chapter23.ppt

Feb 7, 2010Download as PPT, PDF1 like1,011 views

The document discusses minimum spanning trees and Kruskal's algorithm. A minimum spanning tree is a subset of edges in a connected, undirected graph that connects all vertices with the minimum total weight. Kruskal's algorithm finds a minimum spanning tree by growing a forest of trees while ensuring each added edge connects two different trees without forming a cycle. It uses a disjoint-set data structure to keep track of the connected components in the forest as edges are added from lowest to highest weight.

Introduction to Algorithms Second Edition by Cormen, Leiserson, Rivest & Stein Chapter 23

Minimum Spanning Tree We then wish to find an acyclic subset T є E that connects all of the vertices and whose total weight is minimized.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Minimum Spanning Tree Assume that we have a connected, undirected graph G = ( V, E ) with a weight function w : E -> R , and we wish to find a minimum spanning tree for G . At each step, we determine an edge ( u, v ) that can be added to A without violating this invariant, in the sense that A U {( u, v )} is also a subset of a minimum spanning tree.

Minimum Spanning Tree A cut ( S, V - S ) of an undirected graph G = ( V , E ) is a partition of V . We say that an edge ( u , v ) ε E crosses the cut ( S , V - S ) if one of its endpoints is in S and the other is in V - S .

Minimum Spanning Tree We say that a cut respects a set A of edges if no edge in A crosses the cut. An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

Minimum Spanning Tree It finds a safe edge to add to the growing forest by finding, of all the edges that connect any two trees in the forest, an edge ( u , v ) of least weight.

Kruskal’s Algorithm It uses a disjoint-set data structure to maintain several disjoint sets of elements. Each set contains the vertices in a tree of the current forest. The operation FIND-SET( u ) returns a representative element from the set that contains u . Thus, we can determine whether two vertices u and v belong to the same tree by testing whether FIND-SET( u ) equals FIND-SET( v ). The combining of trees is accomplished by the UNION procedure.

The document discusses various graph algorithms and representations including: - Adjacency lists and matrices for representing graphs - Breadth-first search (BFS) which explores edges from a source vertex s level-by-level - Depth-first search (DFS) which explores "deeper" first, producing a depth-first forest - Classifying edges as tree, back, forward, or cross based on vertex colors in DFS - Topological sorting of directed acyclic graphs (DAGs) - Strongly connected components (SCCs) in directed graphs and using the transpose

chapter24.pptTareq Hasan

The document discusses algorithms for solving single-source shortest path problems on weighted, directed graphs. It describes Dijkstra's algorithm, which finds the shortest paths from a single source node to all other nodes in a graph where all edge weights are non-negative. The algorithm works by repeatedly selecting the node with the smallest estimated shortest path distance, adding it to the set of visited nodes, and relaxing all outgoing edges to update neighboring nodes' distances.

Bellman ford Algorithmtaimurkhan803

The document describes the Bellman-Ford algorithm for finding the shortest paths in a graph. It begins by defining the shortest path problem and describing applications that can be modeled as shortest path problems, such as network routing. It then explains that Bellman-Ford can find single-source shortest paths in graphs with positive or negative edge weights, unlike Dijkstra's algorithm which only works for positive edges. The core of the algorithm uses relaxation to iteratively update the shortest path estimates over multiple rounds until convergence. Pseudocode is provided to demonstrate how the relaxation process is repeated for all edges |V|-1 times to find a shortest path from the source node to all other nodes.

Shortest path problemIfra Ilyas

This document presents information about the shortest path problem in graphs. It defines key graph terms like vertices, edges, and discusses weighted, directed, and undirected graphs. It provides an example of finding the shortest path between two vertices in a graph using Dijkstra's algorithm and walks through the steps of running the algorithm on a sample graph to find the shortest path between vertices 1 and 9.

Shortest path (Dijkistra's Algorithm) & Spanning Tree (Prim's Algorithm)Mohanlal Sukhadia University (MLSU)

The document discusses shortest path problems and algorithms for finding shortest paths in graphs. It describes Dijkstra's algorithm for finding the shortest path between two nodes in a graph with non-negative edge weights. Prim's algorithm is presented for finding a minimum spanning tree, which is a subgraph connecting all nodes with minimum total edge weight. An example graph is given and steps are outlined for applying Prim's algorithm to find its minimum spanning tree.

Shortest path algorithmsana younas

The document discusses several shortest path algorithms for graphs, including Dijkstra's algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm. Dijkstra's algorithm finds the shortest path from a single source node to all other nodes in a graph with non-negative edge weights. Bellman-Ford can handle graphs with negative edge weights but is slower. Floyd-Warshall can find shortest paths in a graph between all pairs of nodes.

Spanning treesShareb Ismaeel

Algorithm to count number of disjoint pathsSujith Jay Nair

Unit26 shortest pathalgorithmmeisamstar

The document discusses the shortest path problem and Dijkstra's algorithm for solving it in graphs with non-negative edge weights. It defines the shortest path problem, explains that it is well-defined for non-negative graphs but not graphs with negative edge weights or cycles. It then describes Dijkstra's algorithm, how it works by iteratively finding the shortest path from the source to each vertex, and provides pseudocode for its implementation.

Dijkstra's Algorithm Rashik Ishrak Nahian

Data structure and algorithmsakthibalabalamuruga

This document discusses various algorithms for finding shortest paths in graphs, including Dijkstra's algorithm, breadth-first search (BFS), depth-first search (DFS), and Bellman-Ford algorithm. It provides pseudocode examples and explanations of how each algorithm works. It also covers properties of spanning trees, minimum spanning tree algorithms like Kruskal's and Prim's, and applications of spanning trees like network planning and routing protocols.

djagdeeparora86

This document provides an overview of representing graphs and Dijkstra's algorithm in Prolog. It discusses different ways to represent graphs in Prolog, including using edge clauses, a graph term, and an adjacency list. It then explains Dijkstra's algorithm for finding the shortest path between nodes in a graph and provides pseudocode for implementing it in Prolog using rules for operations like finding the minimum value and merging lists.

Minimum spanning treeSTEFFY D

This document discusses minimum spanning trees and algorithms to find them. It begins by introducing the problem of laying cable in a new neighborhood along certain paths. It then defines spanning trees and minimum spanning trees. Two algorithms are described - Prim's and Kruskal's. Prim's grows the minimum spanning tree by adding one edge at a time, while Kruskal's grows a forest by adding edges until a single tree remains. The document applies Prim's algorithm to find the minimum spanning tree of a sample graph modeling the cable laying problem. It compares the time complexities of the two algorithms and discusses properties of minimum spanning trees.

Spanning trees & applicationsTech_MX

This document discusses minimum spanning trees. It defines a minimum spanning tree as a spanning tree of a connected, undirected graph that has a minimum total cost among all spanning trees of that graph. The document provides properties of minimum spanning trees, including that they are acyclic, connect all vertices, and have n-1 edges for a graph with n vertices. Applications of minimum spanning trees mentioned include communication networks, power grids, and laying telephone wires to minimize total length.

Shortest path algorithmSubrata Kumer Paul

Dijkstra's algorithm is used to find the shortest paths from a source node to all other nodes in a network. It works by marking all nodes as tentative with initial distances from the source set to 0 and others to infinity. It then extracts the closest node, adds it to the shortest path tree, and relaxes distances of its neighbors. This process repeats until all nodes are processed. When applied to the example network, Dijkstra's algorithm finds the shortest path from node A to all others to be A-B=4, A-C=6, A-D=8, A-E=7, A-F=7, A-G=7, and A-H=9.

Dijkstra's AlgorithmArijitDhali

Dijkstra's algorithm allows finding the shortest path between any two vertices in a graph. It works by overestimating the distance of each vertex from the starting point and then visiting neighbors to find shorter paths. The algorithm uses a greedy approach, finding the next best solution at each step. It maintains path distances in an array and maps each vertex to its predecessor in the shortest path. A priority queue is used to efficiently retrieve the closest vertex. The time complexity is O(E Log V) and space is O(V). Applications include social networks, maps, and telephone networks.

Discrete Mathematics PresentationSalman Elahi

This document discusses shortest path algorithms. It begins with the Konigsberg bridge problem solved by Euler that helped develop graph theory. It then discusses the shortest path problem in graph theory and two algorithms to solve it: Dijkstra's algorithm and the A* search algorithm. It explains how these algorithms work and their applications, such as in map routing, network routing, games development, and more.

Aplicaciones de la derivadaJOELSEBASTIANANDRADE

This document discusses applications of calculus derivatives in telecommunications. It presents two examples of using derivatives to minimize costs and maximize areas. The first example finds the optimal cable cost to minimize total wiring costs. The second determines the maximum area that can be fenced given a length of fencing material and rectangular space. Both examples take derivatives, set them equal to zero, and solve to find critical values that optimize the objective function. The conclusion emphasizes how derivatives provide direct, scientific information needed in fields like telecommunications to optimize complex systems.

Dijkstra's algorithm presentationSubid Biswas

KRUSKAL'S algorithm from chaitraguest1f4fb3

The document summarizes Kruskal's algorithm for finding a minimum cost spanning tree in a graph. Kruskal's algorithm uses a heap data structure to extract the lowest cost edge from the graph. It adds edges to the spanning tree only if they do not create cycles. This results in a minimum cost spanning tree being built incrementally. The document provides an example graph and shows the minimum cost spanning tree obtained using Kruskal's algorithm on that graph.

Aplicaciones de la derivadaJOELSEBASTIANANDRADE

This document discusses applications of calculus derivatives in telecommunications. It presents two examples of using derivatives to minimize costs and maximize areas. The first example finds the optimal cable cost to minimize total wiring costs. The second finds the maximum area that can be fenced given a limited amount of fencing material. Both examples take the derivative of a function, set it equal to zero, and solve to find critical values that optimize the objective function. The conclusion emphasizes how derivatives are essential for optimizing complex systems and aiding efficient problem solving across many fields including telecommunications.

All pairs shortest path algorithmSrikrishnan Suresh

This document describes Floyd's algorithm for solving the all-pairs shortest path problem in graphs. It begins with an introduction and problem statement. It then describes Dijkstra's algorithm as a greedy method for finding single-source shortest paths. It discusses graph representations and traversal methods. Finally, it provides pseudocode and analysis for Floyd's dynamic programming algorithm, which finds shortest paths between all pairs of vertices in O(n3) time.

Networks dijkstra's algorithm- pgsrLinawati Adiman

This document describes Dijkstra's algorithm, a greedy algorithm used to find the shortest paths between nodes in a graph. It explains that Dijkstra's algorithm works by assigning permanent labels to nodes starting with the source node, then iteratively assigning temporary labels to neighboring nodes to track the shortest path distances from the source. The algorithm is demonstrated on a sample graph with 6 nodes labeled A through T, showing how it progressively assigns labels to nodes in order to find the shortest path from node S to node T.

Dijkstra s algorithmmansab MIRZA

Dijkstra’s algorithmfaisal2204

The document describes Dijkstra's algorithm for finding the shortest paths between nodes in a graph. It begins with an overview of how the algorithm works by solving subproblems to find the shortest path from the source to increasingly more nodes. It then provides pseudocode for the algorithm and analyzes its time complexity of O(E+V log V) when using a Fibonacci heap. Some applications of the algorithm include traffic information systems and routing protocols like OSPF.

Performance of the Viterbi Algorithm on a Polya ChannelExampleNestor Barraza

he Performance of the Viterbi algorithm with a new Polya contagion noise process is analyzed. The markov channel is the memory channel usually studied in the bibliography. The Polya channel was introduced years ago as a special case of a memory channel. In this work, the Polya contagion scheme is considered as a noise process and a Gauss- Polya noise model is introduced this way in order to perform soft decision decoding. The behavior of the Viterbi algorithm on a given recursive sequential cir- cuit is analyzed under this special case of memor channel,

Biconnected components (13024116056)Akshay soni

This document discusses bi-connected components in graphs. It defines an articulation point as a vertex in a connected graph whose removal would disconnect the graph. A bi-connected component is a maximal subgraph that contains no articulation points. The document presents algorithms for identifying articulation points and bi-connected components in a graph using depth-first search (DFS). It introduces the concepts of tree edges, back edges, forward edges and cross edges in a DFS tree and explains how to use these edge types to determine if a vertex is an articulation point based on the minimum discovery time of its descendant vertices.

Algorithm Design and Complexity - Course 9Traian Rebedea

The document describes Kruskal's algorithm for finding the minimum spanning tree (MST) of a connected, undirected graph. Kruskal's algorithm works by sorting the edges by weight and then greedily adding edges to the MST one by one, skipping any edges that would create cycles. It uses the property that a minimum edge between disjoint components of the partial MST must be part of the overall MST. The runtime is O(ElogE) where E is the number of edges. An example run on a graph is shown step-by-step to demonstrate the algorithm.

lecture 16sajinsc

The document discusses algorithms for finding minimum spanning trees in graphs. It describes two common algorithms: Kruskal's algorithm and Prim's algorithm. Kruskal's algorithm sorts the edges by weight and builds up the spanning tree by adding edges that do not create cycles. Prim's algorithm maintains a set of vertices in the spanning tree and iteratively adds the lowest weight edge connecting a vertex outside the set to one inside. Both run in O((V+E)logV) time using efficient data structures.

19 Minimum Spanning TreesAndres Mendez-Vazquez

More Related Content

What's hot (19)

Unit26 shortest pathalgorithmmeisamstar

Dijkstra's Algorithm Rashik Ishrak Nahian

Data structure and algorithmsakthibalabalamuruga

djagdeeparora86

Minimum spanning treeSTEFFY D

Spanning trees & applicationsTech_MX

Shortest path algorithmSubrata Kumer Paul

Dijkstra's AlgorithmArijitDhali

Discrete Mathematics PresentationSalman Elahi

Aplicaciones de la derivadaJOELSEBASTIANANDRADE

Dijkstra's algorithm presentationSubid Biswas

KRUSKAL'S algorithm from chaitraguest1f4fb3

Aplicaciones de la derivadaJOELSEBASTIANANDRADE

This document discusses applications of calculus derivatives in telecommunications. It presents two examples of using derivatives to minimize costs and maximize areas. The first example finds the optimal cable cost to minimize total wiring costs. The second finds the maximum area that can be fenced given a limited amount of fencing material. Both examples take the derivative of a function, set it equal to zero, and solve to find critical values that optimize the objective function. The conclusion emphasizes how derivatives are essential for optimizing complex systems and aiding efficient problem solving across many fields including telecommunications.

All pairs shortest path algorithmSrikrishnan Suresh

Networks dijkstra's algorithm- pgsrLinawati Adiman

Dijkstra s algorithmmansab MIRZA

Dijkstra’s algorithmfaisal2204

Performance of the Viterbi Algorithm on a Polya ChannelExampleNestor Barraza

Biconnected components (13024116056)Akshay soni

Unit26 shortest pathalgorithmmeisamstar

Dijkstra's Algorithm Rashik Ishrak Nahian

Data structure and algorithmsakthibalabalamuruga

djagdeeparora86

Minimum spanning treeSTEFFY D

Spanning trees & applicationsTech_MX

Shortest path algorithmSubrata Kumer Paul

Dijkstra's AlgorithmArijitDhali

Discrete Mathematics PresentationSalman Elahi

Aplicaciones de la derivadaJOELSEBASTIANANDRADE

Dijkstra's algorithm presentationSubid Biswas

KRUSKAL'S algorithm from chaitraguest1f4fb3

Aplicaciones de la derivadaJOELSEBASTIANANDRADE

All pairs shortest path algorithmSrikrishnan Suresh

Networks dijkstra's algorithm- pgsrLinawati Adiman

Dijkstra s algorithmmansab MIRZA

Dijkstra’s algorithmfaisal2204

Performance of the Viterbi Algorithm on a Polya ChannelExampleNestor Barraza

Biconnected components (13024116056)Akshay soni

Similar to chapter23.ppt (10)

Algorithm Design and Complexity - Course 9Traian Rebedea

lecture 16sajinsc

19 Minimum Spanning TreesAndres Mendez-Vazquez

minimum spanning trees Algorithm sachin varun

The document discusses minimum spanning trees (MSTs). It defines MSTs and provides examples of applications like wiring electronic circuits. It then describes two common algorithms for finding MSTs: Kruskal's algorithm and Prim's algorithm. Kruskal's algorithm finds MSTs by sorting edges by weight and adding edges that connect different components without creating cycles. Prim's algorithm grows an MST from a single vertex by always adding the lowest-weight edge connecting a vertex to the growing tree.

Daa chapter13B.Kirron Reddi

The document discusses algorithms for finding minimum spanning trees (MSTs) in graphs. It describes: 1) The minimum spanning tree problem of finding a subset of edges in a graph that connects all vertices with minimum total weight. 2) Two algorithms - Kruskal's algorithm and Prim's algorithm - for solving this problem by growing an MST one edge at a time. 3) Kruskal's algorithm sorts edges by weight and adds edges between trees until all vertices are connected, running in O(E log V) time. 4) Prim's algorithm grows the MST from an initial vertex by adding minimum weight edges between the current tree and other vertices, using a priority queue

Chapter 24 aoaHanif Durad

This document summarizes key concepts about minimum spanning trees from a lecture by Dr. Muhammad Hanif Durad. It discusses motivation for finding minimum spanning trees, properties of MSTs, and algorithms like Kruskal's algorithm and Prim's algorithm for finding an MST in polynomial time. Kruskal's algorithm finds the MST by greedily adding the minimum weight edge that connects two components in a forest. Prim's algorithm finds the MST by growing a tree by greedily adding the minimum weight edge connecting the growing tree to a vertex not yet included.

Graph algorithmsUniversity of Haripur

Weighted graphsCore Condor

spanningtreesapplications-120903115339-phpapp02 (1).pdfYasirAli74993

The document defines a tree and minimum spanning tree. A tree is a graph with no cycles and is connected. A minimum spanning tree (MST) is a spanning tree of a connected, undirected graph that has minimum total edge weight. The MST contains all vertices and is the shortest possible spanning tree. Properties of MSTs include possible multiplicity if all edge weights are equal, uniqueness if all edge weights are distinct, and cycle and partition properties related to replacing edges. Applications of MSTs include communication networks, power/water/sewage lines, and telephone wiring where the goal is connecting all points with minimum total cost.

lecture 23 algorithm design and analysisbluebirdrish666

Algorithm Design and Complexity - Course 9Traian Rebedea

lecture 16sajinsc

19 Minimum Spanning TreesAndres Mendez-Vazquez

minimum spanning trees Algorithm sachin varun

Daa chapter13B.Kirron Reddi

Chapter 24 aoaHanif Durad

Graph algorithmsUniversity of Haripur

Weighted graphsCore Condor

spanningtreesapplications-120903115339-phpapp02 (1).pdfYasirAli74993

lecture 23 algorithm design and analysisbluebirdrish666

More from Tareq Hasan (20)

Grow Your Career with WordPressTareq Hasan

This document discusses how to grow one's career with WordPress. It provides an overview of WordPress, including that it was founded in 2003 by Matt Mullenweg and Mike Little and currently powers 27% of the web. The document then suggests several career paths one can take with WordPress, such as being a blogger, support specialist, designer, developer, content writer, QA engineer, or internet marketer. It encourages the reader to choose one path, get good at it, and notes that one's thinking capacity is the limit.

Caching in WordPressTareq Hasan

Caching in WordPress provides three main ways to cache content: 1. Page caching stores entire page outputs to serve static content and avoid database queries on repeat visits. 2. Transients cache query and object results, like recent posts, and are stored in the database for retrieval until expiration. 3. Object caching stores content in memory for faster loading, but requires an external persistent data store like Memcached to remain available between page loads.

How to Submit a plugin to WordPress.org RepositoryTareq Hasan

Composer - The missing package manager for PHPTareq Hasan

WordPress Theme & Plugin development best practices - phpXperts seminar 2011Tareq Hasan

08 c++ Operator Overloading.pptTareq Hasan

Operator overloading allows operators like + and << to be used with user-defined types like classes. It is done by defining corresponding operator functions like operator+() and operator<<(). This allows objects to be used with operators in a natural way while providing custom behavior for that type. The rules for overloading include maintaining precedence and associativity of operators. Common operators like +, -, *, /, <<, >>, ==, =, [] and () can be overloaded to allow user-defined types to work with them.

02 c++ Array PointerTareq Hasan

The document discusses arrays and pointers in C++. It covers: - How to declare and initialize arrays - Accessing array elements using subscripts - Using parallel arrays when subscripts are not sequential numbers - Special considerations for strings as arrays of characters - Declaring pointer variables and using pointers to access array elements - Potential issues with pointers like dangling references

01 c++ Intro.pptTareq Hasan

This document provides an overview of C++ programming concepts including: 1. C++ programs consist of functions, with every program containing a main() function. Functions contain declarations, statements, comments, and can call libraries. 2. Variables must be declared with a type and can be used to store values. C++ supports integer, floating point, character, and other variable types. 3. C++ allows selection and decision making using if/else statements, switch statements, logical operators, and loops like while and for. Operators allow comparisons and boolean evaluations.

chapter - 6.pptTareq Hasan

Algorithm.pptTareq Hasan

The document discusses divide and conquer algorithms and merge sort. It provides details on how merge sort works including: (1) Divide the input array into halves recursively until single element subarrays, (2) Sort the subarrays using merge sort recursively, (3) Merge the sorted subarrays back together. The overall running time of merge sort is analyzed to be θ(nlogn) as each level of recursion contributes θ(n) work and there are logn levels of recursion.

chapter-8.pptTareq Hasan

The document discusses lower bounds for sorting algorithms and the counting sort algorithm. It begins by explaining that sorting algorithms can only use comparisons between elements to determine their order. It then discusses decision trees that model the comparisons made by sorting algorithms and how the longest path in the tree represents the worst case number of comparisons. Finally, it provides pseudocode for counting sort and analyzes its running time as θ(k+n) where k is the range of input values and n is the number of elements.

Algorithm: priority queueTareq Hasan

The document discusses priority queues and quicksort. It defines a priority queue as a data structure that maintains a set of elements with associated keys. Heaps can be used to implement priority queues. There are two types: max-priority queues and min-priority queues. Priority queues have applications in job scheduling and event-driven simulation. Quicksort works by partitioning an array around a pivot element and recursively sorting the sub-arrays.

Algorithm: Quick-SortTareq Hasan

The document describes the quicksort algorithm. Quicksort works by: 1) Partitioning the array around a pivot element into two sub-arrays of less than or equal and greater than elements. 2) Recursively sorting the two sub-arrays. 3) Combining the now sorted sub-arrays. In the average case, quicksort runs in O(n log n) time due to balanced partitions at each recursion level. However, in the worst case of an already sorted input, it runs in O(n^2) time due to highly unbalanced partitions. A randomized version of quicksort chooses pivots randomly to avoid worst case behavior.

Java: GUITareq Hasan

Java: InheritanceTareq Hasan

Java: ExceptionTareq Hasan

Java: Introduction to ArraysTareq Hasan

Java: Class Design ExamplesTareq Hasan

Java: Objects and Object ReferencesTareq Hasan

Java: Primitive Data TypesTareq Hasan

Grow Your Career with WordPressTareq Hasan

Caching in WordPressTareq Hasan

How to Submit a plugin to WordPress.org RepositoryTareq Hasan

Composer - The missing package manager for PHPTareq Hasan

WordPress Theme & Plugin development best practices - phpXperts seminar 2011Tareq Hasan

08 c++ Operator Overloading.pptTareq Hasan

02 c++ Array PointerTareq Hasan

01 c++ Intro.pptTareq Hasan

chapter - 6.pptTareq Hasan

Algorithm.pptTareq Hasan

chapter-8.pptTareq Hasan

Algorithm: priority queueTareq Hasan

Algorithm: Quick-SortTareq Hasan

Java: GUITareq Hasan

Java: InheritanceTareq Hasan

Java: ExceptionTareq Hasan

Java: Introduction to ArraysTareq Hasan

Java: Class Design ExamplesTareq Hasan

Java: Objects and Object ReferencesTareq Hasan

Java: Primitive Data TypesTareq Hasan

chapter23.ppt

1. Introduction to Algorithms Second Edition by Cormen, Leiserson, Rivest & Stein Chapter 23

2. Minimum Spanning Tree We then wish to find an acyclic subset T є E that connects all of the vertices and whose total weight is minimized.

4. Minimum Spanning Tree Assume that we have a connected, undirected graph G = ( V, E ) with a weight function w : E -> R , and we wish to find a minimum spanning tree for G . At each step, we determine an edge ( u, v ) that can be added to A without violating this invariant, in the sense that A U {( u, v )} is also a subset of a minimum spanning tree.

6. Minimum Spanning Tree A cut ( S, V - S ) of an undirected graph G = ( V , E ) is a partition of V . We say that an edge ( u , v ) ε E crosses the cut ( S , V - S ) if one of its endpoints is in S and the other is in V - S .

7. Minimum Spanning Tree We say that a cut respects a set A of edges if no edge in A crosses the cut. An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

11. Minimum Spanning Tree It finds a safe edge to add to the growing forest by finding, of all the edges that connect any two trees in the forest, an edge ( u , v ) of least weight.

12. Kruskal’s Algorithm It uses a disjoint-set data structure to maintain several disjoint sets of elements. Each set contains the vertices in a tree of the current forest. The operation FIND-SET( u ) returns a representative element from the set that contains u . Thus, we can determine whether two vertices u and v belong to the same tree by testing whether FIND-SET( u ) equals FIND-SET( v ). The combining of trees is accomplished by the UNION procedure.

chapter23.ppt

Recommended

More Related Content

What's hot (19)

Similar to chapter23.ppt (10)

More from Tareq Hasan (20)

chapter23.ppt