Problem Description

In this problem, you are given a two-dimensional matrix where each cell contains an integer. Starting from the top-left corner of the matrix (0, 0), you need to find a path that leads you to the bottom-right corner (m - 1, n - 1). The only allowed moves are to the right or down from your current position.

Your goal is to find the path that results in the maximum non-negative product of all the numbers on that path. The product of a path is calculated by multiplying all the values of the cells visited on that path. Finally, you'll need to return this maximum product modulo 10^9 + 7. If no such non-negative product exists, you should return -1.

Intuition

The problem can be tackled using dynamic programming. The key observation is that the maximum product at any given cell will either be the result of multiplying the maximum or the minimum product up to that point by the value of the current cell. This is because multiplying a negative value by another negative value could give a positive product, which might be the maximum.

We use a 3-dimensional dynamic programming (dp) array where each cell at (i, j) holds two values:

1. dp[i][j][0]: The minimum product to reach cell (i, j). This is important to keep track of because a negative minimum product can become a positive product if we multiply it by a negative number in the grid.
2. dp[i][j][1]: The maximum product to reach cell (i, j).

For the first row and first column, the maximum and minimum products can only come from one direction, so we just multiply the current cell value by the value in the dp array of the previous cell in the same row or column.

As we iterate through the inner cells of the grid, we calculate the possible minimum and maximum products using the previously filled cells above and to the left. We calculate the new values to store by considering the current cell's value:

• If the current cell's value is non-negative, we achieve the minimum product by multiplying the cell's value by the minimum of the two possible pre-calculated minimum products. We obtain the maximum product by multiplying the cell's value by the maximum of the two possible pre-calculated maximum products.
• If the cell value is negative, the minimum product could result in the largest product (if later multiplied by another negative value), hence we take the maximum of the pre-calculated maximum products and multiply it by the cell's value. Conversely, the maximum value is achieved by taking the minimum of the pre-calculated minimum products and multiplying by the cell's value.

After calculating dp values for the entire grid, we look at the maximum product of the last cell. If the maximum is negative, return -1. Otherwise, return the maximum product modulo 10^9 + 7.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution to this problem involves understanding the intricacies of dynamic programming and how to effectively handle both positive and negative numbers while searching for the maximum product path.

The algorithm uses a 3-dimensional dynamic programming array called dp. Here's the step-by-step breakdown:

1. Initialization:

   • The dp array is initialized with a size of m x n x 2, where m and n are the dimensions of the input grid. Each cell in dp will hold two numbers: the minimum and maximum product up to that point in the grid.
   • The first cell, dp[0][0], is initialized with the value of the first cell of the grid, grid[0][0], as both the minimum and maximum, since there's only one number in the path at this point.

2. Filling the first row and column:

   • Each cell in the first row and column can only be reached from one direction. Thus, for the first row, we iterate over j from 1 to n, and set dp[0][j] equal to the product of dp[0][j - 1][0] (the previous cell) and grid[0][j]. We do the same for the first column, iterating over i from 1 to m, setting each dp[i][0] according to the product of the previous cell and the current grid value.

3. Filling the rest of the dp array:

   • We iterate over the cells of the grid starting from (1, 1) and for each cell at (i, j), we calculate the minimum and maximum product paths to that point. This involves comparisons and multiplications using the values from the top (i - 1, j) and left (i, j - 1) cells.
   • The crux of the approach: If grid[i][j] is non-negative, we:
     • Find dp[i][j][0] (min product) by multiplying grid[i][j] with the smaller of either dp[i - 1][j][0] (above) or dp[i][j - 1][0] (left).
     • Calculate dp[i][j][1] (max product) by multiplying grid[i][j] with the larger of either dp[i - 1][j][1] (above) or dp[i][j - 1][1] (left). If grid[i][j] is negative, we switch the above, since minimum times a negative can lead to a maximum product, and vice versa:
     • Calculate dp[i][j][0] (min product) by multiplying grid[i][j] with the larger of either dp[i - 1][j][1] (above) or dp[i][j - 1][1] (left).
     • Find dp[i][j][1] (max product) by multiplying grid[i][j] with the smaller of either dp[i - 1][j][0] (above) or dp[i][j - 1][0] (left).

4. Getting the result:

   • After filling in the entire dp array, we look at the bottom-right corner (last cell) of the grid dp[m - 1][n - 1]. Specifically, we're interested in the maximum product path, which is stored in dp[m - 1][n - 1][1].
   • If the maximum product path is negative, this indicates there are no non-negative product paths to the bottom-right corner, and thus we return -1.
   • If the maximum product is non-negative, we return the maximum product modulo 10^9 + 7 as per the problem's requirement.

The solution demonstrates the utilization of dynamic programming to maintain and compare potential product paths through a grid with both positive and negative integers, ensuring the computations adhere to the maximum non-negative product principle.

Example Walkthrough

Let's consider a small 3x3 matrix as an example to illustrate the solution approach:

Grid:
[1,  -2, 1]
[1,   3, -2]
[-1, -2, 4]

Here's a step-by-step application of the solution approach:

1. Initialization:

   • Initialize dp as a 3-dimensional array with the dimensions of the grid and an extra dimension to hold both minimum and maximum products. For our 3x3 grid, dp will have dimensions 3 x 3 x 2.
   • Set dp[0][0] to [1, 1] since the first cell of the grid is 1, and there's only one path with one cell at the starting point.

2. Filling the first row and column:

   • First row: Iterate over j from 1 to 2.
     • For j = 1, dp[0][1] is initialized as [1 * (-2), 1 * (-2)] => [-2, -2].
     • For j = 2, dp[0][2] becomes [-2 * 1, -2 * 1] => [-2, -2].
   • First column: Iterate over i from 1 to 2.
     • For i = 1, dp[1][0] is [1 * 1, 1 * 1] => [1, 1].
     • For i = 2, dp[2][0] becomes [1 * (-1), 1 * (-1)] => [-1, -1].

3. Filling the rest of the dp array:

   • Now, we iterate over the inner cells starting from (1, 1).
   • For i = 1 and j = 1 (grid[1][1] is 3, a positive value):
     • Calculate minimum as min(dp[0][1][0], dp[1][0][0]) * 3 => min(-2, 1) * 3 => 1 * 3 => 3.
     • Calculate maximum as max(dp[0][1][1], dp[1][0][1]) * 3 => max(-2, 1) * 3 => 1 * 3 => 3.
     • Set dp[1][1] to [3, 3].
   • For i = 1 and j = 2 (grid[1][2] is -2, a negative value):
     • Calculate the minimum as max(dp[0][2][1], dp[1][1][1]) * -2 => max(-2, 3) * -2 => 3 * -2 => -6.
     • Calculate the maximum as min(dp[0][2][0], dp[1][1][0]) * -2 => min(-2, 3) * -2 => -2 * -2 => 4.
     • Set dp[1][2] to [-6, 4].
   • For i = 2 and j = 1 (grid[2][1] is -2, a negative value):
     • Minimum: max(dp[1][1][1], dp[2][0][1]) * -2 => max(3, -1) * -2 => 3 * -2 => -6.
     • Maximum: min(dp[1][1][0], dp[2][0][0]) * -2 => min(3, -1) * -2 => -1 * -2 => 2.
     • Set dp[2][1] to [-6, 2].
   • For i = 2 and j = 2 (grid[2][2] is 4, a positive value):
     • Minimum: min(dp[1][2][0], dp[2][1][0]) * 4 => min(-6, -6) * 4 => -6 * 4 => -24.
     • Maximum: max(dp[1][2][1], dp[2][1][1]) * 4 => max(4, 2) * 4 => 4 * 4 => 16.
     • Set dp[2][2] to [-24, 16].

4. Getting the result:

   • Looking at the last cell dp[2][2], we have the maximum product as 16. Since it is non-negative, we return 16 % (10^9 + 7), which is 16.

This walk-through demonstrates how the dynamic programming array is filled out in order to maximize the product path from the top-left to the bottom-right of the grid.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the code is determined by the nested loops that iterate over each cell of the matrix grid. Since there are two loops, one going through all rows (m) and the other through all columns (n), and the operations inside the inner loop take O(1) time, the time complexity is O(m * n).

Space Complexity

For space complexity, the code is using a 3-dimensional list dp with dimensions m x n x 2 to store the minimum and maximum product paths up to each cell. The size of this list dictates the space complexity, which is O(m * n) as it's directly proportional to the size of the input grid.

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