# 2D grid Hamiltonian path puzzle solver

The puzzle goes like this: in a rectangular 2D grid there are empty spaces (.), exactly one starting point (S, s) and obstacles (denoted below by X's). The objective of the puzzle is to find a path starting at the starting point and going through each empty space exactly once (a Hamiltonian path). You can't, of course cross the obstacles. You can move horizontally and vertically. A typical puzzle would look like this:

......
.SX...
...X..
.....X
......
XX....
......


And its solution:

11 12 13 14 17 18
10 1  X  15 16 19
9  2  3  X  21 20
8  5  4  23 22 X
7  6  25 24 29 30
X  X  26 27 28 31
37 36 35 34 33 32


I wrote a solver in Python 3 (I later learned that this algorithm is actually a simple DFS). It solves the puzzle above in ~0.9 s, which is very good, but I was wondering if I can perhaps optimize it somehow.

import time

EMPTY_SPACE_SYMBOLS = ['.']
STARTING_POINT_SYMBOLS = ['S', 's']
OBSTACLE_SYMBOL = 'X'
PUZZLE_PATH = "grid.txt"
DIRS = [(-1, 0), (1, 0), (0, 1), (0, -1)]

start_time = time.time()

H = len(grid)
W = len(grid)
assert all(len(row) == W for row in grid), "Grid not rectangular"

def print_solution(coords):
result_grid = [[OBSTACLE_SYMBOL for _ in range(W)] for _ in range(H)]
for i, (r, c) in enumerate(coords, start=1):
result_grid[r][c] = i
str_grid = [[str(item).ljust(3) for item in row] for row in result_grid]
print('\n'.join(' '.join(row) for row in str_grid))

def extend(path, legal_coords):
res = []
lx, ly = path[-1]
for dx, dy in DIRS:
new_coord = (lx + dx, ly + dy)
if new_coord in legal_coords and new_coord not in path:
res.append(path + [new_coord])
return res

start_pos = None
legal = set()
for r, row in enumerate(grid):
for c, item in enumerate(row):
if item in STARTING_POINT_SYMBOLS:
assert start_pos is None, "Multiple starting points"
start_pos = (r, c)
elif item in EMPTY_SPACE_SYMBOLS:

assert start_pos is not None, "No starting point"

TARGET_PATH_LEN = len(legal) + 1
paths = [[start_pos]]
found = False
number_of_solutions = 0

while paths:
cur_path = paths.pop()
if len(cur_path) == TARGET_PATH_LEN:
number_of_solutions += 1
print_solution(cur_path)
print("Solution found in {} s".format(time.time() - start_time))
found = True
paths += extend(cur_path, legal)

print('Total number of solutions found: {} (took: {} s)'.format(number_of_solutions, time.time() - start_time))

• I was thinking of using this code to test with. I wonder why when I make the grid 10x10 it doesn't calculate... Mar 4 at 6:24

### 1. Review

1. Code is easier to test (and measure the performance) if it's organized into functions or classes. In this case you have persistent data (W, H, grid, legal, start_pos) that's shared between the various parts of the code, so a class would be the best way to go here.

2. It would be simpler to write:

EMPTY_SPACE_SYMBOLS = '.'
STARTING_POINT_SYMBOLS = 'Ss'


since Python can iterate over the characters in a string just as easily as over the items in a list.

3. Assertions should be reserved for detecting programming errors, and not used for reporting problems with the input. That's because assertions can be disabled via the -O option to the Python interpreter, or the PYTHONOPTIMIZE environment variable. For the errors in this code, you should raise exceptions.

4. It's more general to have a method format_solution that returns a string, rather than a function print_solution that prints it. That way the caller has the option of deciding what to do with it.

[OBSTACLE_SYMBOL for _ in range(self.w)]


write:

[OBSTACLE_SYMBOL] * self.w

6. Instead of building str_grid from result_grid and then formatting the former, just format result_grid directly:

'\n'.join(' '.join(str(item).ljust(3) for item in row)
for row in result_grid)

7. The number "3" only works for small enough puzzles. To make the formatting work in the general case, it should be one more than the maximum number of digits in any number in the path, that is:

len(str(len(path) + 1)) + 1

8. extend is only called from one place in the code, so it would make sense to inline it at that point.

9. Coordinates are r and c in some places in the code, and x and y in other places. It would be better to be consistent.

### 2. Revised code

import time

EMPTY_SPACE_SYMBOLS = '.'
STARTING_POINT_SYMBOLS = 'Ss'
OBSTACLE_SYMBOL = 'X'
DIRS = [(-1, 0), (1, 0), (0, 1), (0, -1)]

class HamiltonSolver:
"""Solver for a Hamilton Path problem."""

def __init__(self, grid):
"""Initialize the HamiltonSolver instance from a grid, which must be a
list of strings, one for each row of the grid.

"""
self.grid = grid
self.h = h = len(grid)
self.w = w = len(grid)
if any(len(row) != w for row in grid):
raise ValueError("Grid is not rectangular")
self.start = None
self.legal = set()
for r, row in enumerate(grid):
for c, item in enumerate(row):
if item in STARTING_POINT_SYMBOLS:
if self.start is not None:
raise ValueError("Multiple starting points")
self.start = (r, c)
elif item in EMPTY_SPACE_SYMBOLS:
if self.start is None:
raise ValueError("No starting point")

def format_solution(self, path):
"""Format a path as a string."""
grid = [[OBSTACLE_SYMBOL] * self.w for _ in range(self.h)]
for i, (r, c) in enumerate(path, start=1):
grid[r][c] = i
w = len(str(len(path) + 1)) + 1
return '\n'.join(''.join(str(item).ljust(w) for item in row)
for row in grid)

def solve(self):
"""Generate solutions as lists of coordinates."""
target_path_len = len(self.legal) + 1
paths = [[self.start]]
while paths:
path = paths.pop()
if len(path) == target_path_len:
yield path
r, c = path[-1]
for dr, dc in DIRS:
new_coord = r + dr, c + dc
if new_coord in self.legal and new_coord not in path:
paths.append(path + [new_coord])

PUZZLE_GRID = '''
......
.SX...
...X..
.....X
......
XX....
......
'''.split()

def main():
start_time = time.time()
n_solutions = 0
puzzle = HamiltonSolver(PUZZLE_GRID)
for solution in puzzle.solve():
if n_solutions == 0:
print(puzzle.format_solution(solution))
print("First solution found in {} s"
.format(time.time() - start_time))
n_solutions += 1
print("{} solution{} found in {} s"
.format(n_solutions, '' if n_solutions == 1 else 's',
time.time() - start_time))

if __name__ == '__main__':
main()


### 3. Speeding it up

1. Each time extend is called, the list of paths under consideration is extended by up to four new paths, each of which is a copy of the current path plus one new grid coordinate. As the search progresses, this results in many paths being created and stored (up to 20 paths in the example from the post).

But in a depth-first search it should be only be necessary to remember a single path at a time, together with the current position in the iteration over the four directions. (This is the "stack of iterators" pattern.)

2. The code checks to see if the new coordinate is available by calling new_coord in self.legal and new_coord not in path. But path is a list, and testing for membership in a list is not efficient (Python has to compare the new coordinate with every item in the list). For efficient membership testing you need a set.

In this case the code already has a set of legal coordinates. So we can remove a coordinate from this set when we add it to the path, and add it back to the set when we backtrack. This means we only need one membership test, and it also means that the path is complete when the set of legal coordinates is empty.

3. We can speed up the inner loop by caching method lookups like path.append in local variables.

The revised solve method looks like this:

def solve(self):
"""Generate solutions as lists of coordinates."""
path = [self.start]
dirs = [iter(DIRS)]

# Cache attribute lookups in local variables
path_append = path.append
path_pop = path.pop
legal = self.legal
legal_remove = legal.remove
dirs_append = dirs.append
dirs_pop = dirs.pop

while path:
r, c = path[-1]
for dr, dc in dirs[-1]:
new_coord = r + dr, c + dc
if new_coord in legal:
path_append(new_coord)
legal_remove(new_coord)
dirs_append(iter(DIRS))
if not legal:
yield path
break
else:
dirs_pop()


On the example from the post this is more than 2½ times as fast as the original code.

Update You asked in comments how this works. Well, the list path contains the current path reached in the search. The list dirs is the same length as path, and each element is an iterator over the list DIRS. Each iterator remembers where we got to in the list of directions at that point in the search, so that when we backtrack we can pop the dirs list and then continue iterating from where we left off.

The control flow is slightly tricky. When the path is extended it reaches the break statement, which breaks out of the for loop (bypassing the else) and so goes back to the top of the while loop, picking up the new coordinates that we just added to the path. But when no further progress can be made, the for loop runs out of directions and enters the else clause. This causes path and dirs to be popped and then the while loop picks up from the previous point reached in the search.

I hope that's clear. If you're still not quite sure, step through the code in the Python debugger so that you can see exactly what it's doing.

• This revised solve method is brilliant, but could you maybe explain how the for-else loop works in this case? Thank you for your help. Sep 27, 2016 at 8:32
• @shooqie: See updated answer. Sep 27, 2016 at 9:43

I suggest starting from the issues of organization and naming in order to make optimization simpler:

### Organization

The organization of your program could have been cleaner: function definition and top-level blocks of code alternate continuously.

In general all functions are defined, then they are called one by one by the main function that puts everything together.

This organization makes reading the code easier by compartmentalizing it into almost independent blocks.

Some practical examples:

start_pos = None
legal = set()
for r, row in enumerate(grid):
for c, item in enumerate(row):
if item in STARTING_POINT_SYMBOLS:
assert start_pos is None, "Multiple starting points"
start_pos = (r, c)
elif item in EMPTY_SPACE_SYMBOLS:


This piece of code clearly finds the start_pos and the legal (coordinates) so it can be a function:

def find_start_and_legals(grid):
start_pos = None
legal = set()
for r, row in enumerate(grid):
for c, item in enumerate(row):
if item in STARTING_POINT_SYMBOLS:
assert start_pos is None, "Multiple starting points"
start_pos = (r, c)
elif item in EMPTY_SPACE_SYMBOLS:
return start_pos, legal


This other block of code finds the hamilton paths, so it can be a function too:

while paths:
cur_path = paths.pop()
if len(cur_path) == TARGET_PATH_LEN:
number_of_solutions += 1
print_solution(cur_path)
print("Solution found in {} s".format(time.time() - start_time))
found = True
paths += extend(cur_path, legal)


So it can become a function too:

def find_hamilton_paths(grid):
start_pos, legal = find_start_and_legals(grid)

TARGET_PATH_LEN = len(legal) + 1
paths = [[start_pos]]

while paths:
cur_path = paths.pop()
if len(cur_path) == TARGET_PATH_LEN:
yield cur_path
paths += extend(cur_path, legal)


I also decided to remove the print and time statements to better highlight the algorithm.

Timing your program with time.time is not appropriate for larger projects and does not offer benefits in smaller ones neither.

Use time python3 hamilton.py to get over-all runtime and python -m cProfile hamilton.py to get detailed function-by-function profiling.

Separation of code into functions also helps because the profiler clearly shows that some functions take so little time that any time spent optimizing them would be wasted:

        2    0.000    0.000    0.000    0.000 hamilton.py:16(find_start_and_legals)

1    0.000    0.000    0.000    0.000 hamilton.py:9(print_solution)


Profiling works better with smaller functions.

I include my modularized function for your and future optimizers benefit:

EMPTY_SPACE_SYMBOLS = ['.']
STARTING_POINT_SYMBOLS = ['S', 's']
OBSTACLE_SYMBOL = 'X'
PUZZLE_PATH = "grid.txt"
DIRS = [(-1, 0), (1, 0), (0, 1), (0, -1)]

def print_solution(coords, WIDTH, HEIGHT):
result_grid = [[OBSTACLE_SYMBOL for _ in range(WIDTH)] for _ in range(HEIGHT)]
for i, (r, c) in enumerate(coords, start=1):
result_grid[r][c] = i
str_grid = [[str(item).ljust(3) for item in row] for row in result_grid]
print('\n'.join(' '.join(row) for row in str_grid))

def find_start_and_legals(grid):
start_pos = None
legal = set()
for r, row in enumerate(grid):
for c, item in enumerate(row):
if item in STARTING_POINT_SYMBOLS:
assert start_pos is None, "Multiple starting points"
start_pos = (r, c)
elif item in EMPTY_SPACE_SYMBOLS:
return start_pos, legal

def extend(path, legal_coords):
res = []
lx, ly = path[-1]
for dx, dy in DIRS:
new_coord = (lx + dx, ly + dy)
if new_coord in legal_coords and new_coord not in path:
res.append(path + [new_coord])
return res

def find_hamilton_paths(grid):
start_pos, legal = find_start_and_legals(grid)

TARGET_PATH_LEN = len(legal) + 1
paths = [[start_pos]]

while paths:
cur_path = paths.pop()
if len(cur_path) == TARGET_PATH_LEN:
yield cur_path
paths += extend(cur_path, legal)

with open(filename) as f:
HEIGHT, WIDTH = len(grid), len(grid)
assert all(len(row) == WIDTH for row in grid), "Grid not rectangular"
return grid, HEIGHT, WIDTH

def main():
start_pos, legal = find_start_and_legals(grid)

for solution in find_hamilton_paths(grid):
print_solution(solution, WIDTH, HEIGHT)

if __name__ == "__main__":
main()


### Naming

To optimize your program people must understand it very well, so naming becomes even more important because it helps the reader in understanding difficult algorithms:

• r -> row

• c -> column

• start_pos -> start_position

• legal -> legal_coordinates

• res -> result or new_paths

• lx, ly -> ??? I did not understand this.

• dx, dy -> delta_x, delta_y

• H, W -> HEIGHT, WIDTH

Time wasted deducing full names from abbreviations is time that could be spend in understanding the algorithm and optimizing it.

• I for one prefer start_pos, and would use col instead of column, because I don't like to read big words.