Problem Description

The problem presents us with an array called nums with length n, and an integer m. We need to determine if it's possible to divide the array into n non-empty arrays by carrying out a certain process.

The division process allows us to split any existing array into two subarrays, as long as each of the resulting subarrays meets at least one of these two conditions:

1. The subarray length is one, meaning it cannot be split further.
2. The sum of the elements in the subarray is equal to or greater than m.

The overall goal is to determine if we can continue splitting the arrays in this manner until we have exactly n non-empty arrays. We must return true if such a split is possible, or false otherwise. A subarray is defined as a contiguous sequence of elements within the array, meaning the elements of the subarray are next to each other and have not been reordered.

Intuition

To solve this problem, we'll employ the approach of depth-first search (DFS) to explore all possible ways of splitting the array until each resulting subarray meets the specified conditions. The recursive nature of DFS is suitable for this task because we need to try all possible splits and then, for each split, try all possible subsequent splits, and so on.

We begin by precomputing the prefix sums of the array, which helps us efficiently calculate the sum of any subarray in constant time. This optimization is crucial to ensure that we're not recalculating the sum of elements over and over for each potential subarray division.

Now, we define a recursive function dfs(i, j) which checks whether we can split the subarray nums[i:j+1] successfully according to the rules provided.

This recursive function works as follows:

1. If the current subarray length is 1 (i.e., i == j), this means we cannot split it further and the condition is trivially satisfied. We return true.
2. Otherwise, we iterate over all possible split points k from i to j - 1 to divide the array into two parts: nums[i:k] and nums[k+1:j].
3. For each split point k, we check if the sum of elements in both subarrays is greater than or equal to m.
4. If both resulting subarrays satisfy one of the conditions (either they cannot be split further or their sum is greater than or equal to m), we recursively call dfs on them.
5. If the recursive calls return true for both subarrays, it means we can successfully split the array with the current split point k. We then return true.
6. If no split point allows for a successful division, we return false.

The resulting decision tree created by the recursive dfs function will explore all possible ways to split the array and eventually tell us if it's possible or not to split the original array into n non-empty arrays satisfying the given conditions.

We cache the results of function calls to ensure we do not repeat the same computation multiple times for the same subarray, which significantly reduces the complexity and makes the solution feasible within a reasonable time frame.

The final call to dfs(0, len(nums) - 1) kicks off the process by attempting to split the entire original array.

Learn more about Greedy and Dynamic Programming patterns.

Solution Approach

The solution takes a dynamic programming approach using a depth-first search (DFS) pattern along with memoization to avoid redundant computations. The primary data structure used is the s, which is a list of prefix sums of the given nums array. This structure allows us to quickly obtain the sum of any subarray in constant time. By using the accumulate method from the itertools module, we generate this list such that s[i] represents the sum of elements from the start of the array up to but not including element i of nums.

Let's walk through the key parts of the implementation:

1. Prefix Sum Computation:

   s = list(accumulate(nums, initial=0))

   The prefix sums array s is initialized with an additional 0 at the start for simplifying edge cases. This means that the sum of the subarray nums[i:j+1] can be calculated as s[j + 1] - s[i].

2. Depth-First Search Function (DFS): The recursive dfs function is the core of the solution, defined as:

   @cache
   def dfs(i: int, j: int) -> bool:

   This function is decorated with @cache from the functools library, which automatically stores the results of recursive calls, thus implementing memoization.

3. Base Case:

   if i == j:
       return True

   If the subarray has length 1 (i == j), we cannot split it further, and this satisfies the condition.

4. Recursive Case: The loop iterates through all possible split points between i and j exclusively.

   for k in range(i, j):

   The loop checks each potential way to split nums[i:j+1] into nums[i:k] and nums[k+1:j].

5. Condition Checking:

   a = k == i or s[k + 1] - s[i] >= m
   b = k == j - 1 or s[j + 1] - s[k + 1] >= m

   Here, a checks if the left subarray (nums[i:k]) either can't be split further (k == i) or its sum meets the condition (>= m). Similarly, b checks the right subarray (nums[k+1:j]).

6. Recursive Calls and Conjunction:

   if a and b and dfs(i, k) and dfs(k + 1, j):
       return True

   If both subarrays meet their respective conditions (a and b), recursive calls are made to check if the subarrays can be split further following the rules. If both calls return true, the current split is a valid split and the function returns true.

7. Return False:

   return False

   If no valid split is found, the function returns false.

8. Initial Call: The initial call is made to attempt to split the whole array:

   return dfs(0, len(nums) - 1)

   This call begins the recursion, and if it returns true, then the array can be split according to the given conditions.

In summary, the implementation explores all possible divisions of the array into required subarrays using a depth-first search while memorizing intermediate results to reduce the computational overhead. The final return value indicates whether the original array can be split into n non-empty arrays that meet the specified constraints.

Example Walkthrough

Let's illustrate the solution approach using a small example. Consider an array nums = [3, 2, 2, 1] with m = 4.

1. Prefix Sum Computation: Compute the prefix sums:

   s = [0, 3, 5, 7, 8] // [initial 0, sum up to before index 1, 2, 3, 4 of nums]

2. Initial Call: We begin with the function call dfs(0, 3) to split the whole array, nums[0:4].

3. Explore Splits: For split point k = 0, we have:

   • Left subarray: nums[0:0], which is empty and invalid.
   • Move to the next split point.

   For split point k = 1, we have:

   • Left sum: s[1] - s[0] = 3 - 0 = 3, which is less than m, so cannot be used.
   • Move to the next split point.

   For split point k = 2, we have:

   • Left subarray: nums[0:2] ([3, 2])
   • Right subarray: nums[2:4] ([2, 1])
   • Left sum: s[2] - s[0] = 5, which is greater than m.
   • Right sum: s[4] - s[2] = 3, which is less than m.

   But since the right subarray length is not 1, we need to check its possible splits recursively.

   For the left subarray:

   • Call dfs(0, 1) which checks whether nums[0:2] can be split.

     For split point k = 0, we have:

     • Left and right subarray sums are 3 and 2 respectively.
     • Both subarrays cannot be split further (single element arrays).
     • The condition is satisfied (base case). Return true.

   For the right subarray:

   • Call dfs(2, 3) which checks whether nums[2:4] can be split.

     For split point k = 2, we have:

     • Left and right subarray sums are both 2 and 1 respectively.
     • Both cannot be split further (single element arrays).
     • The condition is satisfied (base case), return true.

   Since both left and right splits from the main split k = 2 returned true, our initial dfs(0,3) will also return true.

4. Final Output: The final call to dfs(0, 3) returns true, indicating that it's possible to split the array nums = [3, 2, 2, 1] into subarrays that meet the condition or have a length of 1. The subarrays after valid splits are nums[0:2] = [3, 2] and nums[2:4] = [2, 1].

This example illustrates the recursive and divide-and-conquer nature of the depth-first search approach used to solve the problem, leveraging memoization to avoid redundant computations and improve efficiency.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is primarily governed by the depth-first search implemented in the dfs function. We are exploring all possible ways to split the array nums into subarrays with the constraint that each subarray sum should be no less than m.

Every element in the array could be a potential splitting point, and at each splitting point, we are making two recursive calls to the dfs function (one for the left side of the split and one for the right side). In the worst-case scenario, this recursive process forms a full binary tree because we consider splitting at each point and each of those splits can lead to two more splits, and so on.

However, the use of the @cache decorator (which utilizes memoization) significantly cuts down on redundant calculations by storing previously computed results of the dfs function. This means we won't compute the same state ((i, j) pair) more than once.

If n is the length of nums, in the worst case without memoization, the dfs function could be called in order of O(2^n) times (since each function call can lead to two more). With memoization, each unique pair (i, j) will be calculated only once, leading to O(n^2) possible states.

The for loop within the dfs function iterates from i to j, so in the worst case, it could iterate n times for each recursive call. Therefore, considering memoization and the nested loop, the time complexity would be O(n^3).

Space Complexity

The space complexity of the code comes from the use of recursive calls in the dfs function (which uses the call stack) and memoization (which uses additional space to store the results).

1. Call Stack: In the worst case, the call stack could go as deep as n levels since we might recursively be calling the dfs from the first element to the last. Therefore, the call stack could contribute O(n) space complexity.

2. Memoization Cache: Since we are caching each unique (i, j) pair, and there could be O(n^2) such pairs, the space complexity due to memoization is O(n^2).

The space complexity for the array s, which holds the prefix sums, is O(n).

When combined, the dominating factor is the memoization space, leading to an overall space complexity of O(n^2).

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