and z belongs to C; p, q and r are
    size of A, B and C respectively */
    static int findSplTripletsSum(int []a,
                  int []b, int []c, int p,
                             int q, int r)
    {
        int sum = 0;
        for (int i = 0; i < p; i++) {
            for (int j = 0; j < q; j++) {
                for (int k = 0; k < r; k++)
                {
     
                    // (a[i], b[j], c[k]) is
                    // special if a[i] <= b[j]
                    // and c[k] <= b[j];
                    if (a[i] <= b[j] &&
                                 c[k] <= b[j])
                    {
     
                        // calculate the value
                        // of this special
                        // triplet and add sum
                        // of all values
                        // of such triplets
                        sum += (a[i] + b[j])
                               * (b[j] + c[k]);
                    }
                }
            }
        }
        return sum;
    }
     
    // Driver Code
    public static void main(String args[])
    {
        int []A = { 1, 4, 5 };
        int []B = { 2, 3 };
        int []C = { 2, 1, 3 };
     
        int p = A.length;
        int q = B.length;
        int r = C.length;
     
        System.out.print("Sum of values of all"
                       + " special triplets = "
        + findSplTripletsSum(A, B, C, p, q, r));
    }
}
 
// This code is contributed by Manish Shaw
// (manishshaw1)

Python3




# Python3 Program to find sum of values of 
# all special triplets
 
# Finding special triplets (x, y, z)
# where x belongs to A; y belongs to B
# and z belongs to C; p, q and r are 
# size of A, B and C respectively
def findSplTripletsSum(a, b, c, p, q, r):
    sum = 0
    for i in range(p):
        for j in range(q):
            for k in range(r):
               
                # (a[i], b[j], c[k]) is
                # special if a[i] <= b[j]
                # and c[k] <= b[j];
                if(a[i] <= b[j] and c[k] <= b[j]):
                   
                    # calculate the value
                    # of this special
                    # triplet and add sum
                    # of all values 
                    # of such triplets
                    sum += (a[i] + b[j]) * (b[j] + c[k])
    return sum
 
# Driver Code
A = [1, 4, 5]
B = [2, 3 ]
C = [2, 1, 3]
p = len(A)
q = len(B)
r = len(C)
 
print("Sum of values of all","special triplets =",findSplTripletsSum(A, B, C, p, q, r))
 
# This code is contributed by avanitrachhadiya2155

C#




// C# Program to find sum of values of
// all special triplets
using System;
using System.Collections.Generic;
using System.Linq;
class GFG {
     
    /* Finding special triplets (x, y, z) where
       x belongs to A; y belongs to B and z
       belongs to C; p, q and r are size of
       A, B and C respectively */
    static int findSplTripletsSum(int []a, int []b, int []c,
                                       int p, int q, int r)
    {
        int sum = 0;
        for (int i = 0; i < p; i++) {
            for (int j = 0; j < q; j++) {
                for (int k = 0; k < r; k++) {
     
                    // (a[i], b[j], c[k]) is special if
                    // a[i] <= b[j] and c[k] <= b[j];
                    if (a[i] <= b[j] && c[k] <= b[j]) {
     
                        // calculate the value of this special
                        // triplet and add sum of all values
                        // of such triplets
                        sum += (a[i] + b[j]) * (b[j] + c[k]);
                    }
                }
            }
        }
        return sum;
    }
     
    // Driver Code
    public static void Main()
    {
        int []A = { 1, 4, 5 };
        int []B = { 2, 3 };
        int []C = { 2, 1, 3 };
     
        int p = A.Length;
        int q = B.Length;
        int r = C.Length;
     
        Console.WriteLine("Sum of values of all special triplets = "
                            + findSplTripletsSum(A, B, C, p, q, r));
    }
}
 
// This code is contributed by
// Manish Shaw (manishshaw1)

Javascript




<script>
// Javascript Program to find sum of values of
// all special triplets
     
     /* Finding special triplets (x, y, z)
    where x belongs to A; y belongs to B
    and z belongs to C; p, q and r are
    size of A, B and C respectively */
    function findSplTripletsSum(a,b,c,p,q,r)
    {
        let sum = 0;
        for (let i = 0; i < p; i++) {
            for (let j = 0; j < q; j++) {
                for (let k = 0; k < r; k++)
                {
       
                    // (a[i], b[j], c[k]) is
                    // special if a[i] <= b[j]
                    // and c[k] <= b[j];
                    if (a[i] <= b[j] &&
                                 c[k] <= b[j])
                    {
       
                        // calculate the value
                        // of this special
                        // triplet and add sum
                        // of all values
                        // of such triplets
                        sum += (a[i] + b[j])
                               * (b[j] + c[k]);
                    }
                }
            }
        }
        return sum;
    }
     
    // Driver Code
    let A=[1, 4, 5];
    let B=[ 2, 3 ];
    let C=[2, 1, 3 ];
     
    let p = A.length;
    let q = B.length;
    let r = C.length;
     
    document.write("Sum of values of all"
                       + " special triplets = "
        + findSplTripletsSum(A, B, C, p, q, r));
 
// This code is contributed by patel2127
</script>
Output: 
Sum of values of all special triplets = 81

 

Since we need to iterate through the entire array B and for every element apply binary searches in array A and C, the Time Complexity of this approach is O(Q * (logP + logR)) where P, Q, and R are the sizes of the three arrays A, B, and C respectively.
 


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