题目
题目

MTH-1235-001 - 51223 - 202505 Quiz 6

多项选择题

What is the main objective of Prim's algorithm in graph theory?

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思路分析
The question asks about the main objective of Prim's algorithm in graph theory. First, I’ll acknowledge that the provided data shows the user-supplied answer as: 'd. Determining the minimum spanning tree of a weighted, connected graph.' However, the accompanying answer_options field is empty, so there are no alternative choices given to analyze. Understanding the algorithm: Prim's algorithm is a classic greedy algorithm used on weighted graphs. It b......Login to view full explanation

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Prims_Alg_5 This question relates to the use of the Graph Abstract Data Type (ADT) implemented with an Adjacency Map, as covered in class. Below is the pseudocode for Prim’s Algorithm, which is used to compute a Minimum Spanning Tree (MST) for a connected, undirected, weighted graph:   PrimJarnikMST(Graph g):    d = empty map                      // d[v] = min edge weight to connect v    tree = empty list                 // stores MST edges    pq = AdaptableHeapPriorityQueue()    pqlocator = empty map            // maps vertex to its entry in pq     for each vertex v in g.vertices():        if d is empty:            d[v] = 0        else:            d[v] = ∞        pqlocator[v] = pq.add(d[v], (v, None))     while pq is not empty:        (key, (u, edge)) = pq.remove_min()        remove u from pqlocator        if edge ≠ None:            add edge to tree         for each link in g.incident_edges(u):            v = link.opposite(u)            if v in pqlocator:                weight = link.element()                if weight < d[v]:                    d[v] = weight                    pq.update(pqlocator[v], d[v], (v, link))     return tree Which condition guarantees that Prim’s algorithm terminates after building the MST?

Prims_2 This question relates to the use of the Graph Abstract Data Type (ADT) implemented with an Adjacency Map, as covered in class. We are using Prim’s algorithm to form a Minimum Spanning Tree (MST) for a connected, undirected, weighted graph. Below is the pseudocode for Prim’s algorithm: PrimJarnikMST(Graph g):    d = empty map                        // d[v] = min edge weight to connect v    tree = empty list                   // stores MST edges    pq = AdaptableHeapPriorityQueue()    pqlocator = empty map               // maps vertex to its entry in pq     for each vertex v in g.vertices():        if d is empty:            d[v] = 0                    // start vertex        else:            d[v] = ∞        pqlocator[v] = pq.add(d[v], (v, None))     while pq is not empty:        (key, (u, edge)) = pq.remove_min()        remove u from pqlocator        if edge ≠ None:            add edge to tree         for each link in g.incident_edges(u):            v = link.opposite(u)            if v in pqlocator:                weight = link.element()                if weight < d[v]:                    d[v] = weight                    pq.update(pqlocator[v], d[v], (v, link))     return tree We have just completed visiting vertex 4, as part of executing Prim’s algorithm. Using the current state of the distance and parent tables (shown below), and based on the priority queue selection logic in the pseudocode above: Which vertex will be visited next by the algorithm?

Kruskal_Algo_3 This question relates to the use of the Graph Abstract Data Type (ADT) implemented with an Adjacency Map, as covered in class. You are working with an undirected, connected, weighted graph G, where all edges have non-negative weights. The goal is to compute a Minimum Spanning Tree (MST) of G. The following pseudocode implements Kruskal’s Algorithm using a priority queue to select edges in increasing order of weight and a disjoint set forest to track connected components: KRUSKAL-MST(Graph G):    Initialize an empty list MST to store edges of the minimum spanning tree    Initialize a priority queue PQ with all edges of G, prioritized by weight    Initialize a forest where each vertex is its own component     while MST has fewer than (number of vertices - 1) edges and PQ is not empty:        Remove the edge with the smallest weight from PQ        Let (u, v) be the endpoints of the edge         if u and v are in different components:            Add edge (u, v) to MST            Merge the components of u and v     return MST Why does the algorithm stop when the MST contains exactly (number of vertices - 1) edges?

Prims_Alg_3 This question relates to the use of the Graph Abstract Data Type (ADT) implemented with an Adjacency Map, as covered in class. Below is the pseudocode for Prim’s Algorithm, which is used to compute a Minimum Spanning Tree (MST) for a connected, undirected, weighted graph: PrimJarnikMST(Graph g):    d = empty map                      // d[v] = min edge weight to connect v    tree = empty list                 // stores MST edges    pq = AdaptableHeapPriorityQueue()    pqlocator = empty map            // maps vertex to its entry in pq     for each vertex v in g.vertices():        if d is empty:            d[v] = 0        else:            d[v] = ∞        pqlocator[v] = pq.add(d[v], (v, None))     while pq is not empty:        (key, (u, edge)) = pq.remove_min()        remove u from pqlocator        if edge ≠ None:            add edge to tree         for each link in g.incident_edges(u):            v = link.opposite(u)            if v in pqlocator:                weight = link.element()                if weight < d[v]:                    d[v] = weight                    pq.update(pqlocator[v], d[v], (v, link))     return tree Why does the algorithm update d[v] when it finds a lighter edge (u, v) during the inner loop?

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