'Undirected graph g' diaporamas de présentation

Graph Algorithms

Graph Algorithms. Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar. To accompany the text ``Introduction to Parallel Computing'', Addison Wesley, 2003 . Topic Overview . Definitions and Representation Minimum Spanning Tree: Prim's Algorithm

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Graph Algorithms

Graph Algorithms. Topic Overview . Definitions and Representation Minimum Spanning Tree: Prim's Algorithm Single-Source Shortest Paths: Dijkstra's Algorithm All-Pairs Shortest Paths Transitive Closure Connected Components Algorithms for Sparse Graphs .

917 views • 64 slides

Approximation Algorithms

Approximation Algorithms. Outlines. Why approximation algorithm? Approximation ratio Approximation vertex cover problem Approximation traveling salesman problem (TSP) Other interesting problems. 3 approaches for NP-complete Problems. For small inputs, an exponential algorithm is OK.

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CSCE 3110 Data Structures & Algorithm Analysis

CSCE 3110 Data Structures & Algorithm Analysis. Rada Mihalcea http://www.cs.unt.edu/~rada/CSCE3110 Graphs (I) Reading: Chap.9, Weiss. What is a Graph?. A graph G = ( V ,E) is composed of: V : set of vertices E : set of edges connecting the vertices in V

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MST

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4.5. Minimum Spanning Trees

4.5. Minimum Spanning Trees. Definition of MST Generic MST algorithm Kruskal's algorithm Prim's algorithm. Definition of MST. Let G=(V,E) be a connected, undirected graph. For each edge (u,v) in E , we have a weight w(u,v) specifying the cost (length of edge) to connect u and v.

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Depth-First Search

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NP-complete and NP-hard problems

NP-complete and NP-hard problems. Transitivity of polynomial-time many-one reductions Concept of Completeness and hardness for a complexity class Definition of complexity class NP NP-complete and NP-hard problems. P 1 ? p P 2 & P 2 ? p P 3 ? P 1 ? p P 3.

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Graphs

Denver. Chicago. LA. Miami. Tampa. Graphs. A Graph G = (V, E) consists of V, a nonempty set of vertices (or nodes ) and E, a set of edges . Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints.

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Dynamic Programming

Dynamic Programming. JinJu Lee & Beatrice Seifert CSE 5311 Fall 2005 Week 10 (Nov 1 & 3). Contents. Max Flow Min Cut Theorem Edmond-Karp-Algorithm Bipartite Graph Matching Problem Dynamic Programming Paradigm - Fibonacci Sequence - Longest Common Subsequence Problem

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Graphical Models for Strategic and Economic Reasoning

Graphical Models for Strategic and Economic Reasoning. Michael Kearns Computer and Information Science University of Pennsylvania BNAIC 2003. Joint work with: Sham Kakade, John Langford, Michael Littman, Luis Ortiz, Satinder Singh. Probabilistic Reasoning.

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Minimal Spanning Trees

Minimal Spanning Trees. Spanning Tree. Assume you have an undirected graph G = (V,E) Spanning tree of graph G is tree T = (V,E T E, R) Tree has same set of nodes All tree edges are graph edges Root of tree is R

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Exact Inference Continued

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Graph Theory and Representation

Graph Theory and Representation. Graph Algorithms. Graphs and Theorems about Graphs Graph ADT and implementation Graph Algorithms Shortest paths minimum spanning tree. What can graphs model?. Cost of wiring electronic components together. Shortest route between two cities.

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Odds and Ends

Odds and Ends. HP ? p HC (again) Turing reductions Strong NP-completeness versus Weak NP-completeness Vertex Cover to Hamiltonian Cycle. Example: HP ? p HC. Hamiltonian Path Input: Undirected Graph G = (V,E) Y/N Question: Does G contain a Hamiltonian Path? Hamiltonian Cycle

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Testing for Connectedness and Cycles

Testing for Connectedness and Cycles. Connectedness of Undirected Graphs Implementation of Connectedness detection Algorithm for Undirected Graphs. Implementation of Strong Connectedness Algorithm. Cycles in Directed Graphs. Implementation of Cycle detection Algorithm for Directed Graphs.

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Approximate Algorithms (chap. 35)

Approximate Algorithms (chap. 35). Motivation: Many problems are NP-complete, so unlikely find efficient algorithms Three ways to get around: If input size is small, exponential algorithm is OK. Isolate important special case and find poly algorithms for them.

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Approximate Algorithms (chap. 35)

Approximate Algorithms (chap. 35). Motivation: Many problems are NP-complete, so unlikely find efficient algorithms Three ways to get around: If input size is small, exponential algorithm is OK. Isolate important special case and find poly algorithms for them.

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An NP-completeness Example

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Testing for Connectedness and Cycles

Testing for Connectedness and Cycles. Connectedness of an Undirected Graph Implementation of Connectedness detection Algorithm. Implementation of Strong Connectedness Algorithm. Cycles in a Directed Graph. Implementation of a Cycle detection Algorithm. Review Questions.

245 views • 7 slides