Problem Description

In this problem, we are given a matrix mat that is composed of binary values - 1's and 0's. Every 1 in the matrix represents a soldier, and every 0 represents a civilian. One of the key elements of the setup is the arrangement of the soldiers and civilians in each row; all the soldiers (1's) come before any civilians (0's). This ordering makes it visually similar to a sorted binary array where all 1's are at the start of the array, followed by all 0's.

We are asked to evaluate the "strength" of each row based on the number of soldiers (1's) in it. A row is considered "weaker" if it has fewer soldiers in it than another row, or if it has the same number of soldiers but comes earlier in the matrix (i.e., it has a smaller row index).

The problem requires us to return the indices of the k weakest rows in the matrix ordered from the weakest to the strongest. It's essentially like constructing a "leaderboard" of rows, with the least number of soldiers making a row rank higher (weaker) on this board.

Intuition

Approaching this problem, we observe two tasks: counting the number of soldiers in each row and then sorting the rows according to their "strength." Since the soldiers (1's) are all positioned to the left, the count of soldiers in a row is equal to the number of continuous 1's starting from the first column until the first 0 appears. This can be thought of as finding the first occurrence of 0 in the row.

Since the rows are sorted in the non-increasing order with all the soldiers at the beginning, we can use a binary search technique to quickly find the position of the first civilian (0) which indicates the number of soldiers in the row. The binary search here drastically reduces the time complexity over a linear scan, especially when the rows are long.

We construct a result ans that maps each row to the number of soldiers it has, by applying the binary search on the reversed row. We reverse the row before applying the binary search with bisect_right because this function is typically used to find the insertion point in a sorted array to maintain the order. By reversing, our soldiers 1 come at the end, and the insertion point for 0tells us the number of 1's.

After we have the counts for each row, we create an index list idx which initially is just a list of row indices [0, 1, 2, ..., m-1]. We sort this index list based on the corresponding soldier counts from the ans array. Finally, we use slicing to get the first k elements from this sorted index list which represent the indices of the k weakest rows.

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

Solution Approach

The solution makes effective use of Python's built-in bisect library, which provides support for maintaining a list in sorted order without having to sort the list after each insertion. In particular, the bisect_right function is utilized to implement a binary search through a row to find the count of soldiers in that row.

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

1. Determine the dimensions of the matrix with m as the number of rows and n as the number of columns.
2. An array ans is created to hold the number of soldiers (1's) for each row.
3. We iterate over each row in the matrix, and for each row, we apply bisect_right to the reversed row: bisect_right(row[::-1], 0). This reversed row is a trick used to turn our sorted array of 1's followed by 0's into an array where a binary search can find the insertion point of 0 effectively. The result of this operation is the count of civilians (0's) in the reversed row, which is subtracted from n to get the count of soldiers.
4. Create an index list idx containing the indices of the rows, which starts from 0 up to m-1.
5. The index list idx is then sorted using a lambda function as the key. This lambda function maps each index i to the number of soldiers ans[i], ensuring the sort operation arranges the indices according to the strength of the rows (with ties broken by row index, as per problem statement).
6. We use list slicing to retrieve the first k indices from the sorted idx, giving us the indices of the k weakest rows.

Code Snippet Deconstructed

Let's look at the critical sections of the code:

• ans = [n - bisect_right(row[::-1], 0) for row in mat] This line computes the strength of each row and stores it in ans. It calculates the number of soldiers as the total length n minus the number of civilians at the end of each reversed row.

• idx.sort(key=lambda i: ans[i]) This line sorts the idx list according to the number of soldiers in each corresponding row. The lambda function returns the number of soldiers for each row using ans[i] as the key, ensuring that rows with fewer soldiers come first.

• return idx[:k] Returns the first k elements from the sorted list of indices, which corresponds to the k weakest rows in the matrix ordered from weakest to strongest.

In terms of algorithms and data structures, this solution primarily uses the binary search algorithm (through bisect_right) and basic list operations in Python (list comprehension, sorting, and slicing).

Example Walkthrough

Let's consider a matrix mat and an integer k:

mat = [
  [1, 1, 0, 0],
  [1, 1, 1, 0],
  [1, 0, 0, 0],
  [1, 1, 0, 0],
  [1, 1, 1, 1]
]
k = 3

In this matrix:

• Row 0 has 2 soldiers.
• Row 1 has 3 soldiers.
• Row 2 has 1 soldier (making it the weakest).
• Row 3 has 2 soldiers.
• Row 4 has 4 soldiers (making it the strongest).

We need to find the indices of the k weakest rows in the matrix, which are k = 3 in our example.

Following the solution approach:

1. The dimensions of the matrix are m = 5 rows and n = 4 columns.
2. We'll create an array ans to hold counts of soldiers for each row.

For each row's count of soldiers using bisect_right:

• ans[0] = 4 - bisect_right([0, 0, 1, 1][::-1], 0) = 4 - 2 = 2
• ans[1] = 4 - bisect_right([0, 1, 1, 1][::-1], 0) = 4 - 3 = 1
• ans[2] = 4 - bisect_right([0, 0, 0, 1][::-1], 0) = 4 - 1 = 3
• ans[3] = 4 - bisect_right([0, 0, 1, 1][::-1], 0) = 4 - 2 = 2
• ans[4] = 4 - bisect_right([1, 1, 1, 1][::-1], 0) = 4 - 4 = 0

After applying the count for each row, ans looks like this: [2, 1, 3, 2, 0].

3. Now we have ans = [2, 1, 3, 2, 0] and our index list idx = [0, 1, 2, 3, 4].

4. Sorting the index list idx using the ans array:

After sorting idx based on ans values, we get the order [4, 1, 0, 3, 2] since ans[4] corresponds to the strongest row and ans[2] corresponds to the weakest.

5. Since we want the k weakest rows, we take the first k elements of the sorted index idx list: [4, 1, 0].

The array [4, 1, 0] is then sorted to maintain the row order: [0, 1, 4].

6. The final answer, which represents the indices of the k weakest rows ordered from weakest to stronger, is [0, 1, 4].

This method allows for efficiently determining the weakest rows due to the use of the bisect_right to perform binary searches, rather than linear searches, and thus reduces the average complexity of the solution.

Solution Implementation

Time and Space Complexity

The time complexity of the code primarily consists of two parts: the computation of the soldiers in each row and the sorting of the index based on the number of soldiers.

1. Computing the number of soldiers in each row: This uses a binary search (bisect_right) for each of the m rows. Since the binary search operates on a row of size n, the complexity for this part is O(m * log(n)).

2. Sorting the index list: After having computed the number of soldiers, we sort m indices based on the computed values. The worst-case time complexity for sorting in Python is O(m * log(m)).

Combining both parts, the total time complexity is O(m * log(n) + m * log(m)).

In terms of space complexity:

• The ans list is O(m) because it contains the number of soldiers for each of the m rows.
• The idx list is also O(m) because it includes an index for each row.
• No additional significant space is used.

Hence, the space complexity is O(m) as we only need space proportional to the number of rows.

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