Problem

https://leetcode.com/problems/knight-dialer/description/

A chess knight can move as indicated in the chess diagram below:

 .           

 

This time, we place our chess knight on any numbered key of a phone pad (indicated above), and the knight makes N-1 hops.  Each hop must be from one key to another numbered key.

Each time it lands on a key (including the initial placement of the knight), it presses the number of that key, pressing N digits total.

How many distinct numbers can you dial in this manner?

Since the answer may be large, output the answer modulo 10^9 + 7.

Example 1:

Input: 1
Output: 10

Example 2:

Input: 2
Output: 20

Example 3:

Input: 3
Output: 46

Note:

• 1 <= N <= 5000

Solution: DP

V1

Similar to 花花酱 688. Knight Probability in Chessboard

We can define dp[k][i][j] as # of ways to dial and the last key is (j, i) after k steps

Note: dp[*][3][0], dp[*][3][2] are always zero for all the steps.

Init: dp[0][i][j] = 1

Transition: dp[k][i][j] = sum(dp[k – 1][i + dy][j + dx]) 8 ways of move from last step.

ans = sum(dp[k])

Time complexity: O(kmn) or O(k * 12 * 8) = O(k)

Space complexity: O(kmn) -> O(mn) or O(12*8) = O(1)

V2

define dp[k][i] as # of ways to dial and the last key is i after k steps

init: dp[0][0:10] = 1

transition: dp[k][i] = sum(dp[k-1][j]) that j can move to i

ans: sum(dp[k])

Time complexity: O(k * 10) = O(k)

Space complexity: O(k * 10) -> O(10) = O(1)

Related Problem

• 花花酱 688. Knight Probability in Chessboard
• 花花酱 LeetCode 576. Out of Boundary Paths