python coding test course, K-th shortest path search

Understanding and practicing algorithm problem solving is essential for software developers, especially for students aiming for employment. By solving various types of problems, one must develop problem-solving skills and learn how to practically apply algorithms and data structures. In this course, we will cover the topic of “Finding the Kth Shortest Path” and explain the process of solving this problem in Python in detail.

Problem Description

Given a graph, the problem is to find the Kth shortest path between two specific nodes. The Kth shortest path means the Kth shortest path among the shortest paths between the two nodes. This problem can be approached through variations of shortest path algorithms like Dijkstra’s algorithm.

Problem Summary

  • The graph can be a directed graph with or without weights.
  • A graph is given (number of nodes N, number of edges M).
  • The starting node S and the ending node E are given.
  • The Kth shortest path needs to be found.

Input Format

N M K
a1 b1 c1
a2 b2 c2
...

In the input format above, N is the number of nodes, M is the number of edges, and K is the Kth shortest path you want to find. Each edge is given by three numbers: a is the starting node, b is the ending node, and c is the weight.

Output Format

The length of the Kth shortest path, or -1 if the Kth shortest path does not exist

Examples

Input Example

4 5 2
1 2 4
1 3 2
2 3 5
2 4 1
3 4 8

Output Example

6

Solution Process

To solve this problem, we need to implement an algorithm to find the Kth shortest path. We plan to modify a typical shortest path algorithm, Dijkstra’s algorithm, and devise a way to manage the costs of each path. Here, we will solve it by exploring paths using a Priority Queue.

Step-by-step Approach

Step 1: Representing the Graph

We will represent the graph in the form of an adjacency list. We will use a dictionary or list to hold information about each node and edge. 

from collections import defaultdict
import heapq

def build_graph(edges):
    graph = defaultdict(list)
    for (u, v, w) in edges:
        graph[u].append((v, w))
    return graph

Step 2: Path Exploration Using a Priority Queue

We will use a priority queue to explore the shortest paths. The costs of each path and nodes can be saved and managed in tuple form. At this point, we should prepare a list to store K paths. 

def kth_shortest_path(graph, start, end, k):
    # Initialization
    pq = []
    heapq.heappush(pq, (0, start))  # (cost, node)
    paths = defaultdict(list)

    while pq:
        cost, node = heapq.heappop(pq)
        
        # If the Kth path is found
        if node == end:
            paths[end].append(cost)
            if len(paths[end]) >= k:
                break
        
        for neighbor, weight in graph[node]:
            new_cost = cost + weight
            heapq.heappush(pq, (new_cost, neighbor))
    
    return paths[end][k - 1] if len(paths[end]) >= k else -1

Step 3: Integrating the Overall Algorithm

We will combine the above two functionalities to complete the overall algorithm. We will receive input, create the graph, and implement the logic for finding the Kth shortest path. 

def main():
    N, M, K = map(int, input().split())
    edges = [tuple(map(int, input().split())) for _ in range(M)]
    graph = build_graph(edges)
    result = kth_shortest_path(graph, 1, N, K)
    print(result)

if __name__ == "__main__":
    main()

Conclusion and Wrap-up

In this course, we have covered the Kth shortest path finding problem. We understood the structure of the graph and the variations of Dijkstra’s algorithm, and learned how to explore paths through a priority queue. As algorithm problems can be variously modified, it’s important to practice solving many different problems.

Now, you have laid the foundation to solve the given problem. The methods learned will be applicable not only to the Kth shortest path problem but also to various other graph-related problems, so developing your application skills is important. I hope you continue to practice more algorithms and problem-solving methods.

Happy Coding!