#2202

Medium

auto_awesomeGreedy choice plus invariant validation

LeetCode Problem Workspace

Maximize the Topmost Element After K Moves

Maximize the topmost element in a pile after making exactly k moves using a greedy strategy and invariant checks.

Problem Statement

You are given a 0-indexed integer array nums representing the contents of a pile, where nums[0] is the topmost element of the pile. In one move, you can either remove the topmost element or add back any previously removed element to the pile. You must make exactly k moves, and the goal is to maximize the topmost element after these k moves. If it is impossible to achieve this, return -1.

For example, if nums = [5,2,2,4,0,6] and k = 4, the goal is to figure out the maximum element you can place at the top after 4 moves. If you can't perform the required operations to get a non-empty pile after k moves, return -1.

Examples

Example 1

Input: nums = [5,2,2,4,0,6], k = 4

Output: 5

One of the ways we can end with 5 at the top of the pile after 4 moves is as follows:

Step 1: Remove the topmost element = 5. The pile becomes [2,2,4,0,6].
Step 2: Remove the topmost element = 2. The pile becomes [2,4,0,6].
Step 3: Remove the topmost element = 2. The pile becomes [4,0,6].
Step 4: Add 5 back onto the pile. The pile becomes [5,4,0,6]. Note that this is not the only way to end with 5 at the top of the pile. It can be shown that 5 is the largest answer possible after 4 moves.

Example 2

Input: nums = [2], k = 1

Output: -1

In the first move, our only option is to pop the topmost element of the pile. Since it is not possible to obtain a non-empty pile after one move, we return -1.

Constraints

1 <= nums.length <= 105
0 <= nums[i], k <= 109

Solution Approach

Greedy Choice Strategy

The solution focuses on making the best choice at each step. First, determine which elements can potentially be placed at the top after k moves. This involves evaluating the possibility of removing or adding elements in such a way that the largest element can end up at the top. For every element in nums, check if it can become the topmost element based on the number of moves left.

Invariant Validation

Once a greedy decision is made, validate it using an invariant check. For each element, ensure that if it is possible to make it the topmost element with the remaining moves, it will remain the largest possible option. The number of remaining moves impacts whether elements can be re-added, removed, or if the array reaches a dead end.

Edge Case Handling

Special cases, such as when there is only one element or when no valid topmost element can be achieved, must be handled. In such cases, return -1 if it's impossible to obtain a non-empty pile after exactly k moves. Otherwise, identify the valid topmost element after k moves.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the approach chosen. The most efficient solutions will have a time complexity close to O(n) where n is the length of the nums array. Space complexity depends on the extra data structures used for tracking elements during the moves.

What Interviewers Usually Probe

Can the candidate identify the key greedy choice in this problem?
Does the candidate show an understanding of the invariant conditions required for the greedy approach?
Can the candidate handle edge cases, such as an empty pile or fewer than k elements?

Common Pitfalls or Variants

Common pitfalls

Failing to recognize the greedy choice that maximizes the top element after exactly k moves.
Incorrectly validating the invariant, leading to incorrect conclusions about the top element after k moves.
Not handling edge cases properly, such as returning an incorrect result when the pile is emptied prematurely.

Follow-up variants

What if we were allowed to remove more than one element at a time?
How would the solution change if we could perform additional operations other than removing and adding elements?
How would you approach this problem if k were much larger than the number of elements in nums?

FAQ

What is the key pattern to solve 'Maximize the Topmost Element After K Moves'?

The main pattern involves greedy choice and invariant validation. At each step, check if an element can be the topmost after a set number of moves.

How do we handle cases where the pile is emptied before k moves are made?

If it is impossible to perform k valid moves and end with a non-empty pile, return -1. Edge cases like these should be handled explicitly.

What happens if k is larger than the number of elements in nums?

If k is larger than the number of elements, you may still achieve a solution by removing and re-adding elements as long as there are valid moves left.

Can this problem be solved with a brute-force approach?

A brute-force approach may work, but it will likely be inefficient, especially with large arrays. A greedy strategy ensures optimal performance.

How can I test my solution for edge cases?

Test cases where k is very small, very large, or when nums contains only one element. Also, test with larger arrays where early moves could empty the pile.

terminal

Solution

Solution 1

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class Solution:
    def maximumTop(self, nums: List[int], k: int) -> int:
        if k == 0:
            return nums[0]
        n = len(nums)
        if n == 1:
            if k % 2:
                return -1
            return nums[0]
        ans = max(nums[: k - 1], default=-1)
        if k < n:
            ans = max(ans, nums[k])
        return ans

Continue Practicing

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