Number of Paths with Max Score

Problem Description

You are given a square board of characters. You can move on the board starting at the bottom right square marked with the character 'S'. You need to reach the top left square marked with the character 'E'. The rest of the squares are labeled either with a numeric character '1, 2, ..., 9' or with an obstacle 'X'. In one move you can go up, left or up-left (diagonally) only if there is no obstacle there. Return a list of two integers: the first integer is the maximum sum of numeric characters you can collect, and the second is the number of such paths that you can take to get that maximum sum, taken modulo 10^9 + 7. In case there is no path, return [0, 0].

Key Insights

• The problem can be solved using dynamic programming, where we store the maximum score and the number of paths at each cell.
• We can only move up, left, or diagonally up-left, which means we need to consider transitions from these three possible directions.
• Obstacles ('X') must be avoided, and they block any paths through them.
• The final result is computed from the top-left cell 'E' to the bottom-right cell 'S', traversing the board in reverse.

Space and Time Complexity

Time Complexity: O(n^2), where n is the length of the board (since we potentially inspect each cell). Space Complexity: O(n^2), for storing the maximum scores and counts in a 2D array.

Solution

We utilize a dynamic programming approach to solve the problem. We create two 2D arrays: one for storing the maximum score at each cell and another for counting the number of paths leading to that score. We iterate through the board from the destination to the starting point, updating these arrays based on possible moves (up, left, diagonal).

The algorithm works as follows:

1. Initialize two DP tables: dp_score to store the maximum score at each cell and dp_count to store the number of paths to that score.
2. Start from the 'S' position and fill the tables moving towards 'E'.
3. For each cell, check if moving from the three valid directions leads to a new maximum score. Update the score and count accordingly.
4. Return the values from the top-left cell, ensuring to take the modulo as specified.

Code Solutions