travel_salesman.py

"""
Solves the Travelling Salesman Problem (TSP) using dynamic programming and bitmasking.

Example:
distances = [
    [0, 10, 15, 20],
    [10, 0, 35, 25],
    [15, 35, 0, 30],
    [20, 25, 30, 0]
]
Output: 80
"""

from functools import lru_cache


def tsp(distances: list[list[int]]) -> int:
    """
    Solves TSP using dynamic programming.

    >>> tsp([[0, 10, 15, 20], [10, 0, 35, 25], [15, 35, 0, 30], [20, 25, 30, 0]])
    80
    >>> tsp([[0, 29, 20, 21], [29, 0, 15, 17], [20, 15, 0, 28], [21, 17, 28, 0]])
    69
    >>> tsp([[0, 10, -15, 20], [10, 0, 35, 25], [15, 35, 0, 30], [20, 25, 30, 0]])
    Traceback (most recent call last):
        ...
    ValueError: Distance cannot be negative
    """
    n = len(distances)
    if any(distances[i][j] < 0 for i in range(n) for j in range(n)):
        raise ValueError("Distance cannot be negative")

    visited_all = (1 << n) - 1

    @lru_cache(None)
    def visit(city: int, mask: int) -> int:
        """Recursively calculates the minimum cost of visiting all cities."""
        if mask == visited_all:
            return distances[city][0]  # Return to start

        min_cost = float('inf')
        for next_city in range(n):
            if not mask & (1 << next_city):  # If next_city is unvisited
                new_cost = distances[city][next_city] + visit(
                    next_city, mask | (1 << next_city)
                )
                min_cost = min(min_cost, new_cost)
        return min_cost

    return visit(0, 1)  # Start from city 0 with only city 0 visited


if __name__ == "__main__":
    import doctest
    doctest.testmod()
    print(f"{tsp([[0, 10, 15, 20], [10, 0, 35, 25], [15, 35, 0, 30], [20, 25, 30, 0]]) = }")
    print(f"{tsp([[0, 29, 20, 21], [29, 0, 15, 17], [20, 15, 0, 28], [21, 17, 28, 0]]) = }")