Problem Description

You are provided with an array arr of n positive integers and a positive integer k. An array is defined as K-increasing if for every index i such that k <= i <= n-1, the inequality arr[i-k] <= arr[i] is satisfied. In other words, for any element in the array, if you move k steps backward, you should not find a larger number.

For example, the array [4, 1, 5, 2, 6, 2] is K-increasing when k=2 because every element is greater than or equal to the element which is 2 places before it. However, it is not K-increasing for k=1 because 4 (element at index 0) is greater than 1 (element at index 1).

The task is to convert the array into a K-increasing array by performing the minimum number of operations. In one operation, you can choose an index i and change arr[i] to any positive integer.

The goal is to find out the minimum number of such operations needed to make the array K-increasing given the value of k.

Intuition

The solution utilizes a dynamic programming approach with a twist. The idea is that if you divide the array into k subarrays, where each subarray contains elements that are k indices apart in the original array, you'll notice that for the array to be K-increasing overall, each of these subarrays must be non-decreasing.

Here's the intuition broken down step by step:

1. Subarray Division: Consider the arr=[4, 1, 5, 2, 6, 2] and k=2; we have two subarrays [4, 5, 6] and [1, 2, 2]. If each of these subarrays is non-decreasing, the original array is K-increasing.

2. Longest Increasing Subsequence (LIS): For each subarray, we want to keep it non-decreasing with the minimum number of changes. To achieve this, we need to find the length of the Longest Increasing Subsequence (LIS) of the subarray. The reason behind this is that elements in the LIS do not need to be changed, as they already contribute to making the subarray non-decreasing.

3. Operations Count: Once we have the LIS length, the number of operations required to make the subarray non-decreasing is the total number of elements minus the LIS length. This is because we can keep the LIS as is and change the other elements to fit around it.

4. Summing Up: Since our original array is divided into k subarrays, we apply the LIS strategy for each subarray and sum up the operations required. This will give us the total minimum number of operations needed to make the entire array K-increasing.

The provided solution uses a helper function lis(arr) which calculates the required operations for a given subarray by finding the length of the LIS using binary search (bisect_right). Then it uses list comprehension to sum up the operations for each subarray, which are slices of the original array (arr[i::k] for i in range(k)).

This approach is efficient because it reduces the problem to k individual LIS problems and avoids unnecessary changes to elements that are already part of the LIS, hence minimizing the number of operations.

Learn more about Binary Search patterns.

Solution Approach

The key to implementing the solution is understanding the concept of Longest Increasing Subsequence (LIS) within the context of the given array and how it applies to each of the k subsequences.

The provided Python solution includes a nested function lis, which is the implementation of the LIS algorithm. This is not the traditional dynamic programming solution for LIS with O(n^2) complexity but a more efficient version that uses binary search (bisect_right from the bisect module) and has a time complexity of O(n log n) for each subsequence.

Here's the breakdown of the algorithms and data structures used in the solution:

1. Nested Function lis(arr):

   • This function computes the length of the LIS for a given subarray.
   • It initializes an empty list t that will store the last element of the smallest increasing subsequence of each length found so far.
   • The function iterates over each element x in the subarray:
     • Using binary search (bisect_right), it finds the position idx in t where x could be placed to either extend an existing subsequence or replace an element to create a potential new subsequence.
     • If idx is equal to the length of t, it means x is larger than any element in t, and we can append x to t, effectively extending the longest subsequence seen so far.
     • Otherwise, we replace the element at t[idx] with x, as x might be the start of a new potential subsequence or a smaller end element for a subsequence of length idx.
   • After processing all elements in the subarray, the length of LIS is obtained by subtracting the length of t from the length of the subarray (len(arr) - len(t)).

2. Combining the Results:

   • The main part of the solution is a single line that sums up the operation counts for each k subsequence: sum(lis(arr[i::k]) for i in range(k)).
   • It iterates over each start index from 0 to k-1 and takes every k-th element from the original array using slicing arr[i::k]. This yields k subsequences that must each individually be non-decreasing to satisfy the K-increasing property.
   • For each subsequence, it applies the lis function to find the number of operations needed, which is then accumulated to get the total minimum number of operations required for the entire array.

By applying the LIS algorithm separately to each of the k subsequences, the solution effectively translates the problem of making an array K-increasing into multiple independent subproblems. Each subproblem aims to minimize the adjustments within its subsequence, and the sum of the solutions to these subproblems is the minimum number of operations needed for the whole array.

Example Walkthrough

Let's consider a small example array arr = [3, 9, 4, 6, 7, 5] and k = 3. According to the given solution approach, we need to divide this array into k subarrays where each subarray contains elements that are k indices apart in the original array. Let's do this step by step:

1. Subarray Division:

   • With k = 3, we divide the original array into 3 subarrays: [3, 6], [9, 7], [4, 5].
   • These subarrays are created by taking every third element from the original array starting from indices 0, 1, and 2 respectively.

2. Longest Increasing Subsequence (LIS):

   • We then need to determine the LIS for each subarray:
     • For the first subarray [3, 6], the LIS is [3, 6] itself, and the length is 2.
     • For the second subarray [9, 7], since 7 is not larger than 9, the LIS is [7], and the length is 1.
     • For the third subarray [4, 5], the LIS is [4, 5], and the length is 2.

3. Operations Count:

   • We calculate the number of operations required to make each subarray non-decreasing:
     • For [3, 6], no operations are needed as it is already non-decreasing.
     • For [9, 7], we need 2 - 1 = 1 operation, changing 9 to a number not greater than 7 (e.g., 7 or any smaller number).
     • For [4, 5], no operations are needed as it is already non-decreasing.

4. Summing Up:

   • Sum up the operations required: 0 (for the first subarray) + 1 (for the second subarray) + 0 (for the third subarray) = 1.
   • Therefore, we need a minimum of 1 operation to make the entire array k-increasing.

This walkthrough illustrates the solution steps using a simple example, showcasing how to divide the original problem into smaller ones by finding the LIS of the subarrays and then deducing the minimum number of operations needed to achieve a k-increasing array.

Solution Implementation

Time and Space Complexity

The given code implements a function that, for a given list arr and an integer k, finds the minimum number of elements to change to ensure that every k-th subsequence of the list is non-decreasing.

Time Complexity:

To analyze the time complexity, let’s break down the process:

1. The lis function is called k times, once for each of the k subsequences.
2. Inside the lis function, there's a loop that goes through the elements of a subsequence (which has a length of about n/k, where n is the length of arr).
3. In the worst case, bisect_right performs a binary search, which has a time complexity of O(log m) where m is the size of the temporary list t.
4. The size of t can grow up to the size of the subsequence being considered, in the worst case approximated to n/k.

Putting it all together:

• Single call to lis: O((n/k) * log(n/k))
• lis called k times: O(k * (n/k) * log(n/k)) = O((n * log(n/k)))

Since we have a log term that depends on n/k, the overall time complexity isn't perfectly linear with respect to n. However, as k increases, the time complexity approaches O(n log n) since the subsequences processed by each call get shorter.

Space Complexity:

The space complexity can be evaluated by considering:

1. The temporary list t used inside the lis function, which holds the elements of the longest increasing subsequence (LIS) within a k-th subsequence;
2. t's size is at most n/k for a given k-th subsequence; However, t is reused for each subsequence and does not grow with k.
3. There are no additional data structures that grow with the size of the input, other than the input itself and the function call stack.

Hence, the space complexity is O(n/k), which simplifies to O(n) because we keep a single t for each subsequence.

Please note: since the actual maximum length of t can vary depending on the input arr, the space complexity in practice can be less than O(n) if the subsequences have a strong increasing trend, but in the worst case, it is O(n).

Learn more about how to find time and space complexity quickly using problem constraints.