chess knight tour
Finding a tour of a chess knight such that it visits each cell of the board exactly once. For now we assume the start at position (0,0) (top right corner). This is an instance of Hamiltonian path problem.
The program outputs a board, with each cell marked by a position index of that cell in the knight's tour. For now we only output a single (potentially) open tour (we don't strive to get a cycle).
usage
Either compile directly:
swiftc knight.playground/Sources/knight.swift main.swift -o knight
or run
./snapshot-test.sh
it will create an executable knight, which you can then pass the board dimensions (can be rectangular). f.e.:
./knight 5 5
1 6 15 10 21
14 9 20 5 16
19 2 7 22 11
8 13 24 17 4
25 18 3 12 23
You can also use a playground.
DFS with backtracking vs. bruteforce search
DFS (5^(MN)) is obviously much faster than any bruteforce search (just checking each permutation of cells (MN)!), but still is exponential and thus very slow. But obvioiusly this is Hamiltonian path, humanity does not know a polynomial time algo to solve it (otherwise P==NP). It is optimized using a Warnsdorf's rule heuristic: as a next step we always choose the one with the fewest number of potential next steps coming out of it (taking in account already visited cells of course).
./knight 3 4
1 4 7 10
12 9 2 5
3 6 11 8
./knight 4 5
1 14 5 18 7
10 19 8 13 4
15 2 11 6 17
20 9 16 3 12
./knight 5 6
1 28 13 22 17 30
14 23 16 29 12 21
27 2 25 18 9 6
24 15 4 7 20 11
3 26 19 10 5 8
./knight 6 7
1 32 39 36 25 30 27
40 35 22 31 28 37 24
21 2 33 38 23 26 29
34 41 14 17 6 11 8
15 20 3 12 9 18 5
42 13 16 19 4 7 10
./knight 7 8
1 30 47 50 33 28 45 42
48 51 32 29 46 43 24 27
31 2 49 34 25 22 41 44
52 55 36 21 40 13 26 23
37 18 3 54 35 10 7 12
56 53 16 39 20 5 14 9
17 38 19 4 15 8 11 6
./knight 5 5
1 6 15 10 21
14 9 20 5 16
19 2 7 22 11
8 13 24 17 4
25 18 3 12 23
./knight 6 6
1 28 33 18 23 26
34 17 24 27 32 19
29 2 31 20 25 22
16 35 14 11 8 5
13 30 3 6 21 10
36 15 12 9 4 7
./knight 7 7
1 38 31 48 45 36 43
30 49 40 37 42 33 46
39 2 29 32 47 44 35
28 19 26 41 34 15 6
25 22 3 16 7 12 9
18 27 20 23 10 5 14
21 24 17 4 13 8 11
./knight 8 8
1 16 31 40 3 18 21 56
30 39 2 17 42 55 4 19
15 32 41 46 53 20 57 22
38 29 48 43 58 45 54 5
33 14 37 52 47 60 23 62
28 49 34 59 44 63 6 9
13 36 51 26 11 8 61 24
50 27 12 35 64 25 10 7
The last one uses Warnsdorf's rule heuristic.