Make the Prefix Sum Non-negative

Make the Prefix Sum Non-negative

What is this problem about?

Note: Due to naming variations, this describes the standard "Make Prefix Sum Non-negative" logic problem. You are given an array of integers (positive and negative). You process them from left to right, maintaining a running prefix sum. If the prefix sum drops below 0, you are allowed to take one of the previously processed negative numbers and move it to the very end of the array. Your goal is to find the minimum number of elements you must move to the end to ensure the prefix sum never drops below 0.

Why is this asked in interviews?

This is a classic Greedy algorithm problem utilizing a Priority Queue (Heap). Interviewers ask it because it forces candidates to perform "retroactive optimization." Instead of predicting what will happen, you process the array normally, and only when a failure state occurs do you look back and greedily fix the "worst" mistake you made.

Algorithmic pattern used

This problem is solved using a Greedy approach via a Min Heap (Priority Queue). As you iterate through the array, you add the current number to your prefix sum. If the number is negative, you also push it into a Min Heap. If your prefix sum ever becomes negative, it means you've taken too many negative numbers. To fix it optimally, you pop the absolute smallest (most negative) number from your Min Heap, subtract it from your prefix sum (which effectively adds its absolute value back), and increment your "moves" counter.

Example explanation

Array: [2, -3, -1, 5, -4] Heap: [], Sum: 0, Moves: 0

• Index 0 (2): Sum = 2. Heap [].
• Index 1 (-3): Sum = -1. Heap [-3]. Sum is $< 0$! We must fix it. Pop -3 from Heap. Sum becomes -1 - (-3) = 2. Moves = 1.
• Index 2 (-1): Sum = 1. Heap [-1]. Sum is $\ge 0$.
• Index 3 (5): Sum = 6. Heap [-1].
• Index 4 (-4): Sum = 2. Heap [-1, -4]. Sum is $\ge 0$. End of array. We only needed 1 move (moving the -3 to the end).

Common mistakes candidates make

A frequent mistake is popping the most recently added negative number instead of the most negative number seen so far. Moving the most negative number to the end gives you the largest possible mathematical recovery for the cost of exactly one move. Another mistake is tracking the indices of the moved numbers, which is entirely unnecessary; the problem only asks for the count of moves.

Interview preparation tip

When tackling the Make the Prefix Sum Non-negative interview pattern, recognize "retroactive greedy" problems. If you are traversing data and hit a failure condition, and you have a pool of past choices you can alter, a Priority Queue is the perfect structure to let you instantly undo the "most damaging" past choice.