UNDIRECTED GRAPHS BBM 201 DATA STRUCTURES DEPT. OF COMPUTER ENGINEERING

Transcription

1 Acknowledgement: he course slides are adapted from the slides prepared by R. Sedgewick and K. Wayne of Princeton Uniersity. BBM DAA SRUCURES DEP. O COMPUER ENGINEERING UNDIRECED GRAPHS

2 Undirected Graphs Graph API Depth-first search Breadth-first search ODAY

3 Undirected graphs Graph. Set of ertices connected pairwise by edges. Why study graph algorithms? housands of practical applications. Hundreds of graph algorithms known. Interesting and broadly useful abstraction. Challenging branch of computer science and discrete math.

4 Undirected graphs Graph. Set of ertices connected pairwise by edges. Why study graph algorithms? housands of practical applications. Hundreds of graph algorithms known. Interesting and broadly useful abstraction. Challenging branch of computer science and discrete math.

5 Protein-protein interaction network Reference: Jeong et al, Nature Reiew Genetics

6 he Internet as mapped by the Opte Project

7 Map of science clickstreams

8 million acebook friends "Visualizing riendships" by Paul Butler

9 Graph applications graph ertex edge communication telephone, computer fiber optic cable circuit gate, register, processor wire mechanical joint rod, beam, spring financial stock, currency transactions transportation street intersection, airport highway, airway route internet class C network connection game board position legal moe social relationship person, actor friendship, moie cast neural network neuron synapse protein network protein protein-protein interaction chemical compound molecule bond

10 Graph terminology Path. Sequence of ertices connected by edges. Cycle. Path whose first and last ertices are the same. wo ertices are connected if there is a path between them.

11 Some graph-processing problems Path. Is there a path between s and t? Shortest path. What is the shortest path between s and t? Cycle. Is there a cycle in the graph? Euler tour. Is there a cycle that uses each edge exactly once? Hamilton tour. Is there a cycle that uses each ertex exactly once? Connectiity. Is there a way to connect all of the ertices? MS. What is the best way to connect all of the ertices? Biconnectiity. Is there a ertex whose remoal disconnects the graph? Planarity. Can you draw the graph in the plane with no crossing edges? Graph isomorphism. Do two adjacency lists represent the same graph? Challenge. Which of these problems are easy? difficult? intractable?

12 Graph API Depth-first search Breadth-first search UNDIRECED GRAPHS

13 Graph representation Graph drawing. Proides intuition about the structure of the graph. two drawings of the same graph Caeat. Intuition can be misleading.

14 Graph representation Vertex representation. his lecture: use integers between and V. Applications: conert between names and integers with symbol table. A B C G symbol table D E Anomalies.

15 Graph API: sample client Graph input format. tinyg.txt

16 Set-of-edges graph representation Maintain a list of the edges (linked list or array).

17 Adjacency-matrix graph representation Maintain a two-dimensional V-by-V boolean array; for each edge w in graph: adj[][w] = adj[w][] = true. two entries for each edge

18 Adjacency-list graph representation Maintain ertex-indexed array of lists.

19 Adjacency-list graph representation: Jaa implementation //for each edge call addedge twice addedge(,w); addedge(w,); public oid addedge(int, int w){ node *q; //acquire memory for the new node q=(node*)malloc(sizeof(node)); q->ertex=w; typedef struct node { struct node *next; int ertex; }node; node * adj []; q->next=null; //insert the node to beginning of the linked list q->next=adj[]; adj[]= q; }

20 Graph representations In practice. Use adjacency-lists representation. Algorithms based on iterating oer ertices adjacent to. Real-world graphs tend to be sparse. huge number of ertices, small aerage ertex degree

21 Graph representations In practice. Use adjacency-lists representation. Algorithms based on iterating oer ertices adjacent to. Real-world graphs tend to be sparse. huge number of ertices, small aerage ertex degree representation space add edge edge between and w? iterate oer ertices adjacent to? list of edges E E E adjacency matrix V V adjacency lists E + V degree() degree()

22 Graph API Depth-first search Breadth-first search UNDIRECED GRAPHS

23 Depth-first search Goal. Systematically search through a graph. DS (to isit a ertex ) Mark as isited. Recursiely isit all unmarked ertices w adjacent to. ypical applications. ind all ertices connected to a gien source ertex. ind a path between two ertices.

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25 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. graph G

26 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] graph G

27 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

28 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

29 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

30 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

31 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

32 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

33 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

34 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

35 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

36 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

37 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

38 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

39 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

40 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

41 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

42 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

43 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

44 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

45 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

46 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] isit

47 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

48 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] done

49 Depth-first search o isit a ertex : Mark ertex as isited. Recursiely isit all unmarked ertices adjacent to. marked[] edgeo[] ertices reachable from

50 Depth-first search Goal. ind all ertices connected to s (and a path). Algorithm. Use recursion (ball of string). Mark each isited ertex (and keep track of edge taken to isit it). Return (retrace steps) when no unisited options. Data structures. boolean[] marked to mark isited ertices. int[] edgeo to keep tree of paths. (edgeo[w] == ) means that edge -w taken to isit w for first time

51 Depth-first search oid DS(int i) { node *p = adj[i]; isited[i]=; while(p!= NULL) { i = p->ertex; if(!isited[i]) DS(i); p = p->next; } } typedef struct node { struct node *next; int ertex; }node; //GLOBAL PARAMEERS node * adj []; int isited[];

52 Graph API Depth-first search Breadth-first search UNDIRECED GRAPHS

53 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. graph G

54 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] add to queue

55 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

56 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

57 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

58 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

59 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] done

60 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

61 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

62 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

63 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

64 Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. Breadth-first search dequeue edgeo[] queue

65 Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. Breadth-first search done edgeo[] queue

66 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

67 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

68 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

69 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] done

70 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

71 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

72 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

73 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] done

74 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

75 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

76 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

77 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

78 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] done

79 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

80 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

81 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] dequeue

82 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. queue edgeo[] done

83 Breadth-first search Repeat until queue is empty: Remoe ertex from queue. Add to queue all unmarked ertices adjacent to and mark them. edgeo[] done

84 Breadth-first search Depth-first search. Put unisited ertices on a stack. Breadth-first search. Put unisited ertices on a queue. Shortest path. ind path from s to t that uses fewest number of edges. BS (from source ertex s) Put s onto a IO queue, and mark s as isited. Repeat until the queue is empty: - remoe the least recently added ertex - add each of 's unisited neighbors to the queue, and mark them as isited. Intuition. BS examines ertices in increasing distance from s.

85 Breadth-first search properties Proposition. BS computes shortest path (number of edges) from s in a connected graph in time proportional to E + V. Pf. [correctness] Queue always consists of zero or more ertices of distance k from s, followed by zero or more ertices of distance k +. Pf. [running time] Each ertex connected to s is isited once. standard drawing dist = dist = dist =