How do you solve Number of Paths with Max Score on LeetCode?

Number of Paths with Max Score (LeetCode #1301) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n²) time complexity and O(n²) space complexity. Key insight: Dynamic programming is essential for optimizing pathfinding problems.

What is the brute force solution for Number of Paths with Max Score?

The brute force approach for Number of Paths with Max Score is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute-force approach involves exploring all possible paths from the start point 'S' to the end point 'E'. We will recursively explore each valid move (up, left, or up-left) while keeping track of the score collected along the way. The optimal Optimal Solution approach improves this to O(n²).

What algorithm or data structure is used to solve Number of Paths with Max Score?

Number of Paths with Max Score uses the following concepts: Array, Dynamic Programming, Matrix. The recommended patterns to study are: Dynamic Programming, Grid Traversal.

What are the interview tips for Number of Paths with Max Score?

Clearly explain your thought process while coding. Discuss edge cases, such as all obstacles or no valid paths. Practice dynamic programming problems to get comfortable with state transitions.

What are common mistakes when solving Number of Paths with Max Score?

Not handling obstacles correctly, leading to incorrect path counts. Failing to initialize DP tables properly, which can cause index errors.

#1301

Number of Paths with Max Score

Hard

ArrayDynamic ProgrammingMatrixDynamic ProgrammingGrid Traversal

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(n²)
Space	O(1)	O(n²)

💡

Intuition

Time O(n²)Space O(n²)

The optimal approach uses dynamic programming to efficiently calculate the maximum score and the number of paths to reach 'E'. We maintain two DP tables: one for scores and another for path counts, updating them based on previously computed values.

⚙️

Algorithm

4 steps

1Step 1: Initialize two DP tables: one for maximum scores and another for path counts, both of the same size as the board.
2Step 2: Start filling the DP tables from the bottom-right corner (S) and move towards the top-left corner (E).
3Step 3: For each cell, if it's not an obstacle, update the score and path count based on valid moves from below, left, and diagonally.
4Step 4: Return the maximum score and the number of paths that lead to that score from the top-left corner.

solution.py29 lines

1def maxScorePaths(board):
2    MOD = 10**9 + 7
3    n = len(board)
4    dp_score = [[0] * n for _ in range(n)]
5    dp_count = [[0] * n for _ in range(n)]
6    dp_count[n - 1][n - 1] = 1  # Start from 'S'
7
8    for i in range(n - 1, -1, -1):
9        for j in range(n - 1, -1, -1):
10            if board[i][j] == 'X':
11                continue
12            if i == n - 1 and j == n - 1:
13                continue
14            max_score = 0
15            if i + 1 < n:
16                max_score = max(max_score, dp_score[i + 1][j])
17            if j + 1 < n:
18                max_score = max(max_score, dp_score[i][j + 1])
19            if i + 1 < n and j + 1 < n:
20                max_score = max(max_score, dp_score[i + 1][j + 1])
21            dp_score[i][j] = max_score + (int(board[i][j]) if board[i][j].isdigit() else 0)
22            if max_score == dp_score[i + 1][j]:
23                dp_count[i][j] += dp_count[i + 1][j]
24            if max_score == dp_score[i][j + 1]:
25                dp_count[i][j] += dp_count[i][j + 1]
26            if max_score == dp_score[i + 1][j + 1]:
27                dp_count[i][j] += dp_count[i + 1][j + 1]
28
29    return [dp_score[0][0], dp_count[0][0] % MOD] if dp_score[0][0] > 0 else [0, 0]

ℹ

Complexity note: Both time and space complexities are O(n²) because we are using two 2D arrays (DP tables) to store scores and path counts for each cell in the n x n board.

1Dynamic programming is essential for optimizing pathfinding problems.
2Tracking both scores and counts can help solve complex path problems efficiently.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.