Database System Concepts
Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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For the following questions, assume we run the following version of Prim's on the graph above. **Algorithm: Prim's** **Input:** Weighted, Undirected, Connected Graph G=(V,E) with edge weights w_e **Output:** A Tree T=(V,E'), with E'⊆E that minimizes the edge weight sum - For all u ∈ V: - cost(u) = ∞ - prev(u) = nil - Pick any initial node u₀ - cost(u₀) = 0 **Process:** 1. unvisited = makequeue(V) (priority queue, using cost-values as keys) 2. While unvisited is not empty: - v = deletemin(unvisited) - For each {v, z} ∈ E: - If cost(z) > w(v, z) and unvisited.contains(z): - cost(z) = w(v, z) - prev(z) = v - decreasekey(unvisited, z)
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Transcribed Image Text:For the following questions, assume we run the following version of Prim's on the graph above. **Algorithm: Prim's** **Input:** Weighted, Undirected, Connected Graph G=(V,E) with edge weights w_e **Output:** A Tree T=(V,E'), with E'⊆E that minimizes the edge weight sum - For all u ∈ V: - cost(u) = ∞ - prev(u) = nil - Pick any initial node u₀ - cost(u₀) = 0 **Process:** 1. unvisited = makequeue(V) (priority queue, using cost-values as keys) 2. While unvisited is not empty: - v = deletemin(unvisited) - For each {v, z} ∈ E: - If cost(z) > w(v, z) and unvisited.contains(z): - cost(z) = w(v, z) - prev(z) = v - decreasekey(unvisited, z)
# Plans that Span (MSTs) This image illustrates a weighted graph, which is used in computer science and mathematics to study Minimum Spanning Trees (MSTs). The diagram consists of nodes and edges connecting them, with numbers on the edges representing their weights or costs. ### Nodes: - The graph has 7 nodes, labeled from 0 to 6. ### Edges and Weights: - Nodes 0 and 1 are connected by an edge with a weight of 3. - Nodes 0 and 3 are connected by an edge with a weight of 6. - Nodes 1 and 2 are connected by an edge with a weight of 2. - Nodes 1 and 3 are connected by an edge with a weight of 4. - Nodes 1 and 4 are connected by an edge with a weight of 6. - Nodes 2 and 3 are connected by an edge with a weight of 5. - Nodes 2 and 5 are connected by an edge with a weight of 8. - Nodes 3 and 6 are connected by an edge with a weight of 5. - Nodes 4 and 5 are connected by an edge with a weight of 8. - Nodes 5 and 6 are connected by an edge with a weight of 5. - Nodes 1 and 5 are connected by an edge with a weight of 7. ### Objective: In the context of MSTs, the goal is to find the subset of edges that connects all vertices together without any cycles and with the minimum possible total edge weight. This graph is a foundational concept for solving network optimization problems and is widely used in various fields, including network design, logistics, and transportation.
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Transcribed Image Text:# Plans that Span (MSTs) This image illustrates a weighted graph, which is used in computer science and mathematics to study Minimum Spanning Trees (MSTs). The diagram consists of nodes and edges connecting them, with numbers on the edges representing their weights or costs. ### Nodes: - The graph has 7 nodes, labeled from 0 to 6. ### Edges and Weights: - Nodes 0 and 1 are connected by an edge with a weight of 3. - Nodes 0 and 3 are connected by an edge with a weight of 6. - Nodes 1 and 2 are connected by an edge with a weight of 2. - Nodes 1 and 3 are connected by an edge with a weight of 4. - Nodes 1 and 4 are connected by an edge with a weight of 6. - Nodes 2 and 3 are connected by an edge with a weight of 5. - Nodes 2 and 5 are connected by an edge with a weight of 8. - Nodes 3 and 6 are connected by an edge with a weight of 5. - Nodes 4 and 5 are connected by an edge with a weight of 8. - Nodes 5 and 6 are connected by an edge with a weight of 5. - Nodes 1 and 5 are connected by an edge with a weight of 7. ### Objective: In the context of MSTs, the goal is to find the subset of edges that connects all vertices together without any cycles and with the minimum possible total edge weight. This graph is a foundational concept for solving network optimization problems and is widely used in various fields, including network design, logistics, and transportation.
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