Minimum Spanning Trees
|
|
- Curtis Cunningham
- 6 years ago
- Views:
Transcription
1 Minimum Spanning Trees 1
2 Minimum- Spanning Trees 1. Concrete example: computer connection. Definition of a Minimum- Spanning Tree
3 Concrete example Imagine: You wish to connect all the computers in an office building using the least amount of cable - Each vertex in a graph G represents a computer - Each edge represents the amount of cable needed to connect all computers
4 Spanning Tree A spanning tree of G is a subgraph which is tree (acyclic) and connect all the vertices in V. Spanning tree has only V - 1 edges.
5 Problem: Laying Telephone Wire Central office
6 Wiring: Naive Approach Central office Expensiv e!
7 Wiring: Better Approach Central office Minimize the total length of wire connecting the customers
8 Spanning Trees A spanning tree of a graph is just a subgraph that contains all the vertices and is a tree. A graph may have many spanning trees. Graph A Some Spanning Trees from Graph A o r o r o r
9 Complete Graph All 16 of its Spanning Trees
10 Total Number of Spanning Trees A complete graph with n vertices has n (n-) spanning trees.(cayley's formula) is 1 6 is is 100 million 100 is 1016 Compare: there are 1.76*106 seconds in a year. A nanosecond is one billionth (10-) of a second. (An electrical signal can travel about 0cm in a nanosecond.) There are 1.76*101 nanoseconds in a year. We are not going to be able to find all spanning trees for large graphs even on the fastest computers, at least not in our lifetimes. We have to get smart about trees. 10
11 Minimum Spanning Tree Input: Undirected connected graph G = (V, E) and weight function w : E R, Output: A Minimum spanning tree T : tree that connects all the vertices and minimizes Greedy Algorithms Generic MST algorithm Kruskal s algorithm Prim s algorithm w( T) w( u, v) ( u, v) T 11
12 Hallmark for greedy algorithms Greedy-choice property A locally optimal choice is globally optimal. Theorem. Let T be the MST of G = (V, E), and let A V. Suppose that (u, v) E is the least-weight edge connecting A to V A. Then, (u, v) T. 1 1
13 Growing a Minimum Spanning Tree (MST) Generic algorithm Grow MST one edge at a time Manage a set of edges A, maintaining the following loop invariant: Prior to each iteration, A is a subset of some MST At each iteration, we determine an edge (u, v) that can be added to A without violate this invariant A {(u, v)} is also a subset of a MST (u, v) is called a safe edge for A 1 1
14 GENERIC-MST Loop in lines -4 is executed V - 1 times Any MST tree contains V - 1 edges The execution time depends on how to find a safe edge
15 How to Find A Safe Edge? Theorem.1. Let A be a subset of E that is included in some MST, let (S, V-S) be any cut of G that respects A, and let (u, v) be a light edge crossing (S, V-S). Then edge (u, v) is safe for A Cut (S, V-S): a partition of V Crossing edge: one endpoint in S and the other in V-S A cut respects a set of A of edges if no edges in A crosses the cut A light edge crossing a cut if its weight is the minimum of any edge crossing the cut
16 16 16
17 Illustration of Theorem.1 A={(a,b}, (c, i}, (h, g}, {g, f}} S={a, b, c, i, e}; V-S = {h, g, f, d} many kinds of cuts satisfying the requirements of Theorem.1 (c, f) is the light edges crossing S and V-S and will be a safe edge 17
18 Example: MST 1
19 Example: MST 1
20 Kruskal's Algorithm Edge based algorithm Greedy strategy: From the remaining edges, select a least-cost edge that does not result in a cycle when added to the set of already selected edges Repeat V -1 times 0
21 Kruskal's Algorithm INPUT: edge-weighted graph G = (V, E), with V = n OUTPUT: a spanning tree A of G touches all vertices, has n-1 edges of minimum cost ( = total edge weight) Algorithm: Start with A empty, Add the edges one at a time, in increasing weight order An edge is accepted it if connects vertices of distinct trees (if the edge does not form a cycle in A) until A contains n-1 edges 1
22 Kruskal's Algorithm MST-Kruskal(G,w) 1 A for each vertex v V[G] do Make-Set(v) //creates set containing v (for initialization) 4 sort the edges of E for each (u,v)e do 6 if Find-Set(u) Find-Set(v) then // different component 7 A A {(u,v)} Union(Set(u),Set(v)) // merge return A
23 Data Structures For Kruskal s Algorithm Does the addition of an edge (u, v) to T result in a cycle? Each component of T is a tree. When u and v are in the same component, the addition of the edge (u, v) creates a cycle different components, the addition of the edge (u, v) does not create a cycle
24 Data Structures For Kruskal s Algorithm Each component of T is defined by the vertices in the component. Represent each component as a set of vertices. {1,,, 4}, {, 6}, {7, } Two vertices are in the same component iff they are in the same set of vertices
25 Data Structures For Kruskal s Algorithm When an edge (u, v) is added to T, the two components that have vertices u and v combine to become a single component In our set representation of components, the set that has vertex u and the set that has vertex v are united. {1,,, 4} + {, 6} {1,,, 4,, 6}
26 Kruskal s Algorithm ? 1?
27 Kruskal s Algorithm ? ? ?
28 Kruskal s Algorithm ? ? ?
29 Kruskal s Algorithm 1? ? ?
30 A B 4 6 C D 1 E 4 F 0
31 A B 4 6 C D 1 E 4 F 1
32 A B 4 6 C D 1 E 4 F
33 A B 4 6 C D 1 E 4 F
34 A B 4 6 C D 1 E 4 F 4
35 A B 4 6 C D cycle!! 1 E 4 F
36 A B 4 6 C D 1 E 4 F 6
37 A B 4 6 C D 1 E 4 F 7
38 minimum- spanning tree A B C D E 1 F
39 Kruskal's Algorithm MST-Kruskal(G,w) 1 A for each vertex v V[G] do // takes O(V) Make-Set(v) 4 sort the edges of E // takes O(E lg E) // takes O(E) for each (u,v)e, in nondecreasing of weight do 6 if Find-Set(u) Find-Set(v) then 7 A A {(u,v)} Union(Set(u),Set(v)) return A
40 Running Time of Kruskal s Algorithm Kruskal s Algorithm consists of two stages. Initializing the set A in line 1 takes O(1) time. Sorting the edges by weight in line 4. takes O(E lg E) Performing V MakeSet() operations for loop in lines -. E FindSet(), for loop in lines -. V - 1 Union(), for loop in lines -. which takes O(V + E) The total running time is O(E lg E) We have lg E = O(lg V) because # of E = V-1 So total running time becomes O(E lg V). 40
41 Prim s Algorithm The tree starts from an arbitrary root vertex r and grows until the tree spans all the vertices in V. At each step, Adds only edges that are safe for A. When algorithm terminates, edges in A form MST. Vertex based algorithm. Grows one tree T, one vertex at a time 41
42 Prim s Algorithm MST-Prim(G,w,r) //G: graph with weight w and a root vertex r 1 for each u V[G]{ key[u] p[u] NULL // parent of u } 4 key[r] 0 Q = BuildMinHeap(V,key); // Q vertices out of T 6 while Q do 7 u ExtractMin(Q) // making u part of T for each v Adj[u] do if v Q and w(u,v) key[v] then 10 p[v] u 11 key[v] w(u,v) 4 updating keys For each vertex v, key [v] is min weight of any edge connecting v to a vertex in tree. key [v]= if there is no edge and p [v] names parent of v in tree. When algorithm terminates the min-priority queue Q is empty.
43 Prim s Algorithm Lines 1- set the key of each vertex to (except root r, whose key is set to 0 first vertex processed). Also, set parent of each vertex to NULL, and initialize min-priority queue Q to contain all vertices. Line 7 identifies a vertex u є Q Removing u from set Q adds it to set Q-V of vertices in tree, thus adding (u, p[ u]) to A. The for loop of lines -11 update key and p fields of every vertex v adjacent to u but not in tree. 4
44 Run on example graph
45 Run on example graph key[u] = 1 4
46 Run on example graph 6 4 r 10 0 Pick a start vertex r 1 46
47 Run on example graph 6 4 u Red vertices have been removed from Q 47
48 Run on example graph 6 4 u Red arrows indicate parent pointers 4
49 Run on example graph 6 4 u
50 Run on example graph u 1 Extract_min from Q 0
51 Run on example graph u 1 1
52 Run on example graph u 1
53 Run on example graph u 1
54 Run on example graph u 1 4
55 Run on example graph u 1
56 Run on example graph u 1 6
57 Run on example graph u 1 7
58 Run on example graph u 1
59 Run on example graph u 1
60 Run on example graph u 1 60
61 Run on example graph u
62 Run on example graph u 6
63 Run on example graph u 6
64 Prim s Running Time What is the hidden cost in this code? Extract-Min is executed V times MST-Prim(G,w,r) 1 for each u V[Q] Decrease-Key is executed O( E ) times key[u] p[u] NULL while loop is executed V times 4 key[r] 0 Q = BuildHeap(V,key); //Q vertices out of T 6 while Q do 7 u ExtractMin(Q) // making u part of T for each v Adj[u] do if v Q and w(u,v) < key[v] then 10 p[v] u 11 key[v] w(u,v) DecreaseKey(v, w(u,v)); updating keys 64
65 Prim s Running Time Time complexity depends on data structure Q Binary heap: O(E lg V): BuildHeap takes O(log V) time number of while iterations: V ExtractMin takes O(lg V) time total number of for iterations: E DecreaseKey takes O(lg V) time Hence, Time = log V + V.T(ExtractMin) + E.T(DecreaseKey) Time = O(V lg V + E lg V) = O(E lg V) Since E V 1 (because G is connected) 6
66 Minimum bottleneck spanning tree A bottleneck edge is the highest weighted edge in a spanning tree. A spanning tree is a minimum bottleneck spanning tree (or MBST) if the graph does not contain a spanning tree with a smaller bottleneck edge weight. A MST is necessarily a MBST, but a MBST is not necessarily a MST. 66
Minimum Spanning Trees Outline: MST
Minimm Spanning Trees Otline: MST Minimm Spanning Tree Generic MST Algorithm Krskal s Algorithm (Edge Based) Prim s Algorithm (Vertex Based) Spanning Tree A spanning tree of G is a sbgraph which is tree
More informationGreedy Algorithms. At each step in the algorithm, one of several choices can be made.
Greedy Algorithms At each step in the algorithm, one of several choices can be made. Greedy Strategy: make the choice that is the best at the moment. After making a choice, we are left with one subproblem
More informationPartha Sarathi Manal
MA 515: Introduction to Algorithms & MA5 : Design and Analysis of Algorithms [-0-0-6] Lecture 20 & 21 http://www.iitg.ernet.in/psm/indexing_ma5/y09/index.html Partha Sarathi Manal psm@iitg.ernet.in Dept.
More informationMinimum Spanning Trees
Chapter 23 Minimum Spanning Trees Let G(V, E, ω) be a weighted connected graph. Find out another weighted connected graph T(V, E, ω), E E, such that T has the minimum weight among all such T s. An important
More informationIntroduction to Algorithms. Minimum Spanning Tree. Chapter 23: Minimum Spanning Trees
Introdction to lgorithms oncrete example Imagine: Yo wish to connect all the compters in an office bilding sing the least amont of cable - ach vertex in a graph G represents a compter - ach edge represents
More informationLecture Notes for Chapter 23: Minimum Spanning Trees
Lecture Notes for Chapter 23: Minimum Spanning Trees Chapter 23 overview Problem A town has a set of houses and a set of roads. A road connects 2 and only 2 houses. A road connecting houses u and v has
More informationMinimum Spanning Trees
Minimum Spanning Trees Overview Problem A town has a set of houses and a set of roads. A road connects and only houses. A road connecting houses u and v has a repair cost w(u, v). Goal: Repair enough (and
More information2pt 0em. Computer Science & Engineering 423/823 Design and Analysis of Algorithms. Lecture 04 Minimum-Weight Spanning Trees (Chapter 23)
2pt 0em Computer Science & Engineering 423/823 Design and of s Lecture 04 Minimum-Weight Spanning Trees (Chapter 23) Stephen Scott (Adapted from Vinodchandran N. Variyam) 1 / 18 Given a connected, undirected
More informationMinimum Spanning Trees My T. UF
Introduction to Algorithms Minimum Spanning Trees @ UF Problem Find a low cost network connecting a set of locations Any pair of locations are connected There is no cycle Some applications: Communication
More informationComputer Science & Engineering 423/823 Design and Analysis of Algorithms
Computer Science & Engineering 423/823 Design and Analysis of Algorithms Lecture 05 Minimum-Weight Spanning Trees (Chapter 23) Stephen Scott (Adapted from Vinodchandran N. Variyam) sscott@cse.unl.edu Introduction
More informationMinimum-Spanning-Tree problem. Minimum Spanning Trees (Forests) Minimum-Spanning-Tree problem
Minimum Spanning Trees (Forests) Given an undirected graph G=(V,E) with each edge e having a weight w(e) : Find a subgraph T of G of minimum total weight s.t. every pair of vertices connected in G are
More informationMinimum spanning trees
Minimum spanning trees [We re following the book very closely.] One of the most famous greedy algorithms (actually rather family of greedy algorithms). Given undirected graph G = (V, E), connected Weight
More informationMinimum Spanning Trees Ch 23 Traversing graphs
Next: Graph Algorithms Graphs Ch 22 Graph representations adjacency list adjacency matrix Minimum Spanning Trees Ch 23 Traversing graphs Breadth-First Search Depth-First Search 11/30/17 CSE 3101 1 Graphs
More informationWeek 11: Minimum Spanning trees
Week 11: Minimum Spanning trees Agenda: Minimum Spanning Trees Prim s Algorithm Reading: Textbook : 61-7 1 Week 11: Minimum Spanning trees Minimum spanning tree (MST) problem: Input: edge-weighted (simple,
More informationCS 561, Lecture 9. Jared Saia University of New Mexico
CS 561, Lecture 9 Jared Saia University of New Mexico Today s Outline Minimum Spanning Trees Safe Edge Theorem Kruskal and Prim s algorithms Graph Representation 1 Graph Definition A graph is a pair of
More informationDecreasing a key FIB-HEAP-DECREASE-KEY(,, ) 3.. NIL. 2. error new key is greater than current key 6. CASCADING-CUT(, )
Decreasing a key FIB-HEAP-DECREASE-KEY(,, ) 1. if >. 2. error new key is greater than current key 3.. 4.. 5. if NIL and.
More informationCOP 4531 Complexity & Analysis of Data Structures & Algorithms
COP 4531 Complexity & Analysis of Data Structures & Algorithms Lecture 9 Minimum Spanning Trees Thanks to the text authors who contributed to these slides Why Minimum Spanning Trees (MST)? Example 1 A
More information2 A Template for Minimum Spanning Tree Algorithms
CS, Lecture 5 Minimum Spanning Trees Scribe: Logan Short (05), William Chen (0), Mary Wootters (0) Date: May, 0 Introduction Today we will continue our discussion of greedy algorithms, specifically in
More informationMinimum Spanning Trees and Prim s Algorithm
Minimum Spanning Trees and Prim s Algorithm Version of October 3, 014 Version of October 3, 014 Minimum Spanning Trees and Prim s Algorithm 1 / 3 Outline Spanning trees and minimum spanning trees (MST).
More informationLecture 10. Elementary Graph Algorithm Minimum Spanning Trees
Lecture 10. Elementary Graph Algorithm Minimum Spanning Trees T. H. Cormen, C. E. Leiserson and R. L. Rivest Introduction to Algorithms, 3rd Edition, MIT Press, 2009 Sungkyunkwan University Hyunseung Choo
More informationChapter 23. Minimum Spanning Trees
Chapter 23. Minimum Spanning Trees We are given a connected, weighted, undirected graph G = (V,E;w), where each edge (u,v) E has a non-negative weight (often called length) w(u,v). The Minimum Spanning
More informationAnnouncements Problem Set 5 is out (today)!
CSC263 Week 10 Announcements Problem Set is out (today)! Due Tuesday (Dec 1) Minimum Spanning Trees The Graph of interest today A connected undirected weighted graph G = (V, E) with weights w(e) for each
More informationPart VI Graph algorithms. Chapter 22 Elementary Graph Algorithms Chapter 23 Minimum Spanning Trees Chapter 24 Single-source Shortest Paths
Part VI Graph algorithms Chapter 22 Elementary Graph Algorithms Chapter 23 Minimum Spanning Trees Chapter 24 Single-source Shortest Paths 1 Chapter 22 Elementary Graph Algorithms Representations of graphs
More information22 Elementary Graph Algorithms. There are two standard ways to represent a
VI Graph Algorithms Elementary Graph Algorithms Minimum Spanning Trees Single-Source Shortest Paths All-Pairs Shortest Paths 22 Elementary Graph Algorithms There are two standard ways to represent a graph
More informationMinimum Spanning Trees
CSMPS 2200 Fall Minimum Spanning Trees Carola Wenk Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk 11/6/ CMPS 2200 Intro. to Algorithms 1 Minimum spanning trees Input: A
More informationContext: Weighted, connected, undirected graph, G = (V, E), with w : E R.
Chapter 23: Minimal Spanning Trees. Context: Weighted, connected, undirected graph, G = (V, E), with w : E R. Definition: A selection of edges from T E such that (V, T ) is a tree is called a spanning
More informationIntroduction to Algorithms
Introduction to Algorithms 6.046J/18.401J LECTURE 13 Graph algorithms Graph representation Minimum spanning trees Greedy algorithms Optimal substructure Greedy choice Prim s greedy MST algorithm Prof.
More information1 Start with each vertex being its own component. 2 Merge two components into one by choosing the light edge
Taking Stock IE170: in Systems Engineering: Lecture 19 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University March 16, 2007 Last Time Minimum This Time More Strongly Connected
More informationTheory of Computing. Lecture 10 MAS 714 Hartmut Klauck
Theory of Computing Lecture 10 MAS 714 Hartmut Klauck Data structures: Union-Find We need to store a set of disjoint sets with the following operations: Make-Set(v): generate a set {v}. Name of the set
More informationIntroduction to Algorithms
Introduction to Algorithms 6.046J/18.401J LECTURE 16 Greedy Algorithms (and Graphs) Graph representation Minimum spanning trees Optimal substructure Greedy choice Prim s greedy MST algorithm Prof. Charles
More informationCS60020: Foundations of Algorithm Design and Machine Learning. Sourangshu Bhattacharya
CS60020: Foundations of Algorithm Design and Machine Learning Sourangshu Bhattacharya Graphs (review) Definition. A directed graph (digraph) G = (V, E) is an ordered pair consisting of a set V of vertices
More informationMinimum Spanning Trees
Minimum Spanning Trees Problem A town has a set of houses and a set of roads. A road connects 2 and only 2 houses. A road connecting houses u and v has a repair cost w(u, v). Goal: Repair enough (and no
More informationG205 Fundamentals of Computer Engineering. CLASS 21, Mon. Nov Stefano Basagni Fall 2004 M-W, 1:30pm-3:10pm
G205 Fundamentals of Computer Engineering CLASS 21, Mon. Nov. 22 2004 Stefano Basagni Fall 2004 M-W, 1:30pm-3:10pm Greedy Algorithms, 1 Algorithms for Optimization Problems Sequence of steps Choices at
More informationCIS 121 Data Structures and Algorithms Minimum Spanning Trees
CIS 121 Data Structures and Algorithms Minimum Spanning Trees March 19, 2019 Introduction and Background Consider a very natural problem: we are given a set of locations V = {v 1, v 2,..., v n }. We want
More informationCHAPTER 23. Minimum Spanning Trees
CHAPTER 23 Minimum Spanning Trees In the design of electronic circuitry, it is often necessary to make the pins of several components electrically equivalent by wiring them together. To interconnect a
More informationIntroduction to Algorithms
Introduction to Algorithms, Lecture 1 /1/200 Introduction to Algorithms.04J/1.401J LECTURE 11 Graphs, MST, Greedy, Prim Graph representation Minimum spanning trees Greedy algorithms hallmarks. Greedy choice
More informationComplexity of Prim s Algorithm
The main loop is: Complexity of Prim s Algorithm while ( not ISEMPTY(Q) ): u = EXTRACT-MIN(Q) if p[u]!= NIL: A = A U {(p[u],u)} for v in adjacency-list[u]: if v in Q and w(u,v) < priority[v] : DECREASE-PRIORITY(v,
More informationMinimum Spanning Tree
Minimum Spanning Tree 1 Minimum Spanning Tree G=(V,E) is an undirected graph, where V is a set of nodes and E is a set of possible interconnections between pairs of nodes. For each edge (u,v) in E, we
More information22 Elementary Graph Algorithms. There are two standard ways to represent a
VI Graph Algorithms Elementary Graph Algorithms Minimum Spanning Trees Single-Source Shortest Paths All-Pairs Shortest Paths 22 Elementary Graph Algorithms There are two standard ways to represent a graph
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 10 Implementing MST Algorithms Adam Smith Minimum spanning tree (MST) Input: A connected undirected graph G = (V, E) with weight function w : E R. For now, assume
More informationCS583 Lecture 09. Jana Kosecka. Graph Algorithms Topological Sort Strongly Connected Component Minimum Spanning Tree
CS3 Lecture 0 Jana Kosecka Graph Algorithms Topological Sort Strongly Connected Component Minimum Spanning Tree Many slides here are based on E. Demaine, D. Luebke, Kleinberg-Tardos slides Graph Algs.
More informationCS161 - Minimum Spanning Trees and Single Source Shortest Paths
CS161 - Minimum Spanning Trees and Single Source Shortest Paths David Kauchak Single Source Shortest Paths Given a graph G and two vertices s, t what is the shortest path from s to t? For an unweighted
More informationPartha Sarathi Mandal
MA 5: Data Strctres and Algorithms Lectre http://www.iitg.ernet.in/psm/indexing_ma5/y1/index.html Partha Sarathi Mandal Dept. of Mathematics, IIT Gwahati Idea of Prim s Algorithm Instead of growing the
More informationAlgorithms and Theory of Computation. Lecture 5: Minimum Spanning Tree
Algorithms and Theory of Computation Lecture 5: Minimum Spanning Tree Xiaohui Bei MAS 714 August 31, 2017 Nanyang Technological University MAS 714 August 31, 2017 1 / 30 Minimum Spanning Trees (MST) A
More informationChapter 4. Greedy Algorithms. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 4.5 Minimum Spanning Tree Minimum Spanning Tree Minimum spanning tree. Given a connected
More informationAlgorithms and Theory of Computation. Lecture 5: Minimum Spanning Tree
Algorithms and Theory of Computation Lecture 5: Minimum Spanning Tree Xiaohui Bei MAS 714 August 31, 2017 Nanyang Technological University MAS 714 August 31, 2017 1 / 30 Minimum Spanning Trees (MST) A
More informationWeek 12: Minimum Spanning trees and Shortest Paths
Agenda: Week 12: Minimum Spanning trees and Shortest Paths Kruskal s Algorithm Single-source shortest paths Dijkstra s algorithm for non-negatively weighted case Reading: Textbook : 61-7, 80-87, 9-601
More informationTree. number of vertices. Connected Graph. CSE 680 Prof. Roger Crawfis
Tree Introduction to lgorithms Spanning Trees CSE Prof. Roger Crawfis We call an undirected graph a tree if the graph is connected and contains no cycles. Trees: Not Trees: Not connected Has a cycle Number
More informationCSE331 Introduction to Algorithms Lecture 15 Minimum Spanning Trees
CSE1 Introduction to Algorithms Lecture 1 Minimum Spanning Trees Antoine Vigneron antoine@unist.ac.kr Ulsan National Institute of Science and Technology July 11, 201 Antoine Vigneron (UNIST) CSE1 Lecture
More informationExample of why greedy step is correct
Lecture 15, Nov 18 2014 Example of why greedy step is correct 175 Prim s Algorithm Main idea:» Pick a node v, set A={v}.» Repeat: find min-weight edge e, outgoing from A, add e to A (implicitly add the
More informationSolutions to relevant spring 2000 exam problems
Problem 2, exam Here s Prim s algorithm, modified slightly to use C syntax. MSTPrim (G, w, r): Q = V[G]; for (each u Q) { key[u] = ; key[r] = 0; π[r] = 0; while (Q not empty) { u = ExtractMin (Q); for
More informationUnit 2: Algorithmic Graph Theory
Unit 2: Algorithmic Graph Theory Course contents: Introduction to graph theory Basic graph algorithms Reading Chapter 3 Reference: Cormen, Leiserson, and Rivest, Introduction to Algorithms, 2 nd Ed., McGraw
More informationMinimum Spanning Trees
Minimum Spanning Trees Minimum Spanning Trees Representation of Weighted Graphs Properties of Minimum Spanning Trees Prim's Algorithm Kruskal's Algorithm Philip Bille Minimum Spanning Trees Minimum Spanning
More informationKruskal s MST Algorithm
Kruskal s MST Algorithm CLRS Chapter 23, DPV Chapter 5 Version of November 5, 2014 Main Topics of This Lecture Kruskal s algorithm Another, but different, greedy MST algorithm Introduction to UNION-FIND
More informationCSE 431/531: Analysis of Algorithms. Greedy Algorithms. Lecturer: Shi Li. Department of Computer Science and Engineering University at Buffalo
CSE 431/531: Analysis of Algorithms Greedy Algorithms Lecturer: Shi Li Department of Computer Science and Engineering University at Buffalo Main Goal of Algorithm Design Design fast algorithms to solve
More information15 211: RECITATION7, SECTION M. Hui Han Chin,
15 211: RECITATION7, SECTION M Hui Han Chin, hchin@cmu.edu NOTICE! Homework theory is due tomorrow Lab 3 review Make sure your code compiles. Will provide a dummy test harness. You should try compressing
More informationChapter 9. Greedy Technique. Copyright 2007 Pearson Addison-Wesley. All rights reserved.
Chapter 9 Greedy Technique Copyright 2007 Pearson Addison-Wesley. All rights reserved. Greedy Technique Constructs a solution to an optimization problem piece by piece through a sequence of choices that
More informationMinimum-Cost Spanning Tree. Example
Minimum-Cost Spanning Tree weighted connected undirected graph spanning tree cost of spanning tree is sum of edge costs find spanning tree that has minimum cost Example 2 4 12 6 3 Network has 10 edges.
More informationExample. Minimum-Cost Spanning Tree. Edge Selection Greedy Strategies. Edge Selection Greedy Strategies
Minimum-Cost Spanning Tree weighted connected undirected graph spanning tree cost of spanning tree is sum of edge costs find spanning tree that has minimum cost Example 2 4 12 6 3 Network has 10 edges.
More informationGraphs and Network Flows ISE 411. Lecture 7. Dr. Ted Ralphs
Graphs and Network Flows ISE 411 Lecture 7 Dr. Ted Ralphs ISE 411 Lecture 7 1 References for Today s Lecture Required reading Chapter 20 References AMO Chapter 13 CLRS Chapter 23 ISE 411 Lecture 7 2 Minimum
More informationElementary Graph Algorithms: Summary. Algorithms. CmSc250 Intro to Algorithms
Elementary Graph Algorithms: Summary CmSc250 Intro to Algorithms Definition: A graph is a collection (nonempty set) of vertices and edges A path from vertex x to vertex y : a list of vertices in which
More informationLecture 10: Analysis of Algorithms (CS ) 1
Lecture 10: Analysis of Algorithms (CS583-002) 1 Amarda Shehu November 05, 2014 1 Some material adapted from Kevin Wayne s Algorithm Class @ Princeton 1 Topological Sort Strongly Connected Components 2
More information6.1 Minimum Spanning Trees
CS124 Lecture 6 Fall 2018 6.1 Minimum Spanning Trees A tree is an undirected graph which is connected and acyclic. It is easy to show that if graph G(V,E) that satisfies any two of the following properties
More informationChapter 4. Greedy Algorithms. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 4 Greedy Algorithms Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 4.5 Minimum Spanning Tree Minimum Spanning Tree Minimum spanning tree. Given a connected
More informationMinimum Spanning Tree (5A) Young Won Lim 5/11/18
Minimum Spanning Tree (5A) Copyright (c) 2015 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationAlgorithms for Minimum Spanning Trees
Algorithms & Models of Computation CS/ECE, Fall Algorithms for Minimum Spanning Trees Lecture Thursday, November, Part I Algorithms for Minimum Spanning Tree Sariel Har-Peled (UIUC) CS Fall / 6 Sariel
More informationMinimum Spanning Trees. Minimum Spanning Trees. Minimum Spanning Trees. Minimum Spanning Trees
Properties of Properties of Philip Bille 0 0 Graph G Not connected 0 0 Connected and cyclic Connected and acyclic = spanning tree Total weight = + + + + + + = Applications Network design. Computer, road,
More information23.2 Minimum Spanning Trees
23.2 Minimum Spanning Trees Kruskal s algorithm: Kruskal s algorithm solves the Minimum Spanning Tree problem in O( E log V ) time. It employs the disjoint-set data structure that is similarly used for
More informationWe ve done. Introduction to the greedy method Activity selection problem How to prove that a greedy algorithm works Fractional Knapsack Huffman coding
We ve done Introduction to the greedy method Activity selection problem How to prove that a greedy algorithm works Fractional Knapsack Huffman coding Matroid Theory Now Matroids and weighted matroids Generic
More informationIntroduction: (Edge-)Weighted Graph
Introduction: (Edge-)Weighted Graph c 8 7 a b 7 i d 9 e 8 h 6 f 0 g These are computers and costs of direct connections. What is a cheapest way to network them? / 8 (Edge-)Weighted Graph Many useful graphs
More informationDepth-first Search (DFS)
Depth-first Search (DFS) DFS Strategy: First follow one path all the way to its end, before we step back to follow the next path. (u.d and u.f are start/finish time for vertex processing) CH08-320201:
More informationCS 6783 (Applied Algorithms) Lecture 5
CS 6783 (Applied Algorithms) Lecture 5 Antonina Kolokolova January 19, 2012 1 Minimum Spanning Trees An undirected graph G is a pair (V, E); V is a set (of vertices or nodes); E is a set of (undirected)
More informationTheory of Computing. Lecture 10 MAS 714 Hartmut Klauck
Theory of Computing Lecture 10 MAS 714 Hartmut Klauck Seven Bridges of Königsberg Can one take a walk that crosses each bridge exactly once? Seven Bridges of Königsberg Model as a graph Is there a path
More informationmanaging an evolving set of connected components implementing a Union-Find data structure implementing Kruskal s algorithm
Spanning Trees 1 Spanning Trees the minimum spanning tree problem three greedy algorithms analysis of the algorithms 2 The Union-Find Data Structure managing an evolving set of connected components implementing
More informationCSE 100 Minimum Spanning Trees Prim s and Kruskal
CSE 100 Minimum Spanning Trees Prim s and Kruskal Your Turn The array of vertices, which include dist, prev, and done fields (initialize dist to INFINITY and done to false ): V0: dist= prev= done= adj:
More informationCSE 431/531: Algorithm Analysis and Design (Spring 2018) Greedy Algorithms. Lecturer: Shi Li
CSE 431/531: Algorithm Analysis and Design (Spring 2018) Greedy Algorithms Lecturer: Shi Li Department of Computer Science and Engineering University at Buffalo Main Goal of Algorithm Design Design fast
More informationlooking ahead to see the optimum
! Make choice based on immediate rewards rather than looking ahead to see the optimum! In many cases this is effective as the look ahead variation can require exponential time as the number of possible
More informationMinimum Spanning Trees. Lecture II: Minimium Spanning Tree Algorithms. An Idea. Some Terminology. Dr Kieran T. Herley
Lecture II: Minimium Spanning Tree Algorithms Dr Kieran T. Herley Department of Computer Science University College Cork April 016 Spanning Tree tree formed from graph edges that touches every node (e.g.
More informationUC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 8 Lecturer: David Wagner February 20, Notes 8 for CS 170
UC Berkeley CS 170: Efficient Algorithms and Intractable Problems Handout 8 Lecturer: David Wagner February 20, 2003 Notes 8 for CS 170 1 Minimum Spanning Trees A tree is an undirected graph that is connected
More informationMinimum Spanning Tree (undirected graph)
1 Minimum Spanning Tree (undirected graph) 2 Path tree vs. spanning tree We have constructed trees in graphs for shortest path to anywhere else (from vertex is the root) Minimum spanning trees instead
More informationMinimum Trees. The problem. connected graph G = (V,E) each edge uv has a positive weight w(uv)
/16/015 Minimum Trees Setting The problem connected graph G = (V,E) each edge uv has a positive weight w(uv) Find a spanning graph T of G having minimum total weight T is a tree w T = w uv uv E(T) is minimal
More informationAlgorithms and Data Structures: Minimum Spanning Trees (Kruskal) ADS: lecture 16 slide 1
Algorithms and Data Structures: Minimum Spanning Trees (Kruskal) ADS: lecture 16 slide 1 Minimum Spanning Tree Problem Given: Undirected connected weighted graph (G, W ) Output: An MST of G We have already
More informationCSC 8301 Design & Analysis of Algorithms: Kruskal s and Dijkstra s Algorithms
CSC 8301 Design & Analysis of Algorithms: Kruskal s and Dijkstra s Algorithms Professor Henry Carter Fall 2016 Recap Greedy algorithms iterate locally optimal choices to construct a globally optimal solution
More informationAlgorithm Analysis Graph algorithm. Chung-Ang University, Jaesung Lee
Algorithm Analysis Graph algorithm Chung-Ang University, Jaesung Lee Basic definitions Graph = (, ) where is a set of vertices and is a set of edges Directed graph = where consists of ordered pairs
More informationDesign and Analysis of Algorithms
CSE 101, Winter 2018 Design and Analysis of Algorithms Lecture 9: Minimum Spanning Trees Class URL: http://vlsicad.ucsd.edu/courses/cse101-w18/ Goal: MST cut and cycle properties Prim, Kruskal greedy algorithms
More informationGreedy Algorithms Part Three
Greedy Algorithms Part Three Announcements Problem Set Four due right now. Due on Wednesday with a late day. Problem Set Five out, due Monday, August 5. Explore greedy algorithms, exchange arguments, greedy
More informationWhat is a minimal spanning tree (MST) and how to find one
What is a minimal spanning tree (MST) and how to find one A tree contains a root, the top node. Each node has: One parent Any number of children A spanning tree of a graph is a subgraph that contains all
More informationCS 5321: Advanced Algorithms Minimum Spanning Trees. Acknowledgement. Minimum Spanning Trees
CS : Advanced Algorithms Minimum Spanning Trees Ali Ebnenasir Department of Computer Science Michigan Technological University Acknowledgement Eric Torng Moon Jung Chung Charles Ofria Minimum Spanning
More informationDijkstra s algorithm for shortest paths when no edges have negative weight.
Lecture 14 Graph Algorithms II 14.1 Overview In this lecture we begin with one more algorithm for the shortest path problem, Dijkstra s algorithm. We then will see how the basic approach of this algorithm
More informationCHAPTER 13 GRAPH ALGORITHMS
CHAPTER 13 GRAPH ALGORITHMS SFO LAX ACKNOWLEDGEMENT: THESE SLIDES ARE ADAPTED FROM SLIDES PROVIDED WITH DATA STRUCTURES AND ALGORITHMS IN C++, GOODRICH, TAMASSIA AND MOUNT (WILEY 00) AND SLIDES FROM NANCY
More informationRepresentations of Weighted Graphs (as Matrices) Algorithms and Data Structures: Minimum Spanning Trees. Weighted Graphs
Representations of Weighted Graphs (as Matrices) A B Algorithms and Data Structures: Minimum Spanning Trees 9.0 F 1.0 6.0 5.0 6.0 G 5.0 I H 3.0 1.0 C 5.0 E 1.0 D 28th Oct, 1st & 4th Nov, 2011 ADS: lects
More information1 Minimum Spanning Trees (MST) b 2 3 a. 10 e h. j m
Minimum Spanning Trees (MST) 8 0 e 7 b 3 a 5 d 9 h i g c 8 7 6 3 f j 9 6 k l 5 m A graph H(U,F) is a subgraph of G(V,E) if U V and F E. A subgraph H(U,F) is called spanning if U = V. Let G be a graph with
More informationTaking Stock. Last Time Flows. This Time Review! 1 Characterize the structure of an optimal solution
Taking Stock IE170: Algorithms in Systems Engineering: Lecture 26 Jeff Linderoth Last Time Department of Industrial and Systems Engineering Lehigh University April 2, 2007 This Time Review! Jeff Linderoth
More informationCS : Data Structures
CS 600.226: Data Structures Michael Schatz Dec 7, 2016 Lecture 38: Union-Find Assignment 10: Due Monday Dec 5 @ 10pm Remember: javac Xlint:all & checkstyle *.java & JUnit Solutions should be independently
More informationCSC 1700 Analysis of Algorithms: Minimum Spanning Tree
CSC 1700 Analysis of Algorithms: Minimum Spanning Tree Professor Henry Carter Fall 2016 Recap Space-time tradeoffs allow for faster algorithms at the cost of space complexity overhead Dynamic programming
More informationCS711008Z Algorithm Design and Analysis
CS711008Z Algorithm Design and Analysis Lecture 7. Binary heap, binomial heap, and Fibonacci heap Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 108 Outline
More informationCS711008Z Algorithm Design and Analysis
CS700Z Algorithm Design and Analysis Lecture 7 Binary heap, binomial heap, and Fibonacci heap Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / Outline Introduction
More informationCSE 100: GRAPH ALGORITHMS
CSE 100: GRAPH ALGORITHMS Dijkstra s Algorithm: Questions Initialize the graph: Give all vertices a dist of INFINITY, set all done flags to false Start at s; give s dist = 0 and set prev field to -1 Enqueue
More informationtree follows. Game Trees
CPSC-320: Intermediate Algorithm Design and Analysis 113 On a graph that is simply a linear list, or a graph consisting of a root node v that is connected to all other nodes, but such that no other edges
More informationLecture 4: Graph Algorithms
Lecture 4: Graph Algorithms Definitions Undirected graph: G =(V, E) V finite set of vertices, E finite set of edges any edge e = (u,v) is an unordered pair Directed graph: edges are ordered pairs If e
More information