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This approach involves representing row and column conditions as directed graphs. You perform a topological sort on these graphs to determine valid row and column positions for each number from 1 to k. If cycle detection reveals any contradictions, it indicates that it's impossible to build the matrix.
The algorithm has a time complexity of O(k + n + m), where k is the size of the matrix, n is the number of row conditions, and m is the number of column conditions. The space complexity is O(k + n + m) for storing the conditions and dependencies.
1// C# code omitted for brevityC# implementation uses Dictionaries and queues for Kahn's algorithm to arrange the numbers as per the conditions.
This approach considers using backtracking to build the matrix. For each valid permutation of numbers, it ensures constraints are satisfied and adjusts dynamically by propagating constraints to subsequent steps.
Time complexity: O(k!). Space complexity: O(k^2) due to matrix storage and recursive call stack.
1Utilizing recursive functions and checking constraints before further exploration helps ensure only valid matrices are formed.