The N-Queens 女王 Problem
The n-queens puzzle is the problem of placing n chess queens on an ｎ ｘ ｎ chessboard so that no two queens threaten each other.
Thus, a solution requires that no two queens share the same row, column, or diagonal.
Usage
make or make compile && ./nqueens < test.dat
Edit test.dat to the maximum board size you want to test.
The program will test every board from size 1 to n.
Sample output with test.dat containing n = 5
Profiling
gprof -P -b ./nqueens gmon.out > analysis.txt
You can generate a call graph using gprof2dot and GraphViz.
Recursion
Let's discuss a simple solution to the problem, without implementing any heuristics for optimization, only bruteforce backtracking.
Have we reached the end of the board (last line)?
- YES: Return TRUE
- NO: Continue
Iterate through the current row.
Place a queen at current position [row][i].
Is this queen possible? (See Validation)
-
YES: Calls recursion to next row.
-
NO: Remove queen and continue loop.
If the loop has ended and we couldn't place any queen, it means the previous queen is blocking us.
We backtrack to her and continue the process.
Validation
If there's a queen in:
- Same row
- Same column
- Same diagonals
The function will return FALSE.
Performance
The program asks for a maximum size of board. It'll try to solve every board with increasing size until n.
It doesn't have any restrictions, so it could take hours for a big test case. Use it carefully.
We can observe that the number of attributions and the time needed to solve a board increases a lot with board size. After the last case test, my computer kept running the program for almost an hour, still not producing the output for the 33 x 33 board.
Benchmark Machine:
OS: Antergos
Kernel: x86_64 Linux 4.7.6-1-ARCH
Shell: zsh 5.2
CPU: Intel Core i5-6200U CPU @ 2.7GHz
GPU: Mesa DRI Intel(R) HD Graphics 520 (Skylake GT2)
RAM: 2096MiB / 7854MiB
Run Results
| board size | calls | time |
|---|---|---|
| 0 | 0 | 0.000001s |
| 1 | 1 | 0.000000s |
| 4 | 26 | 0.000005s |
| 5 | 15 | 0.000004s |
| 6 | 171 | 0.000026s |
| 7 | 42 | 0.000009s |
| 8 | 876 | 0.000101s |
| 9 | 333 | 0.000046s |
| 10 | 975 | 0.000117s |
| 11 | 517 | 0.000064s |
| 12 | 3066 | 0.000374s |
| 13 | 1365 | 0.000181s |
| 14 | 26495 | 0.002876s |
| 15 | 20280 | 0.002252s |
| 16 | 160712 | 0.017150s |
| 17 | 91222 | 0.010383s |
| 18 | 743229 | 0.084488s |
| 19 | 48184 | 0.005725s |
| 20 | 3992510 | 0.490726s |
| 21 | 179592 | 0.023457s |
| 22 | 38217905 | 5.092550s |
| 23 | 584591 | 0.081210s |
| 24 | 9878316 | 1.410959s |
| 25 | 1216775 | 0.188140s |
| 26 | 10339849 | 1.599553s |
| 27 | 12263400 | 1.987257s |
| 28 | 84175966 | 14.078644s |
| 29 | 44434525 | 7.684626s |
| 30 | 1692888135 | 298.843353s |
| 31 | 408773285 | 74.617912s |
| 32 | -1495242192 | 526.441956s |
| 33 | 323601164 | 893.228821s |
Number of calls has exceed long int on board 32 x 32.
Algorithm Complexity
Backtracking algorithms have a worst case complexity of O(d^n).
- d = domain (possible values for a variable)
- n = number of variables
For the n-queens problem, we have a domain of 2 (0 or 1) and n² variables.
Consider the 34 * 34 board.
How about it? Without heuristics and a very good implementation, it's insane how much this problem grows.
Memory
The program will allocate a structure named BOARD, that contains a double int pointer (matrix) and it's size. After it attempts to solve the board, the heap memory allocated is destroyed.
Tested with Valgrind, no memory leaks.
Flowchart
To better understand the algorithm, here's a handy flowchart of the N-Queens problem without using heuristics:
Keep in mind this is not following proper flowchart rules, it was drawn just for quick reference before an exam