Maximum Sum Obtained of Any Permutation

Maximum Sum Obtained of Any Permutation

1. What is this problem about?

The Maximum Sum Obtained of Any Permutation problem involves an array of numbers and a list of query ranges. Each query asks for the sum of elements within a specific range [start, end]. You are allowed to rearrange (permute) the numbers in the original array in any way you want. The goal is to find the permutation that maximizes the total sum of all the query results. Essentially, you want to put your largest numbers in the positions that are "requested" most frequently by the queries.

2. Why is this asked in interviews?

This "Maximum Sum Obtained of Any Permutation interview question" is a great test of a candidate's ability to use the Difference Array (or Sweep Line) technique. Companies like PayPal use it to see if you can efficiently count overlapping ranges. Instead of incrementing counts for every index in every query (which would be slow), you use a prefix sum trick to calculate frequencies in linear time. It evaluates both your algorithmic efficiency and your greedy reasoning.

3. Algorithmic pattern used

The "Array, Sorting, Greedy, Prefix Sum interview pattern" is the optimal strategy.

1. Create a frequency array to count how many times each index is visited across all queries. Use the "difference array" trick: for a query [i, j], increment freq[i] and decrement freq[j+1].
2. Compute the prefix sum of the frequency array to get actual counts for each index.
3. Sort both the frequency array and the original numbers array in descending order.
4. Multiply the largest frequency by the largest number, the second largest frequency by the second largest number, and so on. This is the "Greedy" part.

4. Example explanation

Numbers: [1, 2, 3, 4, 5], Queries: [[1, 3], [0, 1]]

• Index 0 is in 1 query.
• Index 1 is in 2 queries.
• Index 2 is in 1 query.
• Index 3 is in 1 query.
• Index 4 is in 0 queries. Frequencies: [1, 2, 1, 1, 0]. Sorted Frequencies: [2, 1, 1, 1, 0]. Sorted Numbers: [5, 4, 3, 2, 1]. Max Sum = (2 * 5) + (1 * 4) + (1 * 3) + (1 * 2) + (0 * 1) = 10 + 4 + 3 + 2 = 19.

5. Common mistakes candidates make

In the "Maximum Sum Obtained of Any Permutation coding problem," the most common mistake is using a nested loop to calculate the frequencies, which leads to O(N * Q) complexity and usually times out. Another error is forgetting to sort both the frequencies and the numbers. Some candidates also forget to apply the modulo operation if the question requires it, or they perform the modulo too late, leading to overflow errors in languages with fixed-size integers.

6. Interview preparation tip

Practice the "Difference Array" technique (sometimes called the +1/-1 trick). It is a very common tool for range-based problems. Once you understand how to turn a range update into two point updates, you can handle many complex array problems. Also, remember the Greedy Choice Property: to maximize a sum of products, you should always multiply the largest available values together. This concept is useful across many different coding challenges.