Lecture 10. Graphs Vertices, edges, paths, cycles Sparse and dense graphs Representations: adjacency matrices and adjacency lists Implementation notes
|
|
- Meryl Harvey
- 6 years ago
- Views:
Transcription
1 Lecture 10 Graphs Vertices, edges, paths, cycles Sparse and dense graphs Representations: adjacency matrices and adjacency lists Implementation notes Reading: Weiss, Chapter 9 Page 1 of 24
2 Midterm exam Date and time: Thu May 7, during lecture time Place: lecture room Closed book, closed notes, no calculators Just bring something to write with, and picture ID Coverage: Lectures up to the exam; Programming assignments 1 and 2; corresponding readings Practice exam: available online Information about using PeerWise available on the class web pages Midterm reviews: Discussion section May 6 Page 2 of 24
3 Kinds of data structures You are familiar with these kinds of data structures: unstructured structures: sets linear, sequential structures: arrays, linked lists hierarchical structures: trees Now we will look at graphs Graphs consist of a collection of elements, called nodes or vertices a set of connections, called edges or links or arcs, between pairs of nodes Graphs are in general not hierarchical or sequential: there is no requirement for a distinguished root node or first node, no requirement that nodes have a unique parent or a unique successor, etc. Page 3 of 24
4 Why graphs? Trees are a generalization of lists (a list is just a special case of a tree)... Graphs are a generalization of of trees (a tree is just a special case of a graph)... So, graphs are very general structures and are very useful in many applications the set of machines on the internet, and network lines between them, form a graph the set of statements in a program, and flow of control between them, form a graph the set of web pages in the world, and HREF links between them, form a graph the set of transistors on a chip, and wires between them, form a graph the set of possible base sequences in a DNA gene, and mutations between them, form a graph the set of possible situations that can arise in solving a problem or playing a game, and moves that get you from one situation to another, form a graph et cetera... We will look at a formal definition of a graph, some ways of representing graphs, and some important algorithms on graphs Page 4 of 24
5 Graphs: some definitions A graph G = (V,E) consists of a set of vertices V and a set of edges E Each edge in E is a pair (v,w) such that v and w are in V. If G is an undirected graph, (v,w) in E means vertices v and w are connected by an edge in G. This (v,w) is an unordered pair If G is a directed graph, (v,w) in E means there is an edge going from vertex v to vertex w in G. This (v,w) is an ordered pair; there may or may not also be an edge (w,v) in E In a weighted graph, each edge also has a weight or cost c, and an edge in E is a triple (v,w,c) When talking about the size of a problem involving a graph, the number of vertices V and the number of edges E will be relevant Page 5 of 24
6 Graphs: an example Here is an unweighted directed graph: V0 V1 V2 V3 V4 V5 V6 V = { } V = E = { } E = Page 6 of 24
7 Graphs: more definitions A path in a graph G=(V,E) is a sequence of vertices v 1, v 2,..., v N in V such that (v i, v i+1 ) is in E for all i = 1,...,N-1. The length of a path is the number of edges in the path (might be zero) The weighted length of a path is the sum of the weights of the edges in the path A simple path is a path in which all the vertices are different (except the first and last can be the same) A cycle in a directed graph is a path of length >= 1 in which the first and last vertices are the same (in an undirected graph, the edges in a cycle must be distinct) A simple cycle is a cycle that is a simple path If a directed graph has no cycles, it is called a directed acyclic graph (DAG) Is the example graph on the previous page a DAG? Note: Every tree is a DAG, but not every DAG is a tree. Example: V1 V2 V3 V4 Page 7 of 24
8 Dense and sparse graphs If a directed graph has V vertices, how many edges can it have? The first vertex can have an edge to every vertex (including itself): V edges The second vertex can have an edge to every vertex (including itself): V edges... and so on for each of the V vertices; and all these edges are distinct So, the maximum total number of edges possible is E = V x V = V 2 A graph with close to V 2 edges is considered dense A graph with closer to V edges is considered sparse Page 8 of 24
9 Representing graphs There are two major techniques for representing graphs: Adjacency matrix Adjacency list Each of these has advantages and we will look at each Page 9 of 24
10 Adjacency matrices An adjacency matrix is a 2D array The [i][j] entry in the matrix encodes connectivity information between vertices i and j For an unweighted graph, the entry is 1 or true if there is an edge, 0 or false if there is no edge For a weighted graph, the entry is the weight of the edge, or infinity if there is no edge For an undirected graph, the matrix will be symmetric (or you could just use an upper-triangular matrix) There are V rows and V columns in an adjacency matrix, and so the matrix has V 2 entries This is space inefficient for sparse graphs Page 10 of 24
11 Adjacency matrix, an example Fill in this adjacency matrix for the example graph: Page 11 of 24
12 Adjacency lists An adjacency list representation uses, well, lists Each vertex in the graph has associated with it a list of the vertices adjacent to it That is, if (v j, v k ) is an edge in the graph, then v j s adjacency list contains (a reference to) v k For a weighted graph, the list entry would also contain the weight of the edge For an undirected graph, if v j s adjacency list contains v k, then v k s adjacency list should contain v j Using an adjacency list representation, each edge in a directed graph is represented by one item in one list; and there are as many lists as there are vertices Therefore the storage required is proportional to V + E, which is much better than V 2 for sparse graphs, and comparable to V 2 for dense graphs Page 12 of 24
13 Adjacency lists, an example Write down the adjacency lists to represent the example graph: V0: V1: V2: V3: V4: V5: V6: Page 13 of 24
14 Vertices, vertex names, and vertex indices In a graph application, the vertices of a graph may have names associated with them by the user as one example, vertices may correspond to cities, and be named with the city name Internally, within the implementation of the graph, it may be more convenient to be able to refer to a vertex using an integer number this number can be used as an index into an adjacency matrix, or an index into an array of adjacency lists, etc. To translate between the internal representation and one that makes sense to the user, you need some way to translate among names, vertices, and vertex index numbers So, in implementing a graph ADT, it is often convenient to have these (in addition to the adjacency matrix or adjacency lists): A dictionary data structure, e.g. a hash table, that associates names with vertex index numbers, and... An array or vector, that associates vertex numbers with vertices themselves, and... An instance variable in each vertex that stores the name associated with that vertex Page 14 of 24
15 Vertices, vertex names, and vertex indices: A design Internally within a Graph object, we will want a vertex to be uniquely identified by its name, or its integer index number, or by the identity of its corresponding Vertex object So, we will want a way to go from any one of those to any other. One way is with these mappings: Name via a Hashtable Index Index via a Vector Vertex Vertex via Vertex instance variable Name Note: only vertex names move through the public interface to the Graph; Vertex objects and indices are strictly part of the implementation Page 15 of 24
16 Graph, Vertex, and Edge Following a natural object-oriented design approach, the most important classes we will define are Graph, Vertex, and Edge A Graph will represent a weighted directed graph. It will permit a user to create an empty graph, and to add edges to the graph by specifying the names of the vertices connected by the edge, and the edge cost. Internally, it uses an adjacency list representation. In this framework, unweighted graphs can be handled by specifying 1 for edge weight; undirected graphs can be handled by having 2 parallel directed edges for each undirected edge A Graph object may also provide operations which implement useful graph algorithms such as shortest path and minimum spanning tree A Vertex will represent a vertex in a directed graph; it is used internally by the Graph class. A Vertex contains a String representation of its name, and its adjacency list (as a LinkedList). It may also contain other instance variables that are useful in implementing various graph operations. An Edge will represent an edge in a weighted directed graph; it is used internally by the Graph class. Edge objects are the elements in a Vertex s adjacency list. As such, an Edge only needs to contain an integer index of the vertex it is connecting to, and the cost of the edge. Page 16 of 24
17 An implementation of a Graph class We will look at parts of an implementation of a Graph class This implementation uses adjacency lists to represent a weighted directed graph The example Graph would be set up like this (the example is an unweighted directed graph, so costs are set equal to 1): Graph g = new Graph(); // create an empty Graph g.addedge("v0","v1",1); //edge from V0 to V1, cost 1 g.addedge("v1","v4",1); //edge from V1 to V4, cost 1 g.addedge("v1","v3",1); //edge from V1 to V3, cost 1 // etc. After the Graph is set up with all its edges, interesting algorithms can be run on the graph (more about these later) Page 17 of 24
18 An Edge class Instances of this class are elements of the adjacency lists in the Graph /** * This class defines the information in each * item in an adjacency list. */ class Edge { // Source vertex of edge is implicit public int dest; // Index of destination vertex of this edge public int cost; // Cost (weight) of this edge } public Edge( int d, int c ) { dest = d; cost = c; } Page 18 of 24
19 A Vertex class /** * This class defines the basic information * stored for each vertex. */ class Vertex { String name; // The name of this Vertex List<Edge> adj; // The adjacency list for this Vertex int dist; // variable for use by algorithms int prev; // variable for use by algorithms int scratch; // variable for use by algorithms } Vertex( String nm ) { name = nm; // name of this Vertex adj = new LinkedList<Edge>( ); // Start an empty adj list } Page 19 of 24
20 A Graph class /** * A class instances of which represent directed weighted graphs. */ public class Graph { // Maps vertex name to number private HashMap<String,Integer> vertexmap; // Vector of Vertexes in Graph private Vector<Vertex> vertexvec; private static final int NULL_VERTEX = -1; public Graph() { vertexvec = new Vector<Vertex>(); vertexmap = new HashMap<String,Integer>(); } Page 20 of 24
21 Adding an edge, given the names of its vertices and its cost /** * Add the edge ( source, dest, cost ) to the graph. */ public void addedge( String source, String dest, int cost ) { // get internal index of source, creating it if necessary Integer sourcenum = vertexmap.get(source); if (sourcenum == null) // new vertex, not seen before; add it sourcenum = addvertex(source); // get internal index of dest, creating it if necessary Integer destnum = vertexmap.get(dest); if (destnum == null) // new vertex, not seen before; add it destnum = addvertex(dest); } // now actually add the edge to the graph internaladdedge( sourcenum, destnum, cost ); Page 21 of 24
22 Adding a vertex, given its name /** * Add a vertex with the given name to the Graph, and * return its internal number as an Integer. * This involves adding entries to the Hashtable and vertex Vector. * PRE: vertexname is not already in the Graph */ private int addvertex( String vertexname ) { // get the next unused index; this will be the internal number int indx = vertexvec.size(); // create a new Vertex, put it at end of vertexvec, // where it will have index indx vertexvec.add( new Vertex(vertexName) ); // associate the vertex name and index in the vertexmap vertexmap.put( vertexname, indx ); } // return the internal vertex number return indx; Page 22 of 24
23 Adding an edge, cont d /** * Add an edge given internal index numbers of its vertices. * PRE: source and dest are indexes of Vertexes in vertexvec */ private void internaladdedge(int source, int dest, int cost) { // get the source Vertex, and add an edge to dest Vertex // to its adjacency list } vertexvec.get( source ).adj.add( new Edge (dest, cost) ); Page 23 of 24
24 Next time Algorithms on graphs Breadth first, depth first searches Shortest path in unweighted graphs Greedy algorithms Djikstra s algorithm for shortest path in weighted graphs Reading: Weiss, Chapter 9, 10 Page 24 of 24
CSE 100: GRAPH ALGORITHMS
CSE 100: GRAPH ALGORITHMS 2 Graphs: Example A directed graph V5 V = { V = E = { E Path: 3 Graphs: Definitions A directed graph V5 V6 A graph G = (V,E) consists of a set of vertices V and a set of edges
More informationGraphs & Digraphs Tuesday, November 06, 2007
Graphs & Digraphs Tuesday, November 06, 2007 10:34 PM 16.1 Directed Graphs (digraphs) like a tree but w/ no root node & no guarantee of paths between nodes consists of: nodes/vertices - a set of elements
More informationGraph Theory. ICT Theory Excerpt from various sources by Robert Pergl
Graph Theory ICT Theory Excerpt from various sources by Robert Pergl What can graphs model? Cost of wiring electronic components together. Shortest route between two cities. Finding the shortest distance
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 informationCS 4407 Algorithms Lecture 5: Graphs an Introduction
CS 4407 Algorithms Lecture 5: Graphs an Introduction Prof. Gregory Provan Department of Computer Science University College Cork 1 Outline Motivation Importance of graphs for algorithm design applications
More informationBasic Graph Definitions
CMSC 341 Graphs Basic Graph Definitions A graph G = (V,E) consists of a finite set of vertices, V, and a finite set of edges, E. Each edge is a pair (v,w) where v, w V. V and E are sets, so each vertex
More information04/03/03 Lecture 20 1
Figure 14.2 Adjacency list representation of the graph shown in Figure 14.1; the nodes in list i represent vertices adjacent to i and the cost of the connecting edge. 04/03/03 Lecture 20 1 Shortest Paths
More informationGraphs. Data Structures and Algorithms CSE 373 SU 18 BEN JONES 1
Graphs Data Structures and Algorithms CSE 373 SU 18 BEN JONES 1 Warmup Discuss with your neighbors: Come up with as many kinds of relational data as you can (data that can be represented with a graph).
More informationThis course is intended for 3rd and/or 4th year undergraduate majors in Computer Science.
Lecture 9 Graphs This course is intended for 3rd and/or 4th year undergraduate majors in Computer Science. You need to be familiar with the design and use of basic data structures such as Lists, Stacks,
More informationFinal Examination CSE 100 UCSD (Practice)
Final Examination UCSD (Practice) RULES: 1. Don t start the exam until the instructor says to. 2. This is a closed-book, closed-notes, no-calculator exam. Don t refer to any materials other than the exam
More informationGraphs. Tessema M. Mengistu Department of Computer Science Southern Illinois University Carbondale Room - Faner 3131
Graphs Tessema M. Mengistu Department of Computer Science Southern Illinois University Carbondale tessema.mengistu@siu.edu Room - Faner 3131 1 Outline Introduction to Graphs Graph Traversals Finding a
More informationCS 3410 Ch 14 Graphs and Paths
CS 3410 Ch 14 Graphs and Paths Sections 14.1-14.3 Pages 527-552 Exercises 1,2,5,7-9 14.1 Definitions 1. A vertex (node) and an edge are the basic building blocks of a graph. Two vertices, (, ) may be connected
More informationInfo 2950, Lecture 16
Info 2950, Lecture 16 28 Mar 2017 Prob Set 5: due Fri night 31 Mar Breadth first search (BFS) and Depth First Search (DFS) Must have an ordering on the vertices of the graph. In most examples here, the
More informationMultidimensional Arrays & Graphs. CMSC 420: Lecture 3
Multidimensional Arrays & Graphs CMSC 420: Lecture 3 Mini-Review Abstract Data Types: List Stack Queue Deque Dictionary Set Implementations: Linked Lists Circularly linked lists Doubly linked lists XOR
More informationGraph definitions. There are two kinds of graphs: directed graphs (sometimes called digraphs) and undirected graphs. An undirected graph
Graphs Graph definitions There are two kinds of graphs: directed graphs (sometimes called digraphs) and undirected graphs start Birmingham 60 Rugby fill pan with water add salt to water take egg from fridge
More informationChapter 9 Graph Algorithms
Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 3 Definitions an undirected graph G = (V, E)
More information11/22/2016. Chapter 9 Graph Algorithms. Introduction. Definitions. Definitions. Definitions. Definitions
Introduction Chapter 9 Graph Algorithms graph theory useful in practice represent many real-life problems can be slow if not careful with data structures 2 Definitions an undirected graph G = (V, E) is
More informationIntroduction to Graphs
Introduction to Graphs CSE21 Winter 2017, Day 10 (B00), Day 6-7 (A00) February 1, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 What is a graph? A (directed) graph G is A nonempty set of vertices V, also
More informationGraph Algorithms. Chapter 22. CPTR 430 Algorithms Graph Algorithms 1
Graph Algorithms Chapter 22 CPTR 430 Algorithms Graph Algorithms Why Study Graph Algorithms? Mathematical graphs seem to be relatively specialized and abstract Why spend so much time and effort on algorithms
More informationLecture 9 Graph Traversal
Lecture 9 Graph Traversal Euiseong Seo (euiseong@skku.edu) SWE00: Principles in Programming Spring 0 Euiseong Seo (euiseong@skku.edu) Need for Graphs One of unifying themes of computer science Closely
More informationModule 11: Additional Topics Graph Theory and Applications
Module 11: Additional Topics Graph Theory and Applications Topics: Introduction to Graph Theory Representing (undirected) graphs Basic graph algorithms 1 Consider the following: Traveling Salesman Problem
More informationLecture Notes. char myarray [ ] = {0, 0, 0, 0, 0 } ; The memory diagram associated with the array can be drawn like this
Lecture Notes Array Review An array in C++ is a contiguous block of memory. Since a char is 1 byte, then an array of 5 chars is 5 bytes. For example, if you execute the following C++ code you will allocate
More informationCS Algorithms and Complexity
CS 350 - Algorithms and Complexity Graph Theory, Midterm Review Sean Anderson 2/6/18 Portland State University Table of contents 1. Graph Theory 2. Graph Problems 3. Uninformed Exhaustive Search 4. Informed
More informationSELF-BALANCING SEARCH TREES. Chapter 11
SELF-BALANCING SEARCH TREES Chapter 11 Tree Balance and Rotation Section 11.1 Algorithm for Rotation BTNode root = left right = data = 10 BTNode = left right = data = 20 BTNode NULL = left right = NULL
More informationCIS 121 Data Structures and Algorithms Midterm 3 Review Solution Sketches Fall 2018
CIS 121 Data Structures and Algorithms Midterm 3 Review Solution Sketches Fall 2018 Q1: Prove or disprove: You are given a connected undirected graph G = (V, E) with a weight function w defined over its
More informationFundamental Algorithms
Fundamental Algorithms Chapter 8: Graphs Jan Křetínský Winter 2017/18 Chapter 8: Graphs, Winter 2017/18 1 Graphs Definition (Graph) A graph G = (V, E) consists of a set V of vertices (nodes) and a set
More informationECE 242. Data Structures
ECE 242 Data Structures Lecture 28 Introduction to Graphs Overview Problem: How do we represent irregular connections between locations? Graphs Definition Directed and Undirected graph Simple path and
More informationGraph Algorithms Using Depth First Search
Graph Algorithms Using Depth First Search Analysis of Algorithms Week 8, Lecture 1 Prepared by John Reif, Ph.D. Distinguished Professor of Computer Science Duke University Graph Algorithms Using Depth
More informationCSE 373 NOVEMBER 20 TH TOPOLOGICAL SORT
CSE 373 NOVEMBER 20 TH TOPOLOGICAL SORT PROJECT 3 500 Internal Error problems Hopefully all resolved (or close to) P3P1 grades are up (but muted) Leave canvas comment Emails tomorrow End of quarter GRAPHS
More informationProblem Score Maximum MC 34 (25/17) = 50 Total 100
Stony Brook University Midterm 2 CSE 373 Analysis of Algorithms November 22, 2016 Midterm Exam Name: ID #: Signature: Circle one: GRAD / UNDERGRAD INSTRUCTIONS: This is a closed book, closed mouth exam.
More informationALGORITHM DESIGN GREEDY ALGORITHMS. University of Waterloo
ALORITHM DSIN RDY ALORITHMS University of Waterloo LIST OF SLIDS - List of Slides reedy Approaches xample: Making Change 4 Making Change (cont.) 5 Minimum Spanning Tree 6 xample 7 Approaches that Don t
More informationCS 314 Final Fall 2012
Points off 1 2A 2B 2C 3 4A 4B 5 Total off Net Score CS 314 Final Fall 2012 Your Name_ Your UTEID Instructions: 1. There are 5 questions on this exam. The raw point total on the exam is 110. 2. You have
More informationToday s Outline. Graph ADT? Graph Definitions. Trees as Graphs. Graphs Chapter 9 in Weiss. More Definitions: Simple Paths and Cycles.
Today s Outline Graphs Chapter 9 in Weiss CSE 6 Data Structures Ruth Anderson /6/00 Announcements Written Homework #6 due NOW Project Code due Mon March by pm Project Benchmarking & Written due Thurs March
More informationReview: Graph Theory and Representation
Review: Graph Theory and Representation Graph Algorithms Graphs and Theorems about Graphs Graph implementation Graph Algorithms Shortest paths Minimum spanning tree What can graphs model? Cost of wiring
More informationGraphs ADT and Traversing
Steven J. Zeil July 31, 2013 Contents 1 A Graph ADT 3 1.1 Vertex...................................... 4 1.2 Edge....................................... 7 1.3 DiGraph....................................
More informationKonigsberg Bridge Problem
Graphs Konigsberg Bridge Problem c C d g A Kneiphof e D a B b f c A C d e g D a b f B Euler s Graph Degree of a vertex: the number of edges incident to it Euler showed that there is a walk starting at
More informationDirect Addressing Hash table: Collision resolution how handle collisions Hash Functions:
Direct Addressing - key is index into array => O(1) lookup Hash table: -hash function maps key to index in table -if universe of keys > # table entries then hash functions collision are guaranteed => need
More informationCSE 100: GRAPH SEARCH
CSE 100: GRAPH SEARCH Announcements PA3 released Checkpoint due Tuesday, May 5 @10:00pm Final submission due Thursday, May 14 @10:00PM Start early! Start early! Start early! Start early! Start early! I
More informationToday s Outline. CSE 326: Data Structures. Topic #15: Cool Graphs n Pretty Pictures. A Great Mathematician. The Bridges of Königsberg
CSE 326: Data Structures Topic #15: Cool Graphs n Pretty Pictures Ashish Sabharwal Autumn, 2003 Today s Outline Admin Project 3 in-progress checkin due tonight! Graph Algorithms Representation Applications
More informationUnit #9: Graphs. CPSC 221: Algorithms and Data Structures. Will Evans 2012W1
Unit #9: Graphs CPSC 1: Algorithms and Data Structures Will Evans 01W1 Unit Outline Topological Sort: Getting to Know Graphs with a Sort Graph ADT and Graph Representations Graph Terminology More Graph
More informationGraphs Introduction and Depth first algorithm
Graphs Introduction and Depth first algorithm Carol Zander Introduction to graphs Graphs are extremely common in computer science applications because graphs are common in the physical world. Everywhere
More informationUndirected Graphs. DSA - lecture 6 - T.U.Cluj-Napoca - M. Joldos 1
Undirected Graphs Terminology. Free Trees. Representations. Minimum Spanning Trees (algorithms: Prim, Kruskal). Graph Traversals (dfs, bfs). Articulation points & Biconnected Components. Graph Matching
More information18: GRAPH DATA STRUCTURES. Introduction
18: GRAPH DATA STRUCTURES Introduction... 1 Describing graphs... Directed Graphs... 3 Traversing a graph... EXERCISE: Traversal... 8 Implementing a Graph... 9 EXERCISE: Looking at a Graph... 1 EXERICISE:
More informationCS200: Graphs. Prichard Ch. 14 Rosen Ch. 10. CS200 - Graphs 1
CS200: Graphs Prichard Ch. 14 Rosen Ch. 10 CS200 - Graphs 1 Graphs A collection of nodes and edges What can this represent? n A computer network n Abstraction of a map n Social network CS200 - Graphs 2
More informationPoints off Total off Net Score. CS 314 Final Exam Spring 2017
Points off 1 2 3 4 5 6 Total off Net Score CS 314 Final Exam Spring 2017 Your Name Your UTEID Instructions: 1. There are 6 questions on this test. 100 points available. Scores will be scaled to 300 points.
More information1. Graph and Representation
Chapter Graphs Tree Root Direction: parent-child relationships No cycles Graph Vertices + Edges No tree restrictions Examples World Wide Web Maps. Graph and Representation A graph G: a pair (V, E) vertices
More informationChapter 9 Graph Algorithms
Chapter 9 Graph Algorithms 2 Introduction graph theory useful in practice represent many real-life problems can be if not careful with data structures 3 Definitions an undirected graph G = (V, E) is a
More informationCS473-Algorithms I. Lecture 13-A. Graphs. Cevdet Aykanat - Bilkent University Computer Engineering Department
CS473-Algorithms I Lecture 3-A Graphs Graphs A directed graph (or digraph) G is a pair (V, E), where V is a finite set, and E is a binary relation on V The set V: Vertex set of G The set E: Edge set of
More informationGraphs and Graph Algorithms. Slides by Larry Ruzzo
Graphs and Graph Algorithms Slides by Larry Ruzzo Goals Graphs: defns, examples, utility, terminology Representation: input, internal Traversal: Breadth- & Depth-first search Three Algorithms: Connected
More informationGraph Algorithms: Chapters Part 1: Introductory graph concepts
UMass Lowell Computer Science 91.503 Algorithms Dr. Haim Levkowitz Fall, 2007 Graph Algorithms: Chapters 22-25 Part 1: Introductory graph concepts 1 91.404 Graph Review Elementary Graph Algorithms Minimum
More informationGraph Theory. Many problems are mapped to graphs. Problems. traffic VLSI circuits social network communication networks web pages relationship
Graph Graph Usage I want to visit all the known famous places starting from Seoul ending in Seoul Knowledge: distances, costs Find the optimal(distance or cost) path Graph Theory Many problems are mapped
More informationCS 314 Final Fall 2011
Points off 1 2A 2B 2C 3 4 5 Total off Net Score CS 314 Final Fall 2011 Your Name_ Your UTEID Instructions: 1. There are 5 questions on this test. 2. You have 3 hours to complete the test. 3. You may not
More informationAlgorithm Design (8) Graph Algorithms 1/2
Graph Algorithm Design (8) Graph Algorithms / Graph:, : A finite set of vertices (or nodes) : A finite set of edges (or arcs or branches) each of which connect two vertices Takashi Chikayama School of
More informationCSE 373 Autumn 2012: Midterm #2 (closed book, closed notes, NO calculators allowed)
Name: Sample Solution Email address: CSE 373 Autumn 0: Midterm # (closed book, closed notes, NO calculators allowed) Instructions: Read the directions for each question carefully before answering. We may
More informationProblem 1. Which of the following is true of functions =100 +log and = + log? Problem 2. Which of the following is true of functions = 2 and =3?
Multiple-choice Problems: Problem 1. Which of the following is true of functions =100+log and =+log? a) = b) =Ω c) =Θ d) All of the above e) None of the above Problem 2. Which of the following is true
More informationCS2 Algorithms and Data Structures Note 9
CS2 Algorithms and Data Structures Note 9 Graphs The remaining three lectures of the Algorithms and Data Structures thread will be devoted to graph algorithms. 9.1 Directed and Undirected Graphs A graph
More informationPoints off Total off Net Score. CS 314 Final Exam Fall 2016
Points off 1 2 3 4 5 6 Total off Net Score CS 314 Final Exam Fall 2016 Your Name Your UTEID Instructions: 1. There are 6 questions on this test. 100 points available. Scores will be scaled to 300 points.
More informationData Structures and Algorithms for Engineers
4-63 Data Structures and Algorithms for Engineers David Vernon Carnegie Mellon University Africa vernon@cmuedu wwwvernoneu Data Structures and Algorithms for Engineers 1 Carnegie Mellon University Africa
More informationGraphs. Carlos Moreno uwaterloo.ca EIT
Carlos Moreno cmoreno @ uwaterloo.ca EIT-4103 http://en.wikipedia.org/wiki/six_degrees_of_kevin_bacon https://ece.uwaterloo.ca/~cmoreno/ece250 Standard reminder to set phones to silent/vibrate mode, please!
More informationData Structures. Giri Narasimhan Office: ECS 254A Phone: x-3748
Data Structures Giri Narasimhan Office: ECS 254A Phone: x-3748 giri@cs.fiu.edu Lecture 18 03/11/04 Vertex and Edge classes Class Edge { public Vertex dest; public double weight; public Edge (Vertex d,
More informationCOP 3530, Course Outcomes
COP 3530, Course Outcomes be familiar with basic techniques of algorithm analysis master analyzing simple non-recursive algorithms be familiar with analyzing recursive algorithms master the implementation
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 4 Graphs Definitions Traversals Adam Smith 9/8/10 Exercise How can you simulate an array with two unbounded stacks and a small amount of memory? (Hint: think of a
More informationMA/CSSE 473 Day 12. Questions? Insertion sort analysis Depth first Search Breadth first Search. (Introduce permutation and subset generation)
MA/CSSE 473 Day 12 Interpolation Search Insertion Sort quick review DFS, BFS Topological Sort MA/CSSE 473 Day 12 Questions? Interpolation Search Insertion sort analysis Depth first Search Breadth first
More information2. True or false: even though BFS and DFS have the same space complexity, they do not always have the same worst case asymptotic time complexity.
1. T F: Consider a directed graph G = (V, E) and a vertex s V. Suppose that for all v V, there exists a directed path in G from s to v. Suppose that a DFS is run on G, starting from s. Then, true or false:
More informationGraphs. CSE 2320 Algorithms and Data Structures Alexandra Stefan and Vassilis Athitsos University of Texas at Arlington
Graphs CSE 2320 Algorithms and Data Structures Alexandra Stefan and Vassilis Athitsos University of Texas at Arlington 1 Representation Adjacency matrix??adjacency lists?? Review Graphs (from CSE 2315)
More informationGraphs. directed and undirected graphs weighted graphs adjacency matrices. abstract data type adjacency list adjacency matrix
Graphs 1 Graphs directed and undirected graphs weighted graphs adjacency matrices 2 Graph Representations abstract data type adjacency list adjacency matrix 3 Graph Implementations adjacency matrix adjacency
More informationData Structures and Algorithms (DSA) Course 9 Lists / Graphs / Trees. Iulian Năstac
Data Structures and Algorithms (DSA) Course 9 Lists / Graphs / Trees Iulian Năstac Recapitulation It is considered the following type: typedef struct nod { ; struct nod *next; } NOD; 2 Circular
More informationChapter 9 Graph Algorithms
Introduction graph theory useful in practice represent many real-life problems can be if not careful with data structures Chapter 9 Graph s 2 Definitions Definitions an undirected graph is a finite set
More informationUNIT Name the different ways of representing a graph? a.adjacencymatrix b. Adjacency list
UNIT-4 Graph: Terminology, Representation, Traversals Applications - spanning trees, shortest path and Transitive closure, Topological sort. Sets: Representation - Operations on sets Applications. 1. Name
More informationCOT 6405 Introduction to Theory of Algorithms
COT 6405 Introduction to Theory of Algorithms Topic 14. Graph Algorithms 11/7/2016 1 Elementary Graph Algorithms How to represent a graph? Adjacency lists Adjacency matrix How to search a graph? Breadth-first
More informationElementary Graph Algorithms. Ref: Chapter 22 of the text by Cormen et al. Representing a graph:
Elementary Graph Algorithms Ref: Chapter 22 of the text by Cormen et al. Representing a graph: Graph G(V, E): V set of nodes (vertices); E set of edges. Notation: n = V and m = E. (Vertices are numbered
More informationGraphs. 04/01/03 Lecture 20 1
Graphs Graphs model networks of various kinds: roads, highways, oil pipelines, airline routes, dependency relationships, etc. Graph G(V,E) VVertices or Nodes EEdges or Links: pairs of vertices Directed
More informationCS/COE 1501 cs.pitt.edu/~bill/1501/ Graphs
CS/COE 1501 cs.pitt.edu/~bill/1501/ Graphs 5 3 2 4 1 0 2 Graphs A graph G = (V, E) Where V is a set of vertices E is a set of edges connecting vertex pairs Example: V = {0, 1, 2, 3, 4, 5} E = {(0, 1),
More information( ) D. Θ ( ) ( ) Ο f ( n) ( ) Ω. C. T n C. Θ. B. n logn Ο
CSE 0 Name Test Fall 0 Multiple Choice. Write your answer to the LEFT of each problem. points each. The expected time for insertion sort for n keys is in which set? (All n! input permutations are equally
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 informationThe Shortest Path Problem
The Shortest Path Problem 1 Shortest-Path Algorithms Find the shortest path from point A to point B Shortest in time, distance, cost, Numerous applications Map navigation Flight itineraries Circuit wiring
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 5 Exploring graphs Adam Smith 9/5/2008 A. Smith; based on slides by E. Demaine, C. Leiserson, S. Raskhodnikova, K. Wayne Puzzles Suppose an undirected graph G is connected.
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 informationOutline. Introduction. Representations of Graphs Graph Traversals. Applications. Definitions and Basic Terminologies
Graph Chapter 9 Outline Introduction Definitions and Basic Terminologies Representations of Graphs Graph Traversals Breadth first traversal Depth first traversal Applications Single source shortest path
More informationLecture Summary CSC 263H. August 5, 2016
Lecture Summary CSC 263H August 5, 2016 This document is a very brief overview of what we did in each lecture, it is by no means a replacement for attending lecture or doing the readings. 1. Week 1 2.
More informationCSI 604 Elementary Graph Algorithms
CSI 604 Elementary Graph Algorithms Ref: Chapter 22 of the text by Cormen et al. (Second edition) 1 / 25 Graphs: Basic Definitions Undirected Graph G(V, E): V is set of nodes (or vertices) and E is the
More informationGraphs and Genetics. Outline. Computational Biology IST. Ana Teresa Freitas 2015/2016. Slides source: AED (MEEC/IST); Jones and Pevzner (book)
raphs and enetics Computational Biology IST Ana Teresa Freitas / Slides source: AED (MEEC/IST); Jones and Pevzner (book) Outline l Motivacion l Introduction to raph Theory l Eulerian & Hamiltonian Cycle
More informationECE 242 HOMEWORK 5. In the figure below, one can go from node B to A but not from A to B.
ECE 242 HOMEWORK 5 Question 1: Define the following terms. For lines with multiple terms,differentiate between the terms. Also draw a figure illustrating each term. (a) Directed graph and undirected graph
More informationLecture 23 Representing Graphs
Lecture 23 Representing Graphs 15-122: Principles of Imperative Computation (Fall 2018) Frank Pfenning, André Platzer, Rob Simmons, Penny Anderson, Iliano Cervesato In this lecture we introduce graphs.
More informationUNIT III TREES. A tree is a non-linear data structure that is used to represents hierarchical relationships between individual data items.
UNIT III TREES A tree is a non-linear data structure that is used to represents hierarchical relationships between individual data items. Tree: A tree is a finite set of one or more nodes such that, there
More informationData Structures Brett Bernstein
Data Structures Brett Bernstein Final Review 1. Consider a binary tree of height k. (a) What is the maximum number of nodes? (b) What is the maximum number of leaves? (c) What is the minimum number of
More informationCS302 - Data Structures using C++
CS302 - Data Structures using C++ Topic: Graphs - Introduction Kostas Alexis Terminology In the context of our course, graphs represent relations among data items G = {V,E} A graph is a set of vertices
More informationCSE100 Practice Final Exam Section C Fall 2015: Dec 10 th, Problem Topic Points Possible Points Earned Grader
CSE100 Practice Final Exam Section C Fall 2015: Dec 10 th, 2015 Problem Topic Points Possible Points Earned Grader 1 The Basics 40 2 Application and Comparison 20 3 Run Time Analysis 20 4 C++ and Programming
More informationIntroduction to Graphs. common/notes/ppt/
Introduction to Graphs http://people.cs.clemson.edu/~pargas/courses/cs212/ common/notes/ppt/ Introduction Graphs are a generalization of trees Nodes or verticies Edges or arcs Two kinds of graphs irected
More informationcsci 210: Data Structures Graph Traversals
csci 210: Data Structures Graph Traversals 1 Depth-first search (DFS) G can be directed or undirected DFS(v) mark v visited for all adjacent edges (v,w) of v do if w is not visited parent(w) = v (v,w)
More informationUNIT 5 GRAPH. Application of Graph Structure in real world:- Graph Terminologies:
UNIT 5 CSE 103 - Unit V- Graph GRAPH Graph is another important non-linear data structure. In tree Structure, there is a hierarchical relationship between, parent and children that is one-to-many relationship.
More informationMaster Theorem, Introduction to Graphs
Master Theorem, Introduction to Graphs CSE21 Winter 2017, Day 10 (B00), Day 6-7 (A00) February 1, 2017 http://vlsicad.ucsd.edu/courses/cse21-w17 Divide & Conquer: General Strategy Divide the problem of
More informationCS61B, Fall 2015 Final Examination (with corrections) P. N. Hilfinger
CS61B, Fall 2015 Final Examination (with corrections) P. N. Hilfinger READ THIS PAGE FIRST. Your exam should contain 14 problems on 16 pages. Officially, it is worth 46 points. This is an open-book test.
More informationGoals! CSE 417: Algorithms and Computational Complexity!
Goals! CSE : Algorithms and Computational Complexity! Graphs: defns, examples, utility, terminology! Representation: input, internal! Traversal: Breadth- & Depth-first search! Three Algorithms:!!Connected
More informationCOSC 2007 Data Structures II Final Exam. Part 1: multiple choice (1 mark each, total 30 marks, circle the correct answer)
COSC 2007 Data Structures II Final Exam Thursday, April 13 th, 2006 This is a closed book and closed notes exam. There are total 3 parts. Please answer the questions in the provided space and use back
More informationGraphs. The ultimate data structure. graphs 1
Graphs The ultimate data structure graphs 1 Definition of graph Non-linear data structure consisting of nodes & links between them (like trees in this sense) Unlike trees, graph nodes may be completely
More information) $ f ( n) " %( g( n)
CSE 0 Name Test Spring 008 Last Digits of Mav ID # Multiple Choice. Write your answer to the LEFT of each problem. points each. The time to compute the sum of the n elements of an integer array is: # A.
More informationGraphs. Motivations: o Networks o Social networks o Program testing o Job Assignment Examples: o Code graph:
Graphs Motivations: o Networks o Social networks o Program testing o Job Assignment Examples: o Code graph: S1: int x S2: If x > 0 then S3: X = x + 2; Else S4: X =x -1; End if S5: While x > 1 do S6: Print
More informationGraphs. The ultimate data structure. graphs 1
Graphs The ultimate data structure graphs 1 Definition of graph Non-linear data structure consisting of nodes & links between them (like trees in this sense) Unlike trees, graph nodes may be completely
More informationLecture 26: Graphs: Traversal (Part 1)
CS8 Integrated Introduction to Computer Science Fisler, Nelson Lecture 6: Graphs: Traversal (Part ) 0:00 AM, Apr, 08 Contents Introduction. Definitions........................................... Representations.......................................
More information