Introduction. Task. Process. Conclusion. Evaluation. Resources. Egyptian Contributions. 2 nd Grade Created by: Catherine H. Harrison. Introduction. Task. Process. Conclusion. Evaluation. Resources. Introduction. Did you ever wonder about Ancient Egypt?

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0. A. 4. 8. 2. 8. 2. 3. 7. 1. B. C. D. 3. 9. 5. 8. 2. 5. E. F. Shortest Paths. Weighted Graphs ( § 12.5). In a weighted graph, each edge has an associated numerical value, called the weight of the edge Edge weights may represent, distances, costs, etc. Example:

Shortest Paths. Distance is the length of the shortest path between two nodes Breadth-first search (BFS) creates one BFS tree rooted at the initial node Just one tree since we only care about nodes connected to the initial node. The rest have infinite distance.

Shortest Paths. Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 9.6 Based on slides from Chuck Allison, Michael T. Goodrich, and Roberto Tamassia By Longin Jan Latecki. BOS. NY. CHI. SF. DEN. ATL. LA. MIA. Weighted Graphs.

Shortest Paths. Text Discrete Mathematics and Its Applications (5 th Edition) Kenneth H. Rosen Chapter 8.6 Based on slides from Chuck Allison, Michael T. Goodrich, and Roberto Tamassia By Longin Jan Latecki. BOS. NY. CHI. SF. DEN. ATL. LA. MIA. Weighted Graphs.

Shortest Paths. CSE 2320 – Algorithms and Data Structures Vassilis Athitsos University of Texas at Arlington. Terminology. A network is a directed graph . We will use both terms interchangeably. The weight of a path is the sum of weights of the edges that make up the path.

Shortest Path. HKOI 2006 March 11, 2006. Graph. Graph G = (V, E) Vertex/Node V Number |V| or simply V Degree deg[ v ],in-deg[ v ],out-deg[ v ] Edge E Number |E| or simply E Direction e = ( u , v ) or { u , v } Weight w e , w uv E ≤ V 2 i.e. deg[ v ] ≤ |V|.

Shortest Paths. CONTENTS Introduction to Shortest Paths (Section 4.1) Applications of Shortest Paths (Section 4.2) Optimality Conditions (Section 5.2) Generic Label-Correcting Algorithm (Section 5.3) Specific Implementations (Section 5.4) Detecting Negative Cycles (Section 5.5)

Shortest Paths. Definitions Single Source Algorithms Bellman Ford DAG shortest path algorithm Dijkstra All Pairs Algorithms Using Single Source Algorithms Matrix multiplication Floyd-Warshall Both of above use adjacency matrix representation and dynamic programming Johnson’s algorithm

Shortest Path. Graph Theory Basics. Introduction. Weighted Graph – Edges have weights Shortest Path problem is finding a path between two vertices such that the sum of the weights of edges along the path is minimized. 2. B. C. 20. 3. A. 4. 8. E. G. 6. 5. D. 2. F. Variations.