The trick here is to name the state correctly, then let the implementation follow. For 935. Knight Dialer, the solution in this repository is mainly a DFS + memoization solution.
Guide
What?
We want to turn the problem statement into a smaller set of decisions the computer can repeat safely. Instead of trying to be clever immediately, read the code as a sequence of questions:
- What state are we keeping?
- How do we move from one state to the next?
- When do we know the answer is already determined?
For this file, the main tools are: DFS + memoization, dynamic programming.
The notes already sitting in the source point us in the right direction:
- recursion + memorization
Guide
When?
This is the kind of solution you want when the problem has structure hiding inside a messy-looking input. The accepted code reduces that pressure by storing exactly the information that remains useful later.
The important function names to track are dfs, knightDialer.
Guide
Why?
The solution works because it narrows the problem until every update has a clear reason to exist.
- The two-dimensional vector is the memory of the solution: grid state, DP state, or adjacency shape.
- The final return is not magic; it is the invariant after the loops or recursion have finished doing their accounting.
Guide
How?
Walk through the solution in this order:
- Read the setup variables first.
- Follow the main loop or recursive helper next.
- Watch where invalid states get skipped.
- Check which value survives to the return statement.
The most important competitive-programming habit here is to trust the invariant. Once the invariant is right, the loops become much less scary.
Guide
Complexity
- Time: O(n) to O(n log n), depending on the dominant loop or data structure operation
- Space: O(n) in the usual case for auxiliary containers or recursion
Guide
C++ Solution
Your submission
The accepted solution
01//recursion + memorization
02//Runtime: 328 ms, faster than 27.43% of C++ online submissions for Knight Dialer.
03//Memory Usage: 23.2 MB, less than 57.37% of C++ online submissions for Knight Dialer.
04class Solution {
05public:
06 int MOD = 1e9+7;
07 vector<vector<int>> edges;
08 vector<vector<int>> memo;
09
10 int dfs(int cur, int N){
11 if(N == 0) return 1;
12 if(N == 1) return edges[cur].size();
13 if(memo[cur][N] != -1) return memo[cur][N];
14
15 memo[cur][N] = 0;
16
17 for(int nei : edges[cur]){
18 memo[cur][N] = (memo[cur][N] + dfs(nei, N-1)) % MOD;
19 }
20 return memo[cur][N];
21 }
22
23 int knightDialer(int N) {
24 int ans = 0;
25 edges = vector<vector<int>>(10);
26
27 edges[0] = {4,6};
28 edges[1] = {6,8};
29 edges[2] = {7,9};
30 edges[3] = {4,8};
31 edges[4] = {3,9,0};
32 edges[5] = {};
33 edges[6] = {1,7,0};
34 edges[7] = {2,6};
35 edges[8] = {1,3};
36 edges[9] = {2,4};
37
38 memo = vector<vector<int>>(10, vector<int>(N, -1));
39
40 for(int i = 0; i <= 9; i++){
41 ans = (ans + dfs(i, N-1)) % MOD;
42 }
43
44 return ans;
45 }
46};
47
48//bottom-up DP
49//Runtime: 344 ms, faster than 26.23% of C++ online submissions for Knight Dialer.
50//Memory Usage: 21.7 MB, less than 58.60% of C++ online submissions for Knight Dialer.
51class Solution {
52public:
53 int MOD = 1e9+7;
54 vector<vector<int>> edges;
55 vector<vector<int>> memo;
56
57 int knightDialer(int N) {
58 int ans = 0;
59 edges = vector<vector<int>>(10);
60
61 edges[0] = {4,6};
62 edges[1] = {6,8};
63 edges[2] = {7,9};
64 edges[3] = {4,8};
65 edges[4] = {3,9,0};
66 edges[5] = {};
67 edges[6] = {1,7,0};
68 edges[7] = {2,6};
69 edges[8] = {1,3};
70 edges[9] = {2,4};
71
72 vector<vector<int>> dp(10, vector<int>(N, 0));
73
74 for(int n = 0; n < N; n++){
75 for(int cur = 0; cur <= 9; cur++){
76 if(n == 0){
77 dp[cur][n] = 1;
78 }else if(n == 1){
79 dp[cur][n] = edges[cur].size();
80 }else{
81 for(int nei : edges[cur]){
82 dp[cur][n] = (dp[cur][n] + dp[nei][n-1]) % MOD;
83 }
84 }
85 // cout << "(" << cur << ", " << n << "): " << dp[cur][n] << endl;
86 }
87 }
88
89 for(int cur = 0; cur <= 9; cur++){
90 ans = (ans + dp[cur][N-1]) % MOD;
91 }
92
93 return ans;
94 }
95};
96
97//DP, space optimized
98//Runtime: 184 ms, faster than 55.96% of C++ online submissions for Knight Dialer.
99//Memory Usage: 6.1 MB, less than 88.12% of C++ online submissions for Knight Dialer.
100class Solution {
101public:
102 int MOD = 1e9+7;
103 vector<vector<int>> edges;
104 vector<vector<int>> memo;
105
106 int knightDialer(int N) {
107 int ans = 0;
108 edges = vector<vector<int>>(10);
109
110 edges[0] = {4,6};
111 edges[1] = {6,8};
112 edges[2] = {7,9};
113 edges[3] = {4,8};
114 edges[4] = {3,9,0};
115 edges[5] = {};
116 edges[6] = {1,7,0};
117 edges[7] = {2,6};
118 edges[8] = {1,3};
119 edges[9] = {2,4};
120
121 vector<vector<int>> dp(2, vector<int>(10, 0));
122
123 for(int n = 0; n < N; n++){
124 for(int cur = 0; cur <= 9; cur++){
125 if(n == 0){
126 dp[n%2][cur] = 1;
127 }else if(n == 1){
128 dp[n%2][cur] = edges[cur].size();
129 }else{
130 dp[n%2][cur] = 0;
131 for(int nei : edges[cur]){
132 dp[n%2][cur] = (dp[n%2][cur] + dp[(n-1)%2][nei]) % MOD;
133 }
134 }
135 }
136 }
137
138 for(int cur = 0; cur <= 9; cur++){
139 ans = (ans + dp[(N-1)%2][cur]) % MOD;
140 }
141
142 return ans;
143 }
144};Cost