C++ Coding Test Course, Bellman-Ford

1. Overview

The coding test is a process that is essential for applying to software development positions. Many companies assess applicants’ problem-solving abilities through various algorithmic problems, and the ability to understand and utilize the Bellman-Ford algorithm is important. This course will introduce the Bellman-Ford algorithm and provide hands-on practice using C++.

2. Algorithm Introduction

The Bellman-Ford Algorithm is designed to find the shortest path from a given starting point to all vertices. This algorithm is capable of finding the shortest path even in graphs that contain edges with negative weights. Therefore, the Bellman-Ford algorithm is an important algorithm that can handle negative weights, unlike Dijkstra’s algorithm.

The key ideas of the Bellman-Ford algorithm are as follows:

• Repeatedly check all edges to update the shortest path.
• Store the shortest distance from the starting point to all vertices.
• If the path is still updated after a maximum of (number of vertices – 1) iterations, a negative cycle exists.

3. Problem Description

We will understand the Bellman-Ford algorithm through the following problem.

Problem: There are N cities and M roads. Each road is represented by two cities A, B and a weight C. Output the weight of the shortest path that can reach all cities starting from one city. If a path does not exist or a negative cycle exists, output “Impossible”.

Input Format

• The first line contains N (the number of cities) and M (the number of roads).
• The next M lines contain the information A, B, C for each road.

Output Format

• Output the shortest distance to each city from the starting point, or “Impossible”.

4. Algorithm Implementation

Now, let’s implement the Bellman-Ford algorithm using C++. Below is the code to solve the problem:

#include <iostream>
#include <vector>
#include <limits>
using namespace std;

struct Edge {
    int u, v, weight;
};

void bellmanFord(int V, int E, vector<Edge> &edges, int start) {
    // Initialize shortest distance
    vector<double> distance(V, numeric_limits<double>::infinity());
    distance[start] = 0;

    // Repeat V-1 times
    for (int i = 1; i < V; i++) {
        for (const auto &edge : edges) {
            if (distance[edge.u] != numeric_limits<double>::infinity() &&
                distance[edge.u] + edge.weight < distance[edge.v]) {
                distance[edge.v] = distance[edge.u] + edge.weight;
            }
        }
    }

    // Check for negative cycles
    for (const auto &edge : edges) {
        if (distance[edge.u] != numeric_limits<double>::infinity() &&
            distance[edge.u] + edge.weight < distance[edge.v]) {
            cout << "Impossible" << endl;
            return;
        }
    }

    // Output results
    for (int i = 0; i < V; i++) {
        if (distance[i] == numeric_limits<double>::infinity()) {
            cout << "INF" << endl;
        } else {
            cout << distance[i] << endl;
        }
    }
}

int main() {
    int N, M;
    cin >> N >> M;

    vector<Edge> edges(M);
    for (int i = 0; i < M; i++) {
        cin >> edges[i].u >> edges[i].v >> edges[i].weight;
    }

    int start = 0;  // Set the starting point to 0
    bellmanFord(N, M, edges, start);

    return 0;
}

5. Code Explanation

The above code implements the Bellman-Ford algorithm in C++. Here’s a detailed explanation of each part.

• Edge structure: A structure to hold road information, including the starting point u, the endpoint v, and the weight.
• bellmanFord function: A function that performs the shortest distance calculation. It comprehensively measures the shortest distance and checks for negative cycles to provide output.
• Initial distance setting: Initializes the shortest distances to infinity and sets the distance of the starting point to 0.
• V-1 repetitions: Checks each road (edge) to perform distance updates.
• Negative cycle check: Checks for distance changes during the Vth iteration to determine whether a negative cycle exists.

6. Complexity Analysis

The time complexity of the Bellman-Ford algorithm is O(V * E). Here, V represents the number of vertices, and E represents the number of edges. This is because the algorithm performs distance updates while iterating through all edges V-1 times. It performs well with a small number of edges and vertices, but performance degradation may occur as the number of vertices and edges increases.

7. Practice Problems

Through the following practice problems, you can gain a deeper understanding of the Bellman-Ford algorithm:

• Problem: Determine if a negative cycle exists in a specific graph and print that cycle.
• Problem: Implement an algorithm to calculate the shortest distance from multiple starting points simultaneously.
• Problem: Create a more complex graph data structure and extend the relevant algorithm to the Bellman-Ford algorithm.

8. Conclusion

In this course, we have looked at the basic concepts of the Bellman-Ford algorithm and how to implement it using C++. I hope this will be helpful in solving complex graph problems in coding tests. Implementing the code yourself and testing various cases will greatly assist in mastering the algorithm.

Keep practicing coding to solve various problems. Thank you!