Hello! Today, I would like to talk about an algorithm problem that can be implemented in Python, The Salesman’s Dilemma.
This problem combines optimization theory and graph theory to calculate the minimum cost, and it is a common topic that frequently appears in many job coding tests.
Problem Description
Problem: A salesman must visit N cities, visiting each city exactly once and then returning to the starting point.
Given the travel costs between each pair of cities, you need to find the number of ways the salesman can visit all the cities at the minimum cost.
Input Example:
- N = 4
- Cost Matrix:
[[0, 10, 15, 20], [10, 0, 35, 25], [15, 35, 0, 30], [20, 25, 30, 0]]
This matrix represents the travel costs between each city, where the value in the ith row and jth column of the matrix indicates the cost of traveling from city i to city j.
Solution Process
The typical algorithms that can be used to solve this problem are brute force and dynamic programming.
Here, we will use dynamic programming to solve the problem more efficiently.
Step 1: Understanding the Problem
By examining the given cost matrix, we can see that the costs between each pair of cities are set differently.
The goal of this problem is to find the minimum path that visits all cities and returns to the starting point.
To achieve this, we need to consider all combinations of cities to find the optimal path.
Step 2: Preparing the Dynamic Programming Table
To represent the set of visited cities, we can use bits to indicate whether a city has been visited or not.
For example, when there are four cities, we can represent cities 0, 1, 2, and 3 as 0, 1, 2, and 3 respectively.
The state of having visited cities 0 and 1 can be represented as 0011 (in binary).
This allows us to effectively manage the state space.
Step 3: Implementing DFS Using Bit Masks
To efficiently calculate the number of ways to visit all cities, we can use depth-first search (DFS) with bit masks.
Each time we visit a city, we add the cost between the current city and the last visited city, recursively calling until we compare the costs of all paths.