#2111

Hard

auto_awesomeBinary search over the valid answer space

LeetCode Problem Workspace

Minimum Operations to Make the Array K-Increasing

This problem requires finding the minimum number of operations to make an array K-increasing using binary search over the answer space.

array binary search

Problem Statement

You are given a 0-indexed array arr of n positive integers and a positive integer k. An array is called K-increasing if for every index i, where k <= i <= n-1, the condition arr[i-k] <= arr[i] holds.

In one operation, you can choose an index i and change arr[i] to any positive integer. The goal is to find the minimum number of operations required to make the array K-increasing.

Examples

Example 1

Input: arr = [5,4,3,2,1], k = 1

Output: 4

For k = 1, the resultant array has to be non-decreasing. Some of the K-increasing arrays that can be formed are [5,6,7,8,9], [1,1,1,1,1], [2,2,3,4,4]. All of them require 4 operations. It is suboptimal to change the array to, for example, [6,7,8,9,10] because it would take 5 operations. It can be shown that we cannot make the array K-increasing in less than 4 operations.

Example 2

Input: arr = [4,1,5,2,6,2], k = 2

Output: 0

This is the same example as the one in the problem description. Here, for every index i where 2 <= i <= 5, arr[i-2] <= arr[i]. Since the given array is already K-increasing, we do not need to perform any operations.

Example 3

Input: arr = [4,1,5,2,6,2], k = 3

Output: 2

Indices 3 and 5 are the only ones not satisfying arr[i-3] <= arr[i] for 3 <= i <= 5. One of the ways we can make the array K-increasing is by changing arr[3] to 4 and arr[5] to 5. The array will now be [4,1,5,4,6,5]. Note that there can be other ways to make the array K-increasing, but none of them require less than 2 operations.

Constraints

1 <= arr.length <= 105
1 <= arr[i], k <= arr.length

Solution Approach

Binary Search over the Answer Space

We perform binary search over the possible number of operations. The key idea is to determine if a given number of operations can result in a K-increasing array. By checking each possible operation count, we find the minimum required.

Divide the Array into Subsequences

The problem can be simplified by dividing the array into non-overlapping subsequences, each of which should satisfy the K-increasing condition. This allows us to apply dynamic programming techniques or greedy approaches to each subsequence.

Dynamic Programming or Greedy Method

For each subsequence formed by the binary search strategy, we can use dynamic programming or a greedy approach to determine the minimum changes needed. The solution involves minimizing the number of operations for each subsequence independently.

Complexity Analysis

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

The time complexity depends on the efficiency of the binary search and subsequence checking approach. Binary search typically operates in O(log n) steps, while subsequence checks may require O(n) time per step. Therefore, the overall complexity can vary based on the specific approach used.

What Interviewers Usually Probe

Can the candidate optimize the problem using binary search?
Is the candidate able to divide the problem into smaller subsequences?
Does the candidate utilize dynamic programming or greedy strategies efficiently?

Common Pitfalls or Variants

Common pitfalls

Ignoring the importance of dividing the array into subsequences and attempting to solve it as a single entity.
Overcomplicating the solution by missing the opportunity to optimize using binary search.
Failing to handle large input sizes efficiently within the given time and space constraints.

Follow-up variants

What if k is greater than the length of the array?
How would the solution change if negative numbers were allowed in the array?
What if we had to return all possible K-increasing arrays with the minimum operations?

FAQ

How can binary search be used in the 'Minimum Operations to Make the Array K-Increasing' problem?

Binary search is used over the number of operations, checking whether a given number of operations can make the array K-increasing. This reduces the problem's complexity.

What is the main challenge in solving the 'Minimum Operations to Make the Array K-Increasing' problem?

The main challenge lies in finding the minimum number of operations, optimizing the solution by using binary search and efficiently dividing the array into subsequences.

What patterns are most relevant to solving the 'Minimum Operations to Make the Array K-Increasing' problem?

The key pattern is binary search over the answer space, which is crucial for minimizing operations and determining whether a solution is feasible within the given constraints.

What is the best approach to handling the input size in the 'Minimum Operations to Make the Array K-Increasing' problem?

Optimizing the solution using binary search and breaking the array into subsequences is essential for handling large input sizes efficiently.

Can dynamic programming be applied to the 'Minimum Operations to Make the Array K-Increasing' problem?

Yes, dynamic programming can be used within the subsequences to minimize the number of operations required for each subsequence independently.

terminal

Solution

Solution 1

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class Solution:
    def kIncreasing(self, arr: List[int], k: int) -> int:
        def lis(arr):
            t = []
            for x in arr:
                idx = bisect_right(t, x)
                if idx == len(t):
                    t.append(x)
                else:
                    t[idx] = x
            return len(arr) - len(t)

        return sum(lis(arr[i::k]) for i in range(k))

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