Maximize Total Cost of Alternating Subarrays

Maximize Total Cost of Alternating Subarrays

What is this problem about?

You are given an integer array. You need to split the array into contiguous subarrays. For each subarray, the "cost" is calculated by taking the first element, subtracting the second, adding the third, subtracting the fourth, and so on (alternating signs). Your goal is to partition the array in a way that maximizes the sum of the costs of all subarrays combined.

Why is this asked in interviews?

This is a Dynamic Programming problem that tests a candidate's ability to track alternating states. Interviewers use it to see if you recognize that splitting an array simply resets the alternating sign sequence back to positive. It evaluates your ability to build a State Machine DP where the decision at index i depends on the sign applied to index i-1.

Algorithmic pattern used

This leverages 1D Dynamic Programming (State Machine). For every element, it can either be assigned a positive sign (added) or a negative sign (subtracted). Let addState be the max cost if the current element is added. Let subState be the max cost if it is subtracted.

• If we ADD the current element arr[i], it means we are either starting a brand new subarray (previous sign doesn't matter, we take the best of previous states) or we are continuing a subarray that previously subtracted. So, new_add = arr[i] + max(addState, subState).
• If we SUBTRACT the current element arr[i], it MUST be continuing a subarray that previously added. So, new_sub = -arr[i] + addState.

Example explanation

Array: [1, -2, 3] Initialize for the first element (it MUST be added since all subarrays start positive): addState = 1, subState = -infinity.

• Process -2:
  • Try to ADD: Start a new subarray or follow a subtraction. new_add = -2 + max(1, -inf) = -1.
  • Try to SUBTRACT: Must follow an addition. new_sub = -(-2) + addState = 2 + 1 = 3.
  • Update: addState = -1, subState = 3.
• Process 3:
  • Try to ADD: new_add = 3 + max(-1, 3) = 6.
  • Try to SUBTRACT: Must follow an addition. new_sub = -3 + addState = -3 + -1 = -4.
  • Update: addState = 6, subState = -4. The maximum across the states at the end is max(6, -4) = 6. This corresponds to partitioning as [1, -2] (cost $1 - (-2) = 3$) and [3] (cost 3). Total = 6.

Common mistakes candidates make

A common mistake is trying to track the lengths of the subarrays. The length of the subarray doesn't actually matter for the state transitions; the only thing that matters is the sign of the previous operation. If the previous operation was a minus, the next MUST be a plus (or start a new array). If the previous was a plus, the next can be a minus OR start a new array. Simplifying the state to just add and subtract is the key.

Interview preparation tip

When tackling alternating sequence problems, immediately draw a State Machine diagram. Draw a node for + and a node for -. Draw arrows indicating valid transitions (e.g., + can go to -, - can go to +, and both can go to New Subarray +). Translating these arrows into DP equations (new_state = cost + previous_valid_state) makes the code trivial to write.