#2659

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Make Array Empty

Solve the "Make Array Empty" problem using binary search to determine the minimum number of operations required.

array binary search greedy binary indexed tree segment tree

Problem Statement

You are given an integer array nums containing distinct numbers. The goal is to determine the minimum number of operations needed to make the array empty. In each operation, you can remove elements based on a specific order, and the number of steps required depends on the optimal sequence of deletions.

The problem can be efficiently solved using binary search over the valid answer space. By determining the order in which indices are removed, you can minimize the number of operations. The solution requires you to think about the greedy strategies involved in the array manipulation process.

Examples

Example 1

Input: nums = [3,4,-1]

Output: 5

Example details omitted.

Example 2

Input: nums = [1,2,4,3]

Output: 5

Example details omitted.

Example 3

Input: nums = [1,2,3]

Output: 3

Example details omitted.

Constraints

1 <= nums.length <= 105
-109 <= nums[i] <= 109
All values in nums are distinct.

Solution Approach

Binary Search over Answer Space

Utilize binary search to determine the minimum number of operations needed. By performing binary search on the possible number of steps, you can narrow down the solution space and find the optimal answer efficiently.

Greedy Approach for Array Removal

In each operation, apply a greedy strategy where the best element (in terms of reducing future operations) is chosen for removal. This ensures the minimum steps are taken while removing elements from the array.

Segment Tree for Range Queries

Leverage a segment tree or binary indexed tree to handle range queries effectively. This data structure allows efficient updates and retrievals during the removal process, optimizing the solution.

Complexity Analysis

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

The time complexity depends on the final approach. Binary search and data structures like segment trees or binary indexed trees can help reduce the time complexity significantly, possibly bringing it to O(log n) for some steps. Space complexity will vary based on the data structure used to store the array indices during the process.

What Interviewers Usually Probe

The candidate should demonstrate understanding of binary search over a valid answer space.
Look for clear reasoning in how the greedy approach is applied to the problem.
Evaluate if the candidate considers advanced data structures like segment trees or binary indexed trees for optimization.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the order of operations and how it affects the number of steps needed.
Overcomplicating the problem by applying unnecessary data structures when binary search is sufficient.
Failure to recognize the importance of efficiently querying and updating the array's state during removal.

Follow-up variants

Implementing a variation with different array constraints, such as non-distinct values.
Modifying the problem to remove elements based on a different criterion, like value-based removal rather than order.
Introducing multiple arrays and determining the combined minimum number of operations across all arrays.

FAQ

What is the optimal approach for the "Make Array Empty" problem?

The optimal approach involves using binary search over the possible answer space and applying greedy strategies for efficient element removal.

How does binary search apply to this problem?

Binary search helps minimize the number of operations by narrowing down the valid answer space, effectively reducing the time complexity.

What are the common pitfalls in solving the "Make Array Empty" problem?

Common pitfalls include misunderstanding the operation order, using unnecessary data structures, or not efficiently handling the array removal process.

What data structures can help solve this problem more efficiently?

Segment trees and binary indexed trees are helpful for efficiently querying and updating the state of the array during element removal.

Can this problem be solved without binary search?

While binary search is the most efficient approach, the problem can be solved with other strategies, though they may be less optimized.

terminal

Solution

Solution 1: Hash Table + Sorting + Fenwick Tree

First, we use a hash table $pos$ to record the position of each element in array $nums$. Then, we sort array $nums$. The initial answer is the position of the minimum element in array $nums$ plus 1, which is $ans = pos[nums[0]] + 1$.

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class Solution:
    def countOperationsToEmptyArray(self, nums: List[int]) -> int:
        pos = {x: i for i, x in enumerate(nums)}
        nums.sort()
        sl = SortedList()
        ans = pos[nums[0]] + 1
        n = len(nums)
        for k, (a, b) in enumerate(pairwise(nums)):
            i, j = pos[a], pos[b]
            d = j - i - sl.bisect(j) + sl.bisect(i)
            ans += d + (n - k) * int(i > j)
            sl.add(i)
        return ans

Solution 2

#### Python3

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class Solution:
    def countOperationsToEmptyArray(self, nums: List[int]) -> int:
        pos = {x: i for i, x in enumerate(nums)}
        nums.sort()
        sl = SortedList()
        ans = pos[nums[0]] + 1
        n = len(nums)
        for k, (a, b) in enumerate(pairwise(nums)):
            i, j = pos[a], pos[b]
            d = j - i - sl.bisect(j) + sl.bisect(i)
            ans += d + (n - k) * int(i > j)
            sl.add(i)
        return ans

Continue Practicing

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