Introduction

In this blog post, we will explore how to solve the LeetCode problem "935" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

The chess knight has a unique movement, it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an L). The possible movements of chess knight are shown in this diagram: A chess knight can move as indicated in the chess diagram below: We have a chess knight and a phone pad as shown below, the knight can only stand on a numeric cell (i.e. blue cell). Given an integer n, return how many distinct phone numbers of length n we can dial. You are allowed to place the knight on any numeric cell initially and then you should perform n - 1 jumps to dial a number of length n. All jumps should be valid knight jumps. As the answer may be very large, return the answer modulo 109 + 7. Example 1: Input: n = 1 Output: 10 Explanation: We need to dial a number of length 1, so placing the knight over any numeric cell of the 10 cells is sufficient. Example 2: Input: n = 2 Output: 20 Explanation: All the valid number we can dial are [04, 06, 16, 18, 27, 29, 34, 38, 40, 43, 49, 60, 61, 67, 72, 76, 81, 83, 92, 94] Example 3: Input: n = 3131 Output: 136006598 Explanation: Please take care of the mod. Constraints: 1 <= n <= 5000

Explanation

Here's the breakdown of the problem and the solution:

• High-Level Approach:

  • Use dynamic programming to store the number of ways to reach a digit i with j moves.
  • Iterate through the number of moves, updating the counts based on valid knight jumps.
  • Sum the counts for all digits after n moves to get the final result, applying the modulo operator at each step.

• Complexity:

  • Runtime: O(n), Space: O(1) because we are storing values only for the current move, space is optimized by using just two arrays instead of nx10 DP array.

Code

    def knightDialer(n: int) -> int:
    """
    Calculates the number of distinct phone numbers of length n that can be dialed using a knight's movements.

    Args:
        n: The length of the phone number.

    Returns:
        The number of distinct phone numbers modulo 10^9 + 7.
    """

    MOD = 10**9 + 7

    # Define the possible knight moves from each digit
    moves = {
        0: [4, 6],
        1: [6, 8],
        2: [7, 9],
        3: [4, 8],
        4: [0, 3, 9],
        5: [],  # 5 has no valid moves
        6: [0, 1, 7],
        7: [2, 6],
        8: [1, 3],
        9: [2, 4],
    }

    # Initialize the counts for each digit for the first move
    counts = [1] * 10

    # Iterate for n - 1 moves
    for _ in range(1, n):
        new_counts = [0] * 10
        for digit in range(10):
            for next_digit in moves[digit]:
                new_counts[next_digit] = (new_counts[next_digit] + counts[digit]) % MOD
        counts = new_counts

    # Sum the counts for all digits to get the total number of distinct phone numbers
    total_count = sum(counts) % MOD
    return total_count