Python Coding Test Course, Bellman-Ford

In the process of preparing for coding tests, algorithms play a very important role. In particular, graph-related algorithms are frequently used in many problems, and among them, the Bellman-Ford algorithm is very efficient in solving the shortest path problem. In this course, we will learn about the Bellman-Ford algorithm in detail and solve problems using it together.

Understanding the Bellman-Ford Algorithm

The Bellman-Ford algorithm is an algorithm that finds the shortest path from one vertex to all other vertices in a directed graph. This algorithm has the following characteristics:

  • The edge weights can be negative, but there must be no negative cycles.
  • The time complexity is O(VE), where V is the number of vertices and E is the number of edges.
  • Unlike Dijkstra’s algorithm, it can calculate the shortest paths from multiple starting vertices.

The basic idea of the Bellman-Ford algorithm is as follows. The process of updating the shortest path for each vertex is repeated. This process is repeated V-1 times to find the shortest path. If the path is still updated after V-1 repetitions, it means that there is a negative cycle.

Algorithm Steps

The basic steps of the Bellman-Ford algorithm are as follows:

  1. Initialize the distance from the starting vertex to 0 and set the distance to all other vertices to infinity.
  2. Repeatedly update the shortest paths for all edges. Repeat for V-1 times.
  3. Finally, if the shortest path is still updated, then a negative cycle exists.

Implementing the Algorithm

Now, let’s implement the Bellman-Ford algorithm in Python. Below is a simple implementation code for the Bellman-Ford algorithm.


def bellman_ford(graph, start):
    # Step 1: Initialization
    distance = {vertex: float('infinity') for vertex in graph}
    distance[start] = 0

    # Step 2: Iteration
    for _ in range(len(graph) - 1):
        for u, edges in graph.items():
            for v, weight in edges.items():
                if distance[u] != float('infinity') and distance[u] + weight < distance[v]:
                    distance[v] = distance[u] + weight

    # Step 3: Negative cycle check
    for u, edges in graph.items():
        for v, weight in edges.items():
            if distance[u] != float('infinity') and distance[u] + weight < distance[v]:
                print("There is a negative cycle in the graph.")
                return None

    return distance

Solving a Practical Problem

Now that we understand the Bellman-Ford algorithm, let’s solve the following problem based on it.

Problem: Finding the Shortest Path

Given the following graph, find the shortest path from A to all other vertices.


A --(1)--> B
A --(4)--> C
B --(2)--> C
C --(-5)--> D

Here, each edge is represented in the form (start vertex) --(weight)--> (end vertex). This graph explores all paths from A to reach C and D. In particular, the weight of the edge from C to D is negative. Let's solve this problem using the Bellman-Ford algorithm.

Problem Solving Process

  1. Define the graph.
  2. Apply the Bellman-Ford algorithm to find the shortest path.
  3. Print the results.

First, let's define the graph in dictionary form:


graph = {
    'A': {'B': 1, 'C': 4},
    'B': {'C': 2},
    'C': {'D': -5},
    'D': {}
}

Now let's write code that finds the shortest path using the Bellman-Ford algorithm:


start_vertex = 'A'
shortest_paths = bellman_ford(graph, start_vertex)

if shortest_paths is not None:
    print("Shortest Path:", shortest_paths)

Result Analysis

Running the above code will yield the following shortest path results:


Shortest Path: {'A': 0, 'B': 1, 'C': 3, 'D': -2}

As a result, the shortest path from A to B is 1, the shortest path from B to C is 3, and the path from C to D is -2.

Conclusion

The Bellman-Ford algorithm is very useful and can be applied to various problems. Through this course, I hope you have enhanced your understanding of the Bellman-Ford algorithm, and that it will greatly assist you in preparing for coding tests. Practicing and utilizing such algorithms is essential.

Continuously practice solving more algorithm problems and thoroughly understand and memorize the characteristics and principles of each algorithm, as this is key to preparing for coding tests.