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Master-Level Programming Assignment Question 1

Question: Implement Dijkstra’s shortest-path algorithm in Python for a weighted graph and return the shortest distance from a selected source vertex to every other vertex.



Expert Solution:

import heapq

def dijkstra(graph, source):
    distances = {node: float('inf') for node in graph}
    distances[source] = 0
    queue = [(0, source)]

    while queue:
        current_dist, node = heapq.heappop(queue)

        if current_dist > distances[node]:
            continue

        for neighbor, weight in graph[node]:
            distance = current_dist + weight

            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(queue, (distance, neighbor))

    return distances

This implementation uses a priority queue to efficiently select the next vertex with the minimum tentative distance. Its time complexity is O((V + E) log V) when a binary heap is used.

Master-Level Programming Assignment Question 2

Question: Write a C++ function that determines whether a directed graph contains a cycle using depth-first search.

Expert Solution:

bool hasCycle(int node, vector<vector<int>>& graph,
              vector<bool>& visited, vector<bool>& active) {
    visited[node] = true;
    active[node] = true;

    for (int next : graph[node]) {
        if (!visited[next] && hasCycle(next, graph, visited, active))
            return true;

        if (active[next])
            return true;
    }

    active[node] = false;
    return false;
}

The active array tracks vertices currently present in the DFS recursion stack. Finding an edge to an active vertex indicates a directed cycle. This approach runs in O(V + E) time.

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