answer:Here is a Python implementation of Prim's algorithm: ```python import sys import heapq class Graph: def __init__(self, vertices): self.vertices = vertices self.adj_list = [[] for _ in range(vertices)] def add_edge(self, u, v, weight): self.adj_list[u].append((v, weight)) self.adj_list[v].append((u, weight)) def prim_mst(self): visited = [False] * self.vertices weights = [sys.maxsize] * self.vertices parent = [-1] * self.vertices weights[0] = 0 pq = [(0, 0)] while pq: weight, vertex = heapq.heappop(pq) if visited[vertex]: continue visited[vertex] = True for neighbor, neighbor_weight in self.adj_list[vertex]: if not visited[neighbor] and neighbor_weight < weights[neighbor]: weights[neighbor] = neighbor_weight parent[neighbor] = vertex heapq.heappush(pq, (neighbor_weight, neighbor)) self.print_mst(parent, weights) def print_mst(self, parent, weights): print(Edge tWeight) for i in range(1, self.vertices): print(f{parent[i]} - {i} t{weights[i]}) # Example usage: graph = Graph(5) graph.add_edge(0, 1, 2) graph.add_edge(0, 3, 6) graph.add_edge(1, 2, 3) graph.add_edge(1, 3, 8) graph.add_edge(1, 4, 5) graph.add_edge(2, 4, 7) print(Minimum Spanning Tree:) graph.prim_mst() ``` **Explanation** 1. The code defines a `Graph` class to represent a weighted, undirected graph using an adjacency list. 2. The `add_edge` method adds an edge between two vertices with a specified weight. 3. The `prim_mst` method implements Prim's algorithm to find the minimum spanning tree. 4. It initializes the following data structures: * `visited`: A boolean array to track visited vertices. * `weights`: An array to store the minimum weight of the edge connecting each vertex to the MST. * `parent`: An array to store the parent vertex of each vertex in the MST. * `pq`: A priority queue to store vertices to be processed, with their corresponding weights. 5. The algorithm starts by setting the weight of the first vertex to 0 and adding it to the priority queue. 6. It then enters a loop where it extracts the vertex with the minimum weight from the priority queue, marks it as visited, and updates the weights and parents of its neighbors if a shorter path is found. 7. Once all vertices have been processed, the `print_mst` method is called to display the edges and weights of the minimum spanning tree. **How it works** * Prim's algorithm starts with an arbitrary vertex (in this case, vertex 0) and grows the MST by adding edges with the minimum weight that connect new vertices to the existing tree. * The algorithm ensures that the tree remains connected and acyclic by only adding edges that do not form a cycle. * The use of a priority queue allows the algorithm to efficiently select the vertex with the minimum weight to process next. * The algorithm terminates when all vertices have been included in the MST. This implementation demonstrates how to use Prim's algorithm to find the minimum spanning tree of a weighted, undirected graph.

answer:Using the distance formula, the distance between two points (x_1, y_1) and (x_2, y_2) is given by sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. In this case, the distance between (0,4) and (3,0) is sqrt{(3-0)^2+(0-4)^2}=sqrt{9+16}=sqrt{25}=boxed{5}. The answer is: 5

answer:To develop efficient study habits, it's essential to establish a structured routine that balances academic responsibilities with personal well-being. Start by setting clear, achievable goals for each study session and the semester as a whole. Break down larger goals into smaller, manageable chunks, focusing on one task at a time to maintain momentum and reduce stress. Begin by creating a schedule that allocates dedicated blocks of time for studying, attending classes, and participating in extracurricular activities. Allocate specific times for focused studying, eliminating distractions during this period by turning off notifications, finding a quiet space, or using a tool like a website blocker to minimize procrastination. It's also crucial to prioritize self-care by incorporating regular breaks, exercise, and sufficient sleep to maintain a healthy work-life balance. To optimize study sessions, implement active learning techniques such as summarizing key points in your own words, creating concept maps, or making flashcards to reinforce new information. Review notes within 24 hours of taking them, filling in gaps and clarifying any misunderstandings to solidify your understanding of the material. Regularly assess your progress, adjusting your study plan as needed to stay on track and address areas of improvement. Additionally, leverage campus resources such as academic advisors, tutors, or study groups to supplement your learning and stay engaged with the material. Cultivate relationships with classmates and instructors to foster a supportive network that can provide valuable guidance, encouragement, and motivation throughout the semester. By establishing a consistent routine, adopting effective learning strategies, and seeking help when needed, you can develop efficient study habits that will serve you well throughout your college career.

answer:Balancing social life and studies requires intentional time management, prioritization, and boundary-setting. Start by categorizing activities into three tiers: academic commitments, personal well-being, and social engagements. Allocate time for each tier, ensuring that academic responsibilities receive sufficient attention while still allowing for relaxation, self-care, and socialization. Set realistic expectations by recognizing that it's impossible to attend every social event or gathering. Learn to say no to non-essential commitments that may interfere with study time or compromise your well-being. Instead, focus on nurturing meaningful relationships with a smaller group of friends who share similar interests and values. Schedule regular check-ins or activities with these friends, such as weekly coffee breaks, study sessions, or group outings, to maintain social connections while minimizing distractions. Make the most of campus resources and events that combine socialization with academic or personal development, such as joining a study group, participating in a club or organization related to your field of study, or attending workshops and seminars. These activities can help you build relationships with like-minded individuals while enriching your academic experience. Establish boundaries by designating specific times for social media, phone use, and other potential distractions. Use tools like calendars, planners, or apps to stay organized and on track, setting reminders and notifications to ensure you meet deadlines and attend important events. By prioritizing your time and energy, you can strike a balance between academic responsibilities and social engagements, creating a fulfilling college experience that nurtures both your mind and your relationships.