Maximum Alternating Subarray Sum

Maximum Alternating Subarray Sum

What is this problem about?

(Note: Alternating subarray variants usually specify signs or operations. Let's address the standard alternating sign subarray problem). You are given an array of integers. You must find the maximum sum of a contiguous subarray where the elements have alternating signs applied to them (e.g., $+arr[0] - arr[1] + arr[2] - arr[3]...$). The subarray can start anywhere, but the first element chosen MUST be positive.

Why is this asked in interviews?

This is a state-machine Dynamic Programming problem. Interviewers ask it to test your ability to maintain multiple parallel states across an array traversal. It evaluates whether you can split Kadane's algorithm into two interlocking tracks: one expecting an addition next, and one expecting a subtraction next.

Algorithmic pattern used

This problem uses 1D Dynamic Programming (State Machine). We maintain two running variables:

• even_state: The maximum alternating sum ending at the current index, assuming the current element was added (a positive operation).
• odd_state: The maximum alternating sum ending at the current index, assuming the current element was subtracted (a negative operation). For each element:
• new_even = max(element, odd_state + element) (Start fresh, or follow a subtraction).
• new_odd = even_state - element (Must follow an addition). Track the global maximum across all new_even and new_odd states.

Example explanation

Array: [3, 1, 1, 4] Init: even = 3, odd = -infinity. Max = 3.

• Process 1:
  • new_even = max(1, odd + 1) = 1.
  • new_odd = even - 1 = 3 - 1 = 2.
  • Max = max(3, 2) = 3. even = 1, odd = 2.
• Process 1:
  • new_even = max(1, odd + 1) = 2 + 1 = 3.
  • new_odd = even - 1 = 1 - 1 = 0.
  • Max = max(3, 3) = 3. even = 3, odd = 0.
• Process 4:
  • new_even = max(4, odd + 4) = 0 + 4 = 4.
  • new_odd = even - 4 = 3 - 4 = -1.
  • Max = max(3, 4) = 4. The maximum alternating sum is 4.

Common mistakes candidates make

Candidates frequently try to use a single DP array and multiply the element by -1 based on the index's parity (e.g., if i % 2 == 0 add, else subtract). This logic fails because a subarray can start at an odd index! If you lock the signs to the absolute array indices, you miss valid subarrays that start off-beat. The State Machine approach mathematically decouples the signs from the array indices.

Interview preparation tip

When dealing with alternating states (like buy/sell stocks, or +/- signs), always define your variables explicitly by their semantic state (e.g., state_after_adding, state_after_subtracting), not by their array index. This makes the transition formulas highly intuitive: an "add" state must always pull from a previous "subtract" state, and vice versa.