#3510

Hard

auto_awesomeArray scanning plus hash lookup

LeetCode Problem Workspace

Minimum Pair Removal to Sort Array II

The problem asks to find the minimum number of operations to make an array non-decreasing by removing pairs of elements.

array hash table linked list heap priority queue simulation

Problem Statement

Given an array nums, you can perform the following operation any number of times: remove one pair of elements that are out of order. Return the minimum number of operations needed to make the array non-decreasing.

An array is said to be non-decreasing if each element is greater than or equal to its previous element (if it exists). This means you need to remove pairs where nums[i] > nums[i+1] until the array satisfies the non-decreasing condition.

Examples

Example 1

Input: nums = [5,2,3,1]

Output: 2

The array nums became non-decreasing in two operations.

Example 2

Input: nums = [1,2,2]

Output: 0

The array nums is already sorted.

Constraints

1 <= nums.length <= 105
-109 <= nums[i] <= 109

Solution Approach

Array Scanning and Hash Lookups

The key to solving this problem is efficiently scanning the array and using hash lookups to track the operations. By checking pairs of elements and using a hash table to record pairs for removal, you can determine the minimum operations needed.

Simulation of Pair Removal

Simulate the removal process using data structures such as heaps or ordered sets to identify the elements that should be removed. By simulating the removal of elements, you can efficiently calculate the minimum number of operations.

Optimized Handling of Edge Cases

Consider edge cases such as arrays that are already non-decreasing or arrays with only one element. These special cases can simplify the logic by avoiding unnecessary operations.

Complexity Analysis

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

The time and space complexity depend on the final approach, which includes factors such as the number of elements in the array and the specific data structures used. Typically, the time complexity is O(n) or O(n log n) depending on the optimization chosen, and space complexity is proportional to the number of elements being stored for processing (O(n)).

What Interviewers Usually Probe

The candidate should be able to describe the correct approach of using array scanning and hash lookups.
The candidate should demonstrate an understanding of when to simulate pair removal and which data structures are efficient for this.
The candidate must identify edge cases and explain how they handle them during the simulation.

Common Pitfalls or Variants

Common pitfalls

Not optimizing the scanning process and hash lookup steps, resulting in higher time complexity.
Failing to handle edge cases, such as already sorted arrays or arrays of length one.
Using inefficient data structures that increase space complexity unnecessarily.

Follow-up variants

Consider a variation where the array is sorted in descending order initially.
What if the array contains negative numbers as well? This would add complexity to the solution.
Explore a variant where only specific pairs are allowed for removal, adding constraints to the operations.

FAQ

What is the primary approach to solving the Minimum Pair Removal to Sort Array II problem?

The primary approach is to use array scanning combined with hash lookups to identify pairs for removal, minimizing the number of operations.

How do data structures like heaps and ordered sets help in solving this problem?

Heaps and ordered sets help efficiently simulate the pair removal process by allowing you to quickly identify and remove out-of-order pairs.

What are the edge cases I need to handle when solving this problem?

Edge cases include already sorted arrays, arrays with one element, and arrays with negative numbers.

Can the solution to this problem be optimized further?

Yes, by choosing the appropriate data structures and optimizing the scanning and lookup processes, you can reduce the time complexity.

How does GhostInterview assist in preparing for the Minimum Pair Removal to Sort Array II problem?

GhostInterview provides step-by-step assistance in practicing the solution approach, offering examples and a variety of problem variants to help master the technique.

terminal

Solution

Solution 1: Sorted Set

We define a sorted set $\textit{sl}$ to store tuples $(\textit{s}, i)$ of the sum of all adjacent element pairs and their left index, define another sorted set $\textit{idx}$ to store the indices of remaining elements in the current array, and use the variable $\textit{inv}$ to record the number of inversions in the current array. Initially, we traverse the array $\textit{nums}$, add tuples of the sum of all adjacent element pairs and their left index to the sorted set $\textit{sl}$, and calculate the number of inversions $\textit{inv}$.

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class Solution:
    def minimumPairRemoval(self, nums: List[int]) -> int:
        n = len(nums)
        sl = SortedList()
        idx = SortedList(range(n))
        inv = 0
        for i in range(n - 1):
            sl.add((nums[i] + nums[i + 1], i))
            if nums[i] > nums[i + 1]:
                inv += 1
        ans = 0
        while inv:
            ans += 1
            s, i = sl.pop(0)
            pos = idx.index(i)
            j = idx[pos + 1]
            if nums[i] > nums[j]:
                inv -= 1
            if pos > 0:
                h = idx[pos - 1]
                if nums[h] > nums[i]:
                    inv -= 1
                sl.remove((nums[h] + nums[i], h))
                if nums[h] > s:
                    inv += 1
                sl.add((nums[h] + s, h))
            if pos + 2 < len(idx):
                k = idx[pos + 2]
                if nums[j] > nums[k]:
                    inv -= 1
                sl.remove((nums[j] + nums[k], j))
                if s > nums[k]:
                    inv += 1
                sl.add((s + nums[k], i))

            nums[i] = s
            idx.remove(j)
        return ans

Continue Practicing

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