Announcement New exam dates: Graphs Exam 1 Monday, Oct 6 th - PDF document

Announcement • New exam dates: Graphs – Exam 1 – Monday, Oct 6 th – Exam 2 – Monday, Nov 3 rd Terminology & Implementation Announcement Announcement • The ACM is looking for a few good • Actually…so is Google programmers • Google Code Jam – ACM Programming Contest – 1 st Round: Oct 1-Oct 15 • 3 Programmers per Team • Prelims – Sept 26 th • If interested, contact Paul Tymann (ptt@cs.rit.edu) – See by Wednesday. • http://topcoder.com/pl/?&module=Static&d1=googl e&d2=google_overview • Did I mention free food? Before we begin Graphs • Any questions? • A graph consists – Of a collection of points sometimes called vertices – Some pairs of points are connected by a line segment sometimes called an edge • Edges – May have a direction associated with them in which case the graph is called a directed graph ( digraph ) – A graph that contains edges that do not have a direction associated with them is called an undirected graph

Graphs Graph edges A D • In addition to being directed or undirected, A D edges can be weighted or un-weighted. C C B B – A weighted edge has a value associated with it F F – The weight often measures the cost of using the E E edge to go from one node to another • A vertex may also have data associated with it. Graphs are usually drawn using points for vertices and lines for Edges, but a graph is defined independent of its representation Formal Definition Formal Definition – Edges • In an undirected graph • A graph G = ( V , E ), consists of – each edge e = < v 1 , v 2 > is an unordered pair of distinct – V, a set of vertices vertices – E, a set of edges where each edge is a pair of • In a directed graph distinct elements from V – each edge e = < v 1 , v 2 > is an ordered pair of distinct vertices – < v 1 , v 2 > ≠ < v 2 , v 1 > • For weighted edges, weight can have a scalar value, w, associated with it – < v 1 , v 2 , w > Formal definition example Formal definition example • V = { A, B, C, D, E, • V = { A, B, C, D, E, F} F} 10 • E = { <A, B>, • E = { <A, B, 5>, A D A D 5 • <B,C>, 15 • <B,C, 20>, 25 C C B B • <A,C>, • <A,C, 15>, 20 F F • <A,D>, • <A,D, 10>, 30 E E • <C,D>, • <C,D, 25>, • <E,F> } • <E,F, 30> }

Terminology Terminology – Paths and Cycles • The degree of a vertex x is the number of • Two different vertices, x and y , in a graph edges e in which x is one of the endpoints of are said to be adjacent if an edge connects x edge e to y • The neighbors of a vertex v , are the vertices – I.e. if <x,y> is an edge, x and y are adjacent that are directly connected to v Terminology – Paths and Cycles Paths and Cycles -- Example A D • A path is a sequence of vertices in which each • ADCFDC is a path vertex is adjacent to the next one • ADC is a simple path C B – V 1 , V 2 , …, V n and <V i , V i+1 > is an edge. • A simple path is a path in which no vertex is F • ADCA is a cycle repeated E • A cycle is a path of length greater than one that • ADFECBA is a begins and ends at the same vertex Hamiltonian Cycle. • A Hamiltonian cycle is a cycle that visits all vertices exactly once. Connectedness Connectedness – Example A D • A graph is connected if A D – For every pair of nodes A, B, there is a simple C C B B path from A to B. F F E E • A subset of nodes that are connected define a connected subgraph Not connected Connected {A,B,C,D} and {E,F} form connected subgraphs

Terminology Building a Graph • Vertex, Edges (weighted and unweighted) import java.util.*; • Directed vs. undirected public interface DiGraph { • Paths and Cycles // Methods to build the graph public void addVertex( Object key, Object data ); public void addEdge( Object fromKey, Object toKey, • Connectedness Object data ) throws NoSuchVertexException; ... } // DiGraph • Questions so far? Building a Graph Graph Implementations • Adjacency Matrix addVertex (“A”, “A”); 2 • Adjacency List A B addVertex (“B”, “B”); addVertex (“C”, “C”); addEdge (“B”, “C”, new Integer(5)); 6 7 5 addEdge (“A”, “B”, new Integer(2)); addVertex (“D”, “D”); addEdge (“A”, “D”, new Integer(6)); D C addEdge (“B”, “D”, new Integer(7)); Adjacency Matrix Adjacency Matrix • Assumes vertices are numbered from 0 – n- • For unweighted edges, array elements are 1. boolean. 3 • Graph represented by a N x N, 2 0 1 2 3 4 dimensional array. 0 F F T F T 2 4 1 F F F F F • Array element (i, j) indicate in there is an 2 T F F T T edge between vertex i and vertex j 3 F F T F T 4 T F T T F 1 0 T = edge F = no edge

Adjacency Matrix Adjacency Matrix • For weighted edges, array elements can • Array elements might also contain objects hold weights. of an Edge class. 3 3 6.1 6.1 0 1 2 3 4 0 1 2 3 4 1.5 1.5 NULL 1.0 NULL 3.7 0 ∞ ∞ 1.0 ∞ 3.7 NULL 0 4.2 4.2 NULL NULL 2 4 2 4 NULL NULL NULL 1 ∞ ∞ ∞ ∞ ∞ 1 2 1.0 2 1.0 ∞ ∞ 6.1 4.2 1.0 3.7 1.0 3.7 NULL NULL 6.1 4.2 NULL 1.5 3 ∞ ∞ 6.1 ∞ 1.5 NULL NULL 6.1 3 NULL 4.2 1.5 4 3.7 1 0 1 0 NULL 4 3.7 ∞ 4.2 1.5 ∞ ∞ indicates no edge NULL indicates no edge Adjacency Matrix Adjacency List • Things to note: • Assumes vertices are numbered from 0 – n- 1. – For an undirected graph, only half of the matrix is needed. • Graph is represented by an array of lists, – Lots of wasted space if not a lot of edges in the one list for each vertex. graph. • Elements of list corresponding to a vertex indicates edges to neighboring vertices. Adjacency List Adjacency List • For unweighted edges, list nodes simply • For weighted edges, list nodes give the give the neighboring vertex neighboring vertex and the weight 3 3 6.1 0 2 4 0 2 4 1.5 1.0 3.7 2 4 1 1 4.2 2 4 3 0 4 2 3 0 4 2 6.1 1.0 4.2 3.7 3 3 1.0 2 4 2 4 6.1 1.5 4 4 3 2 0 3 2 0 1 0 1.5 4.2 3.7 1 0

Adjacency List Graph Implementations • List elements might also contain objects of • Graph Vertices an Edge class. – Very often, other data may be associated with a vertex 3 6.1 2 4 0 – It is often useful to create a Vertex class that 1.5 1.0 3.7 holds all data. 1 4.2 2 4 2 3 0 4 – Each graph may, in addition to edge 6.1 1.0 4.2 3.7 representation, contain a set of Vertex class 1.0 3 2 4 6.1 1.5 objects. 4 3 2 0 1.5 4.2 3.7 1 0 Graph Implementations Graph Implementations • Comparison – space • Comparison Time – Graph of 50 Vertices – 4bytes / entry Operation Adj. Matrix Adj List Edges Adj. Matrix Adj List Insert Node O (1) O (E / N) 49 10,000 792 Insert Edge O (1) O (E / N) 100 10,000 1,200 Delete Node O (N) O (E) 250 10,000 2,400 Delete Edge O (1) O (E / N) 500 10,000 4,400 Find Neighbors O(N) O (E / N) 1,000 10,000 8,400 N = number of nodes 1,200 10,000 10,000 1,500 10,000 12,400 E = number of edges 2,000 10,000 16,400 E / N = avg number of edges per vertex 2,500 10,000 20,400 Graph Implementations Summary • When implementing CS3DiGraph • Graphs • Terminology – Choose implementation best suited for TSP • Building a Graph – Feel free to use nested classes for aux classes (if required) • Implementing a Graph – Feel free to use Java Collections Framework • Tomorrow: Graph Traversals • Questions?