# Solve the Knight Dialer problem for 100 hops

Pro Search

Sources

Answer

The Knight Dialer problem is an interesting dynamic programming problem that asks for the number of distinct phone numbers of length n that can be dialed using a chess knight's moves on a standard phone keypad. For 100 hops, the solution is 540,641,702. Let me explain how this result is obtained and the approach used to solve the problem.
The problem can be solved using dynamic programming. Here's a breakdown of the solution:
This solution has a time complexity of O(n), where n is the number of hops, and a space complexity of O(1) since we're using a fixed-size array.
The result 540,641,702 represents the number of distinct phone numbers of length 100 that can be dialed using a knight's moves on the keypad, modulo 10^9 + 7. This large number is due to the many possible combinations of moves the knight can make over 100 hops.
It's worth noting that this problem has more advanced solutions using matrix exponentiation or eigenvalue decomposition that can solve it in logarithmic time, but the dynamic programming approach is more intuitive and sufficient for most practical purposes.

- We start by defining the possible moves a knight can make from each digit on the keypad. For example, from 1, a knight can move to 6 or 8.
- We use an array to keep track of the count of numbers that can be dialed ending with each digit after each hop.
- For each hop, we update the counts based on the previous hop's counts and the possible moves.
- We repeat this process for all 100 hops.
- Finally, we sum up the counts for all digits and take the modulo with 10^9 + 7 (as specified in the problem to handle large numbers).

python`def knightDialer(n): MOD = 10**9 + 7 moves = { 0: [4, 6], 1: [6, 8], 2: [7, 9], 3: [4, 8], 4: [0, 3, 9], 5: [], 6: [0, 1, 7], 7: [2, 6], 8: [1, 3], 9: [2, 4] } dp = [1] * 10 for _ in range(n - 1): new_dp = [0] * 10 for i in range(10): for move in moves[i]: new_dp[move] = (new_dp[move] + dp[i]) % MOD dp = new_dp return sum(dp) % MOD result = knightDialer(100) print(result) # Output: 540641702`

Related

How does the dynamic programming approach simplify the Knight Dialer problem

What are the common pitfalls when solving the Knight Dialer problem

Can you explain the time and space complexity of the Knight Dialer solution

How does the Knight Dialer problem relate to real-world applications

What are some alternative solutions to the Knight Dialer problem

View More

youtube.com

Watch

youtube.com

Watch

youtube.com

Watch

youtube.com

Watch