Introduction to Finding Maximum Value of an Expression for Any Possible Pair in an Array

In this article, we will discuss a problem statement and its solution where we need to find the maximum value of an expression for any possible pair in an array. The given expression is:

(arr[i] * arr[j]) + (arr[j] – arr[i])

We will understand the problem statement, discuss the approach to solve it, and provide an implementation in Python.

Problem Statement

Given an array arr of integers, we need to find the maximum value of the expression (arr[i] * arr[j]) + (arr[j] – arr[i]) for any possible pair (i, j), where 0 <= i, j < n and n is the length of the array.

Approach

To find the maximum value of the expression, we need to maximize the value of (arr[i] * arr[j]) and (arr[j] - arr[i]).

Observations:

• The larger the arr[i] and arr[j] values, the larger the value of (arr[i] * arr[j]) can be.
• The difference (arr[j] - arr[i]) will be maximum when the value of arr[j] is the largest and arr[i] is the smallest.

Based on these observations, our approach will be:

1. Sort the given array in non-decreasing order.
2. Calculate the maximum value of the expression using the first and last elements of the sorted array.

Implementation

Here is the Python implementation of the above approach:

def find_max_value(arr):
    arr.sort()  # Sorting the array in non-decreasing order
    n = len(arr)
    max_value = (arr[n-1] * arr[0]) + (arr[n-1] - arr[0])
    return max_value

Complexity Analysis

• Sorting the array takes O(n log n) time complexity.
• Calculating the maximum value of the expression requires constant time.
• Therefore, the overall time complexity of the solution is O(n log n).

Conclusion

In this article, we discussed how to find the maximum value of the expression (arr[i] * arr[j]) + (arr[j] – arr[i]) for any possible pair in a given array. We also provided a step-by-step approach and a Python implementation of the solution. By sorting the array and considering the first and last elements, we can efficiently find the maximum value.