Problem Description

The given problem presents a challenge where we are dealing with a matrix called mat with dimensions m x n, where each row is sorted in a non-decreasing order. The task here is to create arrays by choosing exactly one element from each row of the matrix. The aim is to find the kth smallest sum that can be obtained from all the possible arrays created in this way. An array's sum is the total of all the elements within that array.

Intuition

To approach this problem, we utilize a step by step strategy that builds upon the smallest possible sums array by array, while looking ahead by only considering the top k smallest sums at each step to reduce computational complexity.

Here's how we conceptualize our approach:

1. Begin Small: Initialize a list, let's call it pre, with a single element 0, which represents the smallest initial sum (with no elements selected).

2. Build Up: For each row in the matrix, we combine the values in pre with the new values from the current row. This helps us to generate all possible sums with one more element added to the arrays.

3. Stay Within Bounds: To maintain performance and not generate an unnecessarily large number of sums, we only keep the top k smallest sums at any given step. This is done by sorting the generated sums and slicing the list to retain only the first k elements.

4. Iterate Through Rows: Repeat step 2 and 3 for each row of the matrix. With each iteration, pre grows to contain the smallest sums resulting from picking exactly one element from each of the rows processed so far.

5. Final Answer: After processing all the rows, the last element in pre will be the kth smallest sum, because pre is always maintained to be sorted with only k elements. Thus, pre[-1] yields the desired result.

By employing this incremental approach and limiting the consideration to k elements at each step, we efficiently find the desired sum without having to explore all possible combinations, which would be infeasible for large matrices.

Learn more about Binary Search and Heap (Priority Queue) patterns.

Solution Approach

The solution employs an efficient approach utilizing simple data structures and Python's list comprehension to handle the potentially large search space in a way that is both manageable and scalable to larger datasets.

Here's a step-by-step breakdown of the implementation:

1. Initialization: We begin by creating a list called pre, initialized with a single element, 0. This list represents the sum of elements chosen so far, and with no elements having been considered, the sum is simply 0.

   pre = [0]

2. Iterative Combination: For every row cur in the matrix mat, we use a nested loop through list comprehension to create new sums. This is where we explore every combination of the previously accumulated sums in pre with the new elements from the current row cur.

   for cur in mat:
       pre = [a + b for a in pre for b in cur[:k]]

   Inner Loop Explanation: For every element a in pre and for every element b in the top k elements of the current cur, compute a new sum a + b. The slicing cur[:k] ensures that we don't consider more elements than necessary, which is crucial for performance optimization.

3. Sorting and Trimming: After combining sums from pre and the current row, this new list could be quite large, larger than k. We need to only keep the smallest k sums. We do this by first sorting the new sums and then slicing the list to retain only the first k elements. The sorting operation here not only ensures that we are keeping the smallest sums but also prepares the pre list for the next iteration.

   pre = sorted(pre)[:k]

4. Repeat Process: This process is repeated iteratively for each row of the matrix, each time updating the pre list with the smallest sums obtained from choosing exactly one element from each row processed up to that point.

5. Extracting Result: After the last iteration, the final list pre will be sorted and contain the k smallest sums of all possible arrays. The kth smallest value, which is the answer we need to find, will be the last element of this list.

   return pre[-1]

By utilizing this approach, we effectively minimize the number of possible sums we have to consider at each step, which is essential for handling inputs where the number of combinations can be extraordinarily high. The algorithm leverages the sorted property of the matrix's rows and Python's efficient list operations to deliver a solution that meets the problem's constraints.

Example Walkthrough

Let's consider a matrix mat with two rows and three columns, and suppose we want to find the 2nd smallest sum by choosing exactly one element from each row.

Let mat be:

[
  [1, 3, 4],
  [2, 5, 7]
]

We want the 2nd smallest sum when picking one element from each row, meaning k=2 in this context.

Step 1: Initialization We initialize pre with a single element, 0, representing the empty sum.

pre = [0]

Step 2: Iterative Combination Starting with the first row [1, 3, 4], combine pre with this row.

# First row
pre = [a + b for a in pre for b in [1, 3, 4][:2]] # We only use the first 2 elements for k=2.
# pre becomes [0+1, 0+3] = [1, 3]

We now have all possible sums that can be generated by choosing one element from the first row with pre only keeping the smallest 2 sums.

Step 3: Sorting and Trimming Sort pre and keep only the first k elements.

pre = sorted(pre)[:2]
# Since pre is already [1, 3], no change is made in this case.

Repeat the iterative combination with the next row [2, 5, 7].

Step 4: Repeat Process Take the next row, and combine each element with the sums in pre while considering only the first k elements.

# Second row
pre = [a + b for a in pre for b in [2, 5, 7][:2]] # Again, take the first 2 elements only for k=2.
# pre becomes [1+2, 1+5, 3+2, 3+5] = [3, 6, 5, 8] (combinations of sums)

Again, sort pre and keep only the first k elements.

pre = sorted(pre)[:2]
# After sorting, pre becomes [3, 5], so we keep these as the smallest sums.

Step 5: Extracting Result After processing all rows, the final list pre contains the k smallest sums.

# The 2nd smallest sum is hence
return pre[-1]
# pre[-1] gives 5, which is the 2nd smallest sum possible.

Therefore, by following the approach, we end up with a pre list that contains [3, 5], and the 2nd smallest sum is the last element in this list, which is 5. This sum comes from the array [1, 4], the second smallest array sum by choosing one element from each row of the given mat.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is determined by the operations executed in the nested loops.

• The outer loop iterates over each row of the matrix mat, which we can denote as m, where m is the number of rows in the matrix.
• The inner loop essentially computes all pairwise sums between elements in pre and cur[:k]. If n represents the number of columns in the matrix, we have a maximum of k elements considered in cur. Thus, in the inner loop, the number of sums will be on the order of len(pre) * k. Initially, len(pre) = 1 and can grow up to k after sorting and truncation.
• Sorting the list pre can take O(k log k) time at most since we only keep the smallest k elements.

The worst-case computational load at each iteration is dominated by the sorting step, which is O(k log k). Thus, the total time complexity of the algorithm is O(m * k * log k).

Space Complexity

The space complexity of the code is mainly due to the storage of the list pre, which maintains a length of at most k elements through the iterations.

• At each step of the outer loop, pre is replaced by a new list of length at most k.
• The generation of pairwise sums (a + b for a in pre for b in cur[:k]) creates an intermediary list of size up to len(pre) * k, but since it's immediately sorted and truncated to size k, the space consumption does not accumulate over iterations.

Hence, the space complexity of the code is O(k).

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