Winter 2016 COMP-250: Introduction to Computer Science. Lecture 16, March 10, 2016

Transcription

1 Winter : Introduction to Computer Science Lecture 16, March 10, 2016

2 GRAPHS Definitions Examples The Graph ADT PVD HNL LAX DFW STL FTL

3 What is a Graph? Agraph 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 c b E= {(a,b),(a,c),(a,d), (b,e),(c,d),(c,e), (d,e)} d e

4 electronic circuits Applications CS start find the path of least resistance to CS16 250

5 Applications electronic networks (roads, circuitsflights, communications) CS16 PVD HNL start LAX STL DFW find the path of least resistance to CS16 FTL

6 scheduling (project planning) wake up cs16 meditation A typical student day eat work more cs16 play make cookies for cs16 HTA cs16 program sleep battletris FACEBOOK dream of cs16

7 scheduling (project planning) wake up cs16 meditation A typical student day eat work more cs16 play make cookies for cs16 HTA Hawaii Tourism Authority? cs16 program sleep battletris FACEBOOK dream of cs16

8 scheduling (project planning) wake up cs16 meditation A typical student day eat work more cs16 play make cookies for cs16 HTA Hawaii Tourism Authority? Hunter and Trappers Association? cs16 program sleep battletris FACEBOOK dream of cs16

9 scheduling (project planning) wake up cs16 meditation A typical student day eat work more cs16 play make cookies for cs16 HTA Hawaii Tourism Authority? Hunter and Trappers Association? History Teachers' Association? cs16 program sleep battletris FACEBOOK dream of cs16

10 scheduling (project planning) wake up cs16 meditation A typical student day eat work more cs16 play make cookies for cs16 HTA Hawaii Tourism Authority? Hunter and Trappers Association? History Teachers' Association? Hierarchical Task Analysis! cs16 program sleep battletris FACEBOOK dream of cs16

11 Graph electronic circuits Applications Nodes Edges transportation street intersections highways CS16 communication computers fiber optic cables World Wide Web web pages hyperlinks social people relationships start food web species predator-prey software systems functions function calls scheduling tasks precedence constraints find the path of least resistance to CS16 circuits gates wires

12 Graph Terminology adjacent vertices: connected by an edge degree (of a vertex): # of adjacent vertices 3 2 Σ deg(v) = 2(# edges) v V Since adjacent vertices each count the adjoining edge, it will be counted twice

13 Graph Terminology adjacent path: sequence vertices: of connected vertices v 1 by,v 2 an,.. edge.v k such that degree consecutive (of a vertices vertex): v i # and of adjacent v i+1 are adjacent. vertices a b a Σ b deg(v) = 2(# edges) 3 2 v V c 3 d e 3 a b e d c 3 Since adjacent c vertices each count the adjoining edge, it will d be counted twice e b e d c

14 More Graph Terminology simple path: no repeated vertices a b c b e c d e

15 More Graph Terminology simple cycle: path: simpleno path, repeated except vertices that the last vertex is the same as the first vertex a b a b c c b e c a c d a d d e e

16 Even More Terminology connected graph: any two vertices are connected by some path connected not connected

17 More Graph Terminology simple subgraph: path: subset no repeated of vertices vertices and edges forming a graph connected a component: maximal b connected subgraph. E.g., the graph below has 3 connected components. b e c c d e

18 Yet another Terminology Slide! (free) tree - connected graph without cycles forest - collection of trees tree tree forest tree tree

19 Let n = #vertices m = #edges Connectivity - 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

20 n = #vertices m = #edges For a tree m = n - 1 More Connectivity n = 5 m = 4 If m < n - 1, G is not connected n = 5 m = 3

21 Spanning Tree Aspanning 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)

22 Roberto wants to call the TA s to suggest an extension for the next program... TA But Plant-Ops accidentally cuts a phone cable!!! TA TA TA TA One fault will disconnect part of graph!! One fault will disconnect part of graph!! A cycle would be more fault tolerant and only A requires cycle would n edges be more fault tolerant and only requires n edges

23 Koenigsberg Gilligan s Isle? C Pregal River A D 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 Consider want to retrace if you your were steps. UPS driver, and you didn t want to retrace your steps. In 1736, Euler proved that this is not possible

24 Graph 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

25 The Graph ADT The Graph ADT is a positional container whose positions are the vertices and the edges of the graph. - size() Return the number of vertices plus the number of edges of G. - isempty() - elements() - positions() - swap() - replaceelement()

26 The Graph ADT (contd.) Notation: - directededges() Graph G; Vertices v, w; Edge e; Object o - numvertices() Return an enumeration of all directed edges Return inthe G. number of vertices of G. - undirectededges() numedges() Return an the enumeration number of edges of allof G. - vertices() undirected Return an enumeration edges in G. of the vertices incidentedges(v) of G. - edges() Return Return an an enumeration enumeration of of the all edges edges of G.

27 The 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.

28 The Graph ADT (contd.) - directededges() opposite(v, e) Return an enumeration endpoint of e of distinct all directed from v - degree(v) edges in G. - undirectededges() Return the degree of v. - indegree(v) Return an enumeration of all undirected Return the in-degree edges G. of v. - incidentedges(v) outdegree(v) Return an the enumeration out-degree of of v. all edges

29 More 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.

30 More Methods... - adjacentvertices(v) endvertices(e) Return an enumeration array of size 2 of storing the vertices the adjacent end vertices to v. of e. - inadjacentvertices(v) origin(e) Return an the enumeration end vertex from of the which vertices e adjacent leaves. to v along incoming edges. - outadjacentvertices(v) destination(e) Return an the enumeration end vertex at of which the verticese adjacent arrives. to v along outgoing edges. - isdirected(e) Return true iff e is directed.

31 Update 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.

32 Update Methods - makeundirected(e) insertedge(v, w, o) Set Insert e to and be return an undirected an undirected edge. edge - reversedirection(e) between v and w, storing o at this Switch position. the origin and destination - insertdirectededge(v, vertices of e. w, o) - setdirectionfrom(e, Insert and v) return a directed edge Sets between the direction v and w, storing of e away o at from thisv, one of position. its end vertices. - setdirectionto(e, insertvertex(o) v) Sets Insert the and direction return a of new e toward (isolated) v, one of its vertex end storing vertices. o at this position. - removeedge(e) Remove edge e.

33 DATA STRUCTURES FOR GRAPHS Edge list Adjacency lists Adjacency matrix 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

34 To start with, we store the vertices and the edges into two containers, and each edge object has references to the vertices it connects. Data Structures for Graphs SFO LAX UA 120 AA 49 UA 877 DFW AA 411 TW 45 ORD DL 335 AA 1387 AA 523 JFK AA 903 MIA NW 35 BOS DL 247 Additional structures can be used to perform efficiently the methods of the Graph ADT

35 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 Edge List 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

36 Performance of the Edge List Structure Operation size, isempty, replaceelement, swap numvertices, numedges vertices edges, directededges, undirectededges elements, positions endvertices, opposite, origin, destination, isdirected incidentedges, inincidentedges, outincidentedges, adjacentvertices, inadjacentvertices, outadjacentvertices, areadjacent, degree, indegree, outdegree insertvertex, insertedge, insertdirected- Edge, removeedge, makeundirected, reversedirection, setdirectionfrom, setdirectionto removevertex Time O(1) O(1) O(n) O(m) O(n+m) O(1) O(m) O(1) O(m)

37 adjacency list of a vertex v: sequence of vertices adjacent to v Adjacency List (traditional) represent the graph by the adjacency lists of all the vertices a b c d a d b a c c e c d a d e a e b e e b c d Space = Θ(N + Σdeg(v)) = Θ(N + M)

38 The adjacency list structure extends the edge list structure by adding incidence containers to each vertex. The space requirement is O(n + m). NW 35 DL 247 AA 49 DL 335 AA 1387 AA 523 AA 411 UA 120 AA 903 UA 877 TW 45 Adjacency List (modern) BOS LAX DFW JFK MIA ORD SFO in out in out in out in out in out in out in out NW 35 AA 49 UA 120 AA1387 DL335 NW 35 AA1387 DL 247 AA523 UA 120 UA 877 TW 45 DL 247 AA 411 UA 877 AA 49 AA 903 AA 903 AA 411 DL 335 AA 523 TW 45

39 Performance of the Adjacency List Structure Operation size, isempty, replaceelement, swap numvertices, numedges vertices edges, directededges, undirectededges elements, positions endvertices, opposite, origin, destination, isdirected, degree, indegree, out- Degree incidentedges(v), inincidentedges(v), outincidentedges(v), adjacentvertices(v), inadjacentvertices(v), outadjacentvertices(v) areadjacent(u, v) insertvertex, insertedge, insertdirected- Edge, removeedge, makeundirected, reversedirection, removevertex(v) Time O(1) O(1) O(n) O(m) O(n+m) O(1) O(deg(v)) O(min(deg(u), deg(v))) O(1) O(deg(v))

40 Adjacency Matrix (traditional) a d c F T T T F T F F F T T F F T T T F T F T F T T T F matrix M with entries for all pairs of vertices b e a b c d e M[i,j] = true means that there is an edge (i,j) in the graph. M[i,j] = false means that there is no edge (i,j) in the graph. a b c d e There is an entry for every possible edge, therefore: Space = Θ(N 2 )

41 The adjacency matrix structures augments the edge list structure with a matrix where each row and column corresponds to a vertex. BOS DFW JFK LAX MIA ORD SFO The space requirement is O(n 2 + m) Adjacency Matrix (modern) Ø Ø NW 35 Ø 1 Ø Ø Ø AA 49 2 Ø AA 1387 DL 247 Ø Ø Ø AA 903 Ø DL Ø Ø Ø Ø Ø UA Ø AA Ø UA 877 Ø AA 411 Ø Ø Ø TW 45 Ø Ø Ø Ø Ø Ø Ø Ø Ø 6 Ø Ø Ø Ø Ø Ø Ø

42 Performance of the Adjacency Matrix Structure Operation size, isempty, replaceelement, swap numvertices, numedges vertices edges, directededges, undirectededges elements, positions endvertices, opposite, origin, destination, isdirected, degree, indegree, outdegree incidentedges, inincidentedges, outincidentedges, adjacentvertices, inadjacentvertices, outadjacentvertices, areadjacent insertedge, insertdirectededge, removeedge, makeundirected, reversedirection, setdirectionfrom, setdirectionto Time O(1) O(1) O(n) O(m) O(n+m) O(1) O(n) O(1) O(1) insertvertex, removevertex O(n 2 )

43 Winter : Introduction to Computer Science Lecture 16, March 10, 2016