Problem Description

This problem provides a matrix grid of size n * m and requires us to construct a new matrix p, also of size n * m. The new matrix p is defined as the product matrix of grid, where each element p[i][j] is the product of all elements in grid excluding the element grid[i][j], with each product taken modulo 12345.

To elaborate, if we were to calculate p[i][j], we would take the product of all the numbers in the matrix grid except for grid[i][j], and then take the result modulo 12345. Our task is to construct the entire matrix p that satisfies this condition for every single element.

Intuition

Constructing the product matrix p can appear challenging at first because of the requirement to exclude the current element from the product. A naive approach might involve calculating the product of all elements for each position separately, which would lead to a time-consuming process because of the repeated multiplications. However, this problem can be efficiently solved by decomposing the operations into prefix and suffix products. This approach allows us to calculate the desired product for each element without redundancy.

The intuition for the solution is to first calculate the suffix product matrix, which stores the product of all elements that are below and to the right of the current position. This is done by traversing the matrix from the bottom right corner upwards. Following this, we can calculate the prefix product for each element by traversing the matrix from the top left corner downwards, multiplying the already calculated suffix product by the prefix and taking the modulo at each step. By doing these two separate traversals, we cleverly circumvent the need to directly exclude the current element (since it was never included in suf or pre, to begin with).

Ultimately, the p[i][j] equals to the multiplication of the suffix product (calculated in the first traversal) and the prefix product (calculated in the second traversal). Both of these are calculated modulo 12345 as per the problem statement. As we are making use of previously computed results, computational efficiency is greatly improved over the naive approach.

Learn more about Prefix Sum patterns.

Solution Approach

The problem dictates a classic use of Algorithmic Patterns in which the prefix and suffix multiplications are used to simplify the process of excluding a particular element from the product. The solution employs two important concepts: suffix and prefix product decomposition. To implement the solution, we initialize a product matrix p and two variables, suf and pre, which will hold the suffix and prefix products, respectively.

Step 1: Initialize the suf variable (suffix product) to 1. Here, suf acts as a running product of all elements to the right and below the current element, excluding it. Starting from the bottom right corner of the matrix, iterate over the matrix in reverse order. For each element grid[i][j], we will set p[i][j] to the current suf value. Then, we update the suf by multiplying it by the current element in grid and taking the modulo 12345 as prescribed by the problem.

Step 2: The next traversal calculates the prefix product. Initialize the pre variable (prefix product) to 1. Starting from the top left corner of the matrix, iterate over the matrix in normal order. For each element grid[i][j], we multiply the p[i][j] (which at this moment only contains the suffix product) by the current pre, take the modulo 12345, and then update the pre with the current element in grid, also with a modulo operation.

The processes indicated in Step 1 and Step 2 ensure that each p[i][j] is calculated by multiplying the prefix product. Up to but not including grid[i][j], with the suffix product of all elements after grid[i][j], effectively excluding grid[i][j] from the product as required.

The reason we can do this in two separate steps is because of the mathematical property that allows us to separate the product of a series of numbers into separate parts, compute them independently, and then combine them to get the final product. This separation into prefix and suffix helps us avoid including the current element in either product. The modulo operation is an integral part of every product calculation due to the requirement to take each product modulo 12345.

By employing this specific order of operation—suffix product calculation followed by prefix product multiplication—we avoid the use of division, which is beneficial when dealing with modulus operations as division is not well-defined under modulus without additional steps (like using a multiplicative inverse). Therefore, this technique not only meets the problem constraints but also adheres to the computational limitations when dealing with modular arithmetic.

Finally, after the completion of these two passes, the matrix p is returned, containing the desired product matrix. This approach efficiently computes the solution in O(nxm) time complexity with O(nxm) space complexity, where n and m are the dimensions of the input matrix grid.

Example Walkthrough

Let's walk through the solution with a small example. Consider the following grid matrix of size 3x3:

grid = [
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
]

To construct the new matrix p, we will follow the solution approach detailed in the content.

Step 1: Initialize suf to 1 and traverse the matrix in reverse order. Let's start at the bottom-right corner:

Iteration 1: (i=2, j=2) => p[2][2] = suf (1) => p = [
  [?, ?, ?],
  [?, ?, ?],
  [?, ?, 1]
]
suf = suf * grid[2][2] % 12345 => suf = 1 * 9 % 12345 = 9

(This process repeats for the remaining elements of the matrix from the bottom-right to the top-left.)

After the first traversal, the p matrix will have the suffix products:

p = [
  [1, 1, 1],
  [1, 1, 2],
  [1, 9, 1]
]

Step 2: Initialize pre to 1 and traverse the matrix in normal order:

Iteration 1: (i=0, j=0) => p[0][0] = suf * pre % 12345 => p[0][0] = 1 * 1 % 12345 = 1
pre = pre * grid[0][0] % 12345 => pre = 1 * 1 % 12345 = 1

(In a similar fashion, we update each element in p for the whole matrix in normal order.)

After the second traversal, we fully calculate the p matrix:

p = [
  [(5*6*7*8*9) % 12345, (4*6*7*8*9) % 12345, (4*5*7*8*9) % 12345],
  [(2*3*7*8*9) % 12345, (1*3*7*8*9) % 12345, (1*2*7*8*9) % 12345],
  [(2*3*5*6) % 12345, (1*3*5*6) % 12345, (1*2*5*6) % 12345]
]

With some arithmetic, we find:

p = [
  [15120 % 12345, 60480 % 12345, 30240 % 12345],
  [60480 % 12345, 30240 % 12345, 40320 % 12345],
  [360 % 12345, 180 % 12345, 60 % 12345]
]

And, finally, the resulting p matrix is:

p = [
  [2775, 10530, 7895],
  [10530, 7895, 10080],
  [360, 180, 60]
]

The new p matrix is the product matrix of grid, where each entry is the product of all other elements except itself, modulo 12345. Thus, by using the suffix and prefix product computation, we've successfully found the solution without directly excluding each grid[i][j] from the multiplication, proving the efficiency of this approach.

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(n * m), where n is the number of rows and m is the number of columns in the input grid. This complexity arises because the code contains two nested loops, each of which iterates over every cell in the matrix exactly once.

As for the space complexity, aside from the space needed for the output matrix p, the space complexity is O(1). This is because there are only a finite number of additional variables (n, m, suf, pre, and mod) that do not depend on the size of the input matrix, therefore occupying constant space.

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