Earliest Second to Mark Indices II

Earliest Second to Mark Indices II

What is this problem about?

The Earliest Second to Mark Indices II coding problem is an advanced variation of its predecessor. You are given an array nums and an array changeIndices. Each second, you can perform one of four operations: decrease a value in nums by 1, set a value in nums to 0 if the current second matches the one in changeIndices, mark an index as complete if its value is 0, or do nothing. Your goal is to find the minimum second required to mark all indices as complete.

Why is this asked in interviews?

This is a high-difficulty question asked by companies like Google and MathWorks. It tests a candidate's ability to integrate multiple algorithmic concepts: binary search interview pattern, greedy strategy, and priority queues. It requires identifying that "setting to 0" is a powerful but time-constrained operation, and you must strategically decide which indices to "zero out" to save the most time.

Algorithmic pattern used

This problem is solved using Binary Search on the Answer combined with a Greedy strategy and a Min-Heap.

1. Use binary search to find the minimum time T.
2. To validate time T, use a greedy approach:
   • Identify the first occurrence of each index in changeIndices within the range [1, T].
   • Iterate backwards from T to 1.
   • If the current second is the first occurrence of an index and nums[index] > 1, consider zeroing it out.
   • Use a Min-Heap to store the nums[index] values you've zeroed out. If you run out of time to mark these indices, remove the smallest value from the heap (the least "helpful" zeroing).
3. Check if the saved time is enough to mark all indices.

Example explanation

Suppose nums = [10, 10] and changeIndices = [1, 2]. Check T = 4:

1. First occurrences: index 1 at sec 1, index 2 at sec 2.
2. At sec 2: Zero out nums[1]. (Costs 1 sec to zero, 1 sec to mark). Heap: [10].
3. At sec 1: Zero out nums[0]. (Costs 1 sec to zero, 1 sec to mark). Heap: [10, 10].
4. Total time used for zeroing and marking: 4 seconds.
5. All indices marked. T = 4 is valid.

Common mistakes candidates make

• Not using First Occurrences: In this version, the "set to 0" operation is most useful at the earliest possible time, unlike the "mark as complete" operation in version I.
• Incorrect Heap logic: Forgetting that each zeroing operation also requires a subsequent "mark" operation, consuming two seconds total.
• Complexity: Attempting a direct simulation without binary search, which leads to exponential or high-order polynomial time.

Interview preparation tip

When a problem involves choosing the "best" set of items under a time constraint, always consider using a Priority Queue (Heap). It allows you to dynamically adjust your choices as you discover better options.