Python3 Program to Count triplets with sum smaller than a given value

Given an array of distinct integers and a sum value. Find count of triplets with sum smaller than given sum value. The expected Time Complexity is O(n²).
Examples:

Input : arr[] = {-2, 0, 1, 3}
        sum = 2.
Output : 2
Explanation :  Below are triplets with sum less than 2
               (-2, 0, 1) and (-2, 0, 3) 

Input : arr[] = {5, 1, 3, 4, 7}
        sum = 12.
Output : 4
Explanation :  Below are triplets with sum less than 12
               (1, 3, 4), (1, 3, 5), (1, 3, 7) and 
               (1, 4, 5)

A Simple Solution is to run three loops to consider all triplets one by one. For every triplet, compare the sums and increment count if the triplet sum is smaller than the given sum. 

Python 3

# A Simple Python 3 program to count triplets with sum smaller
# than a given value

def countTriplets(arr, n, sum):

    # Initialize result
    ans = 0

    # Fix the first element as A[i]
    for i in range( 0 ,n-2):

        # Fix the second element as A[j]
        for j in range( i+1 ,n-1):

            # Now look for the third number
            for k in range( j+1, n):
                if (arr[i] + arr[j] + arr[k] < sum):
                    ans+=1

    return ans

# Driver program
arr = [5, 1, 3, 4, 7]
n = len(arr)
sum = 12
print(countTriplets(arr, n, sum))

#Contributed by Smitha

Output: 

4

Time Complexity: O(n³)

Auxiliary Space: O(1)

As constant extra space is used.

An Efficient Solution can count triplets in O(n²) by sorting the array first, and then using method 1 of this post in a loop.

1) Sort the input array in increasing order.
2) Initialize result as 0.
3) Run a loop from i = 0 to n-2.  An iteration of this loop finds all
   triplets with arr[i] as first element.
     a) Initialize other two elements as corner elements of subarray
        arr[i+1..n-1], i.e., j = i+1 and k = n-1
     b) Move j and k toward each other until they meet, i.e., while (j= sum
                then k--
            // Else for current i and j, there can (k-j) possible third elements
            // that satisfy the constraint.
            (ii) Else Do ans += (k - j) followed by j++ 

Below is the implementation of the above idea. 

Python3

# Python3 program to count triplets with
# sum smaller than a given value


# Function to count triplets with sum smaller
# than a given value
def countTriplets(arr,n,sum):

    # Sort input array
    arr.sort()

    # Initialize result
    ans = 0

    # Every iteration of loop counts triplet with
    # first element as arr[i].
    for i in range(0,n-2):

        # Initialize other two elements as corner elements
        # of subarray arr[j+1..k]
        j = i + 1
        k = n-1

        # Use Meet in the Middle concept
        while(j < k):

            # If sum of current triplet is more or equal,
            # move right corner to look for smaller values
            if (arr[i]+arr[j]+arr[k] >=sum):
                k = k-1

            # Else move left corner
            else:

                # This is important. For current i and j, there
                # can be total k-j third elements.
                ans += (k - j)
                j = j+1

    return ans

# Driver program
if __name__=='__main__':
    arr = [5, 1, 3, 4, 7]
    n = len(arr)
    sum = 12
    print(countTriplets(arr, n, sum))

# This code is contributed by
# Yatin Gupta

Output: 

4

Time Complexity: O(n³)

Auxiliary Space: O(1)

As constant extra space is used.

Please refer complete article on Count triplets with sum smaller than a given value for more details!