1605. Find Valid Matrix Given Row and Column Sums

Problem

Question

You are given two arrays rowSum and colSum of non-negative integers where rowSum[i] is the sum of the elements in the ith row and colSum[j] is the sum of the elements of the jth column of a 2D matrix. In other words, you do not know the elements of the matrix, but you do know the sums of each row and column.

Find any matrix of non-negative integers of size rowSum.length x colSum.length that satisfies the rowSum and colSum requirements.

Return a 2D array representing any matrix that fulfills the requirements. It’s guaranteed that at least one matrix that fulfills the requirements exists.

Example 1:

Input: rowSum = [3,8], colSum = [4,7] Output: [[3,0], [1,7]] Explanation: 0th row: 3 + 0 = 3 == rowSum[0] 1st row: 1 + 7 = 8 == rowSum[1] 0th column: 3 + 1 = 4 == colSum[0] 1st column: 0 + 7 = 7 == colSum[1] The row and column sums match, and all matrix elements are non-negative. Another possible matrix is: [[1,2], [3,5]]

Example 2:

Input: rowSum = [5,7,10], colSum = [8,6,8] Output: [[0,5,0], [6,1,0], [2,0,8]]

Constraints:

• 1 <= rowSum.length, colSum.length <= 500
• 0 <= rowSum[i], colSum[i] <= 108
• sum(rowSum) == sum(colSum)

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Solutions

🏆 Greedy

Time Complexity: $O(N\ast M)$ | Space Complexity: $O(N\ast M)$

The solution is simpler than we might thing at the beginning. The solution uses a greedy approach to construct the required matrix. We start by initializing a matrix of zeros with dimensions derived from rowSum and colSum. For each cell (i, j), we assign the minimum value between the remaining sum of the i-th row and the j-th column. After assigning a value to the cell, we update the rowSum and colSum accordingly by subtracting the assigned value.

class Solution:
    def restoreMatrix(self, rowSum: List[int], colSum: List[int]) -> List[List[int]]:
        matrix = [[0] * len(colSum) for _ in range(len(rowSum))]
 
        for i in range(len(rowSum)):
            for j in range(len(colSum)):
                matrix[i][j] = min(rowSum[i], colSum[j])
                rowSum[i] -= matrix[i][j]
                colSum[j] -= matrix[i][j]
 
        return matrix
 

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Before arriving at the solution above, I experimented with more complex approaches. Initially, I theorized that we should start filling the matrix by prioritizing the largest sums in colSum and rowSum. Besides, I created additional dictionaries to keep track of colSum and rowSum changes.

• Version 1:

class Solution:
    def restoreMatrix(self, rowSum: List[int], colSum: List[int]) -> List[List[int]]:
    
        heap = []
        memoColSum = defaultdict(int)
        memoRowSum = defaultdict(int)
 
        for rowIn, row in enumerate(rowSum):
            for colIn, col in enumerate(colSum):
 
                heapq.heappush(heap, (-(row + col), (row, col), (rowIn, colIn)))
 
        matrix = [[0 for _ in range(len(colSum))] for _ in range(len(rowSum))]
 
        while heap:
            _, (rowSum, colSum), (row, col) = heapq.heappop(heap)
            value = min(rowSum - memoRowSum[row], colSum - memoColSum[col])
            value = max(0, value)
            matrix[row][col] = value
            memoColSum[col] += value
            memoRowSum[row] += value
 
        return matrix
 

Version 2:

class Solution:
    def restoreMatrix(self, rowSum: List[int], colSum: List[int]) -> List[List[int]]:
    
        array = []
        memoColSum = defaultdict(int)
        memoRowSum = defaultdict(int)
 
        for rowIn, row in enumerate(rowSum):
            for colIn, col in enumerate(colSum):
                array.append((row, col, rowIn, colIn))
 
        matrix = [[0 for _ in range(len(colSum))] for _ in range(len(rowSum))]
 
        for rowSum, colSum, row, col in array:
            value = min(rowSum - memoRowSum[row], colSum - memoColSum[col])
            matrix[row][col] = value
            memoColSum[col] += value
            memoRowSum[row] += value
 
        return matrix
 

• Version 3:

class Solution:
    def restoreMatrix(self, rowSum: List[int], colSum: List[int]) -> List[List[int]]:
        matrix = [[0] * len(colSum) for _ in range(len(rowSum))]
 
        for i in range(len(rowSum)):
            for j in range(len(colSum)):
                matrix[i][j] = min(rowSum[i], colSum[j])
                rowSum[i] -= matrix[i][j]
                colSum[j] -= matrix[i][j]
 
        return matrix