Problem Description

The given problem presents an array nums of integers and permits operations where you select a subarray and replace it with its sum. The task is to return the maximum length of a non-decreasing array that you can achieve by performing any number of these operations. A non-decreasing array means each element is greater than or equal to the previous element.

Intuition

To solve this problem, the intuition is to use dynamic programming along with binary search. Dynamic programming will store intermediate solutions to subproblems, while binary search will help us quickly find positions in the array relevant to our calculations.

We'll keep track of a running sum of the array and construct a DP array f where f[i] represents the maximum length of the non-decreasing subarray ending at position i. We'll also maintain a pre array where pre[i] holds the highest index j < i such that nums[j] <= 2 * nums[i].

While iterating over the array, we'll use binary search to find the index j where we can replace a subarray [x...j] that doubles the element nums[x]. We then set pre[j] to the current index i if it's not already set to a larger index. By doing this, we can efficiently extend the non-decreasing subarray as we move through nums.

The solution returns f[n], which by the end of the algorithm, holds the maximum length of a non-decreasing array that can be created.

Learn more about Stack, Queue, Binary Search, Dynamic Programming, Monotonic Queue and Monotonic Stack patterns.

Solution Approach

The implementation of the solution uses dynamic programming along with a binary search to effectively tackle the problem. Here's a step-by-step breakdown of how the algorithm and its components – arrays f and pre, the sum array s, and binary search – work together in the solution:

1. Start by initializing an array s to keep the prefix sums of the nums array. This allows us to determine the sum of any subarray quickly.

2. Initialize two arrays – f which will hold the dynamic programming states representing the maximum length of the non-decreasing subarray, and pre which helps track the indices relevant for our subarray replacements.

3. Iterate through the array nums with i going from 1 to n, where n is the length of nums. For each i, we perform the following steps:

   • Set pre[i] to the maximum of pre[i] and pre[i - 1], ensuring pre[i] always holds the largest index so far where subarray replacement can start.

   • Calculate f[i] as f[pre[i]] + 1. This essentially adds the length of the optimal subarray before the current index plus one for the current element.

   • Perform a binary search on the prefix sums array s to find the least index j where s[j] >= s[i] * 2 - s[pre[i]]. The sought j represents the potential end of a replacement subarray.

   • Update the pre[j] index to i, but only if i is greater than the current value of pre[j]. This prepares the pre array for the next elements to make decisions on whether they can extend the current non-decreasing subarray.

4. The dynamic programming array f accumulates the information about the maximum length of a non-decreasing array that can be achieved up to each index i.

After iterating over the whole array nums, f[n] will contain the length of the maximum non-decreasing subarray possible after performing the allowed subarray replacement operations. The final result is then simply f[n].

The combination of prefix sums, binary search, and dynamic programming allows this algorithm to efficiently compute the maximum length of the non-decreasing subarray in an otherwise complex problem.

Example Walkthrough

Let us illustrate the solution approach with a small example:

Suppose our array nums is [1, 2, 4, 1, 2, 4, 1].

1. We first compute the prefix sums array s. The prefix sum at index i is the sum of all nums[j] where j <= i:

   nums: [1, 2, 4, 1, 2, 4, 1]
   s:    [1, 3, 7, 8, 10, 14, 15] (each `s[i]` is the sum of all elements up to and including `i` in `nums`)

2. Initialize the f and pre arrays of the same length as nums, with initial values of 0 for f and indices of 0 for pre.

   f:   [0, 0, 0, 0, 0, 0, 0]
   pre: [0, 0, 0, 0, 0, 0, 0]

3. Start iterating through nums:

   • For i = 1, pre[1] is set to the maximum of pre[0] and 0, yielding pre[1] = pre[0]. For i = 1, the non-decreasing subarray can only include nums[i], so f[1] = 1.

   nums: [1, 2, 4, 1, 2, 4, 1]
   f:    [0, 1, 0, 0, 0, 0, 0]
   pre:  [0, 0, 0, 0, 0, 0, 0]

   • For i = 2, since nums[2] is greater than or equal to nums[1], we can extend the current non-decreasing subarray. Thus, f[2] = f[1] + 1 = 2.

   nums: [1, 2, 4, 1, 2, 4, 1]
   f:    [0, 1, 2, 0, 0, 0, 0]
   pre:  [0, 0, 0, 0, 0, 0, 0]

   • At i = 3, since nums[3] is less than nums[2], we need to perform a replacement operation. We perform a binary search to find the least index j where s[j] >= s[i] * 2 - s[pre[i]]. We find that j is 3 itself. We set pre[3] to the current index i and calculate f[3].

   nums: [1, 2, 4, 1, 2, 4, 1]
   f:    [0, 1, 2, 2, 0, 0, 0]
   pre:  [0, 0, 0, 3, 0, 0, 0]

   • For i = 4, since nums[4] is greater than or equal to nums[3], we can extend the current non-decreasing subarray. Thus, f[4] = f[3] + 1 = 3.

   nums: [1, 2, 4, 1, 2, 4, 1]
   f:    [0, 1, 2, 2, 3, 0, 0]
   pre:  [0, 0, 0, 3, 3, 0, 0]

   • For i = 5, since nums[5] is greater than or equal to nums[4], we can extend the current non-decreasing subarray. Thus, f[5] = f[4] + 1 = 4.

   nums: [1, 2, 4, 1, 2, 4, 1]
   f:    [0, 1, 2, 2, 3, 4, 0]
   pre:  [0, 0, 0, 3, 3, 4, 0]

   • At i = 6, since nums[6] is less than nums[5], we need to perform a replacement operation. We perform a binary search to find the least index j where s[j] >= s[i] * 2 - s[pre[i]]. We find that j is 6 itself. We set pre[6] to the current index i and calculate f[6].

   nums: [1, 2, 4, 1, 2, 4, 1]
   f:    [0, 1, 2, 2, 3, 4, 4]
   pre:  [0, 0, 0, 3, 3, 4, 6]

4. The final result is the largest value in f, which in this case is 4. This indicates the maximum length of a non-decreasing subarray that we can achieve after performing these operations is 4.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code consists of several parts:

1. Accumulating the sum: The accumulate function is applied to the list nums, which takes O(n) time for a list of n elements.

2. Iterative Calculation:

   • There is a for loop that runs from 1 to n. Inside the loop, the max function is called, and access/update operations on arrays are performed. These operations are O(1).
   • The bisect_left function is called within the for loop, which performs a binary search on the prefix sum array s. This takes O(log n) time. Since this is within the for loop, it contributes to O(n log n) time over the full loop.

Combining these, the overall time complexity is O(n) + O(n log n). Since O(n log n) dominates O(n), the final time complexity is O(n log n).

Space Complexity

The space complexity of the code is due to the storage used by several arrays:

1. Prefix Sum Array s: Stores the prefix sums, requiring O(n) space.
2. Array f: An array of length n+1, also requiring O(n) space.
3. Array pre: Another array of length n+2, contributing O(n) space.

The total space complexity sums up to O(n) because adding the space requirements for s, f, and pre does not change the order of magnitude.

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