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What is a Graph?

What is a Graph | download ppt free
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Published Date:09-07-2017
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GRAPHS • Definitions • The Graph ADT • Data structures for graphs PVD LAX LAX STL HNL DFW FTL Graphs 1What is a Graph? •A graph G = (V,E) is composed of: V: set of vertices E: set of edges connecting the vertices in V •An edge e = (u,v) is a pair of vertices • Example: V= a,b,c,d,e a b E= (a,b),(a,c),(a,d), (b,e),(c,d),(c,e), c (d,e) e d Graphs 2Applications • electronic circuits CS16 start find the path of least resistance to CS16 • networks (roads, flights, communications) PVD LAX LAX STL HNL DFW FTL Graphs 3mo’ better examples A Spike Lee Joint Production • scheduling (project planning) A typical student day wake up eat cs16 meditation work more cs16 play cs16 program cxhextris make cookies for cs16 HTA sleep dream of cs16 Graphs 4Graph Terminology • adjacent vertices: connected by an edge • degree (of a vertex): of adjacent vertices Σ deg(v) = 2( edges) 3 2 v∈V • Since adjacent vertices each count the 3 adjoining edge, it will be counted twice 3 3 path: sequence of vertices v ,v ,. . .v such that 1 2 k consecutive vertices v and v are adjacent. i i+1 a a b b c c e d e d a b e d c b e d c Graphs 5More Graph Terminology • simple path: no repeated vertices a b b e c c e d • cycle: simple path, except that the last vertex is the same as the first vertex a b a c d a c e d Graphs 6Even More Terminology • connected graph: any two vertices are connected by some path connected not connected • subgraph: subset of vertices and edges forming a graph • connected component: maximal connected subgraph. E.g., the graph below has 3 connected components. Graphs 7¡Caramba Another Terminology Slide • (free) tree - connected graph without cycles • forest - collection of trees tree tree forest tree tree Graphs 8Connectivity Let n = vertices m = edges - complete graph - all pairs of vertices are adjacent m= (1/2)Σdeg(v) = (1/2)Σ(n - 1) = n(n-1)/2 v∈V v∈V • Each of the n vertices is incident to n - 1 edges, however, we would have counted each edge twice Therefore, intuitively, m = n(n-1)/2. n = 5 m = (5 4)/2 = 10 • Therefore, if a graph is not complete, m n(n-1)/2 Graphs 9More Connectivity n = vertices m = edges • For a tree m = n - 1 n = 5 m = 4 •If m n - 1, G is not connected n = 5 m = 3 Graphs 10Spanning Tree •A spanning tree of G is a subgraph which - is a tree - contains all vertices of G G spanning tree of G • Failure on any edge disconnects system (least fault tolerant) Graphs 11AT&T vs. RT&T (Roberto Tamassia & Telephone) • Roberto wants to call the TA’s to suggest an extension for the next program... TA TA But Plant-Ops ‘accidentally’ cuts a phone cable TA TA TA • One fault will disconnect part of graph • A cycle would be more fault tolerant and only requires n edges Graphs 12Euler and the Bridges of Koenigsberg Gilligan’s Isle? C D Pregal River A B Can one walk across each bridge exactly once and return at the starting point? • Consider if you were a UPS driver, and you didn’t want to retrace your steps. • In 1736, Euler proved that this is not possible Graphs 13Graph Model(with parallel edges) C A D B • Eulerian Tour: path that traverses every edge exactly once and returns to the first vertex • Euler’s Theorem: A graph has a Eulerian Tour if and only if all vertices have even degree • Do you find such ideas interesting? • Would you enjoy spending a whole semester doing such proofs? Well, look into CS22 if you dare... Graphs 14The Graph ADT • The Graph ADT is a positional container whose positions are the vertices and the edges ofthe graph. - size() Return the number of vertices plus the number of edges of G. - isEmpty() - elements() - positions() - swap() - replaceElement() Notation: Graph G; Vertices v, w; Edge e; Object o - numVertices() Return the number of vertices of G. - numEdges() Return the number of edges of G. - vertices() Return an enumeration of the vertices of G. - edges() Return an enumeration of the edges of G. Graphs 15The Graph ADT (contd.) - directedEdges() Return an enumeration of all directed edges in G. - undirectedEdges() Return an enumeration of all undirected edges in G. - incidentEdges(v) Return an enumeration of all edges incident on v. - inIncidentEdges(v) Return an enumeration of all the incoming edges to v. - outIncidentEdges(v) Return an enumeration of all the outgoing edges from v. - opposite(v, e) Return an endpoint of e distinct from v - degree(v) Return the degree of v. - inDegree(v) Return the in-degree of v. - outDegree(v) Return the out-degree of v. Graphs 16More Methods ... - adjacentVertices(v) Return an enumeration of the vertices adjacent to v. - inAdjacentVertices(v) Return an enumeration of the vertices adjacent to v along incoming edges. - outAdjacentVertices(v) Return an enumeration of the vertices adjacent to v along outgoing edges. - areAdjacent(v,w) Return whether vertices v and w are adjacent. - endVertices(e) Return an array of size 2 storing the end vertices of e. - origin(e) Return the end vertex from which e leaves. - destination(e) Return the end vertex at which e arrives. - isDirected(e) Return true iff e is directed. Graphs 17Update Methods - makeUndirected(e) Set e to be an undirected edge. - reverseDirection(e) Switch the origin and destination vertices of e. - setDirectionFrom(e, v) Sets the direction of e away from v, one of its end vertices. - setDirectionTo(e, v) Sets the direction of e toward v, one of its end vertices. - insertEdge(v, w, o) Insert and return an undirected edge between v and w, storing o at this position. - insertDirectedEdge(v, w, o) Insert and return a directed edge between v and w, storing o at this position. - insertVertex(o) Insert and return a new (isolated) vertex storing o at this position. - removeEdge(e) Remove edge e. Graphs 18AA 523 AA 411 Data Structures for Graphs • A Graph How can we represent it? • To start with, we store the vertices and the edges into two containers, and we store with each edge object references to its endvertices TW 45 BOS BOS ORD ORD JFK JFK SFO SFO DFW DFW LAX LAX MIA MIA • Additional structures can be used to perform efficiently the methods of the Graph ADT Graphs 19 AA 49 AA 1387 UA 120 NW 35 DL 247 UA 877 DL 335 AA 903Edge List • The edge list structure simply stores the vertices and the edges into unsorted sequences. • Easy to implement. • Finding the edges incident on a given vertex is inefficient since it requires examining the entire edge sequence E NW 35 DL 247 AA 49 DL 335 AA 1387 AA 523 AA 411 UA 120 AA 903 UA 877 TW 45 BOS LAX DFW JFK MIA ORD SFO V Graphs 20