Data Structures Lecture 14

Transcription

1 Fall 2018 Fang Yu Software Security Lab. ept. Management Information Systems, National hengchi University ata Structures Lecture 14

2 Graphs efinition, Implementation and Traversal

3 Graphs Formally speaking, a graph is a pair (V, ), where V is a set of nodes, called vertices is a collection of pairs of vertices, called edges Vertices and edges are positions and store elements

4 Graphs xample: vertex represents an airport and stores the three-letter airport code n edge represents a flight route between two airports and stores the mileage of the route SFO 1843 OR 849 PV LG HNL 2555 LX 1233 FW 1120 MI

5 dge Types irected edge ordered pair of vertices (u,v) first vertex u is the origin second vertex v is the destination e.g., a flight OR flight 1206 PV Undirected edge unordered pair of vertices (u,v) e.g., a flight route OR 849 miles PV irected graph all the edges are directed e.g., route network Undirected graph all the edges are undirected e.g., flight network

6 pplications lectronic circuits Printed circuit board Integrated circuit cslab1a cs.brown.edu cslab1b math.brown.edu Transportation networks Highway network Flight network omputer networks Local area network Internet Web atabases ntity-relationship diagram att.net Paul cox.net brown.edu avid John qwest.net

7 Terminology nd vertices (or endpoints) of an edge U and V are the endpoints of a dges incident on a vertex a, d, and b are incident on V djacent vertices U and V are adjacent egree of a vertex X has degree 5 U a c V d W b e X g h i Z j Parallel edges h and i are parallel edges f Y Self-loop j is a self-loop

8 Terminology (cont.) Path sequence of alternating vertices and edges begins with a vertex ends with a vertex each edge is preceded and followed by its endpoints Simple path path such that all its vertices and edges are distinct U a c d V P 2 W b e P 1 X g h Z xamples P 1 =(V,b,X,h,Z) is a simple path P 2 =(U,c,W,e,X,g,Y,f,W,d,V) is a path that is not simple f Y

9 Terminology (cont.) ycle circular sequence of alternating vertices and edges each edge is preceded and followed by its endpoints Simple cycle cycle such that all its vertices and edges are distinct xamples 1 =(V,b,X,g,Y,f,W,c,U,a, ) is a simple cycle 2 =(U,c,W,e,X,g,Y,f,W,d,V,a, ) is a cycle that is not simple U a c d V 2 W f e b X Y 1 g h Z

10 Properties Notation n number of vertices m number of edges deg(v) degree of vertex v xample n n = 4 n m = 6 n deg(v) = 3

11 Properties Property 1 Σ v deg(v) = 2m Proof: each edge is counted twice Property 2 In an undirected graph with no self-loops and no multiple edges m n (n 1)/2 Proof: each vertex has degree at most (n 1) What is the bound for a directed graph?

12 Main Methods of the Graph T Vertices and edges are positions store elements ccessor methods endvertices(e): an array of the two endvertices of e opposite(v, e): the vertex opposite of v on e aredjacent(v, w): true iff v and w are adjacent replace(v, x): replace element at vertex v with x replace(e, x): replace element at edge e with x

13 Main Methods of the Graph T Update methods insertvertex(o): insert a vertex storing element o insertdge(v, w, o): insert an edge (v,w) storing element o removevertex(v): remove vertex v (and its incident edges) removedge(e): remove edge e Iterable collection methods incidentdges(v): edges incident to v vertices(): all vertices in the graph edges(): all edges in the graph

14 dge List Structure Vertex object element reference to position in vertex sequence dge object element origin vertex object destination vertex object reference to position in edge sequence u a c b d v w z u v w z Vertex sequence sequence of vertex objects dge sequence sequence of edge objects a b c d

15 djacency List Structure dge list structure Incidence sequence for each vertex sequence of references to edge objects of incident edges a v b u w u v w ugmented edge objects references to associated positions in incidence sequences of end vertices a b

16 djacency Matrix Structure dge list structure ugmented vertex objects Integer key (index) associated with vertex u a v b w 2-array adjacency array Reference to edge object for adjacent vertices Null for non nonadjacent vertices The old fashioned version just has 0 for no edge and 1 for edge 0 u 1 v 2 w a b

17 Performance n vertices, m edges no parallel edges no self-loops dge List djacency List djacency Matrix Space incidentdges(v) aredjacent (v, w) insertvertex(o) insertdge(v, w, o) removevertex(v) removedge(e)

18 Performance n vertices, m edges no parallel edges no self-loops dge List djacency List djacency Matrix Space n + m n + m n 2 incidentdges(v) m deg(v) n aredjacent (v, w) m min(deg(v), deg(w)) 1 insertvertex(o) 1 1 n 2 insertdge(v, w, o) removevertex(v) m deg(v) n 2 removedge(e) 1 1 1

19 Graph Traversal How to visit all vertices? epth-first Search

20 Subgraphs subgraph S of a graph G is a graph such that The vertices of S are a subset of the vertices of G The edges of S are a subset of the edges of G Subgraph spanning subgraph of G is a subgraph that contains all the vertices of G Spanning subgraph

21 onnectivity graph is connected if there is a path between every pair of vertices connected component of a graph G is a maximal connected subgraph of G onnected graph Non connected graph with two connected components

22 Trees and Forests (free) tree is an undirected graph T such that T is connected T has no cycles This definition of tree is different from the one of a rooted tree Tree forest is an undirected graph without cycles The connected components of a forest are trees Forest

23 Spanning Trees and Forests spanning tree of a connected graph is a spanning subgraph that is a tree spanning tree is not unique unless the graph is a tree Graph Spanning trees have applications to the design of communication networks spanning forest of a graph is a spanning subgraph that is a forest Spanning tree

24 epth-first Search epth-first search (FS) is a general technique for traversing a graph FS traversal of a graph G Visits all the vertices and edges of G etermines whether G is connected omputes the connected components of G omputes a spanning forest of G FS on a graph with n vertices and m edges takes O (n + m ) time FS can be further extended to solve other graph problems Find and report a path between two given vertices Find a cycle in the graph epth-first search is to graphs what uler tour is to binary trees

25 FS lgorithm The algorithm uses a mechanism for setting and getting labels of vertices and edges lgorithm FS(G) Input graph G Output labeling of the edges of G as discovery edges and back edges for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR FS(G, v) lgorithm FS(G, v) Input graph G and a start vertex v of G Output labeling of the edges of G in the connected component of v as discovery edges and back edges setlabel(v, VISIT) for all e G.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) FS(G, w) else setlabel(e, K)

26 xample visited vertex unexplored vertex unexplored edge discovery edge back edge

27 xample (cont.)

28 FS and Maze Traversal The FS algorithm is similar to a classic strategy for exploring a maze We mark each intersection, corner and dead end (vertex) visited We mark each corridor (edge ) traversed We keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack)

29 Properties of FS Property 1 FS(G, v) visits all the vertices and edges in the connected component of v Property 2 The discovery edges labeled by FS(G, v) form a spanning tree of the connected component of v

30 nalysis of FS Setting/getting a vertex/edge label takes O(1) time ach vertex is labeled twice once as UNXPLOR once as VISIT ach edge is labeled twice once as UNXPLOR once as ISOVRY or K Method incidentdges is called once for each vertex FS runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that Σ v deg(v) = 2m

31 Path Finding We can specialize the FS algorithm to find a path between two given vertices u and z using the template method pattern We call FS(G, u) with u as the start vertex We use a stack S to keep track of the path between the start vertex and the current vertex s soon as destination vertex z is encountered, we return the path as the contents of the stack lgorithm pathfs(g, v, z) setlabel(v, VISIT) S.push(v) if v = z return S.elements() for all e G.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) S.push(e) pathfs(g, w, z) S.pop(e) else setlabel(e, K) S.pop(v)

32 ycle Finding We can specialize the FS algorithm to find a simple cycle using the template method pattern We use a stack S to keep track of the path between the start vertex and the current vertex s soon as a back edge (v, w) is encountered, we return the cycle as the portion of the stack from the top to vertex w lgorithm cyclefs(g, v, z) setlabel(v, VISIT) S.push(v) for all e G.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) S.push(e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) pathfs(g, w, z) S.pop(e) else T new empty stack repeat o S.pop() T.push(o) until o = w return T.elements() S.pop(v)

33 readth-first Search Traverse the graph level by level L 0 L 1 L 2 F

34 readth-first Search readth-first search (FS) is a general technique for traversing a graph FS traversal of a graph G Visits all the vertices and edges of G etermines whether G is connected omputes the connected components of G omputes a spanning forest of G FS on a graph with n vertices and m edges takes O (n + m ) time FS can be further extended to solve other graph problems Find and report a path with the minimum number of edges between two given vertices Find a simple cycle, if there is one

35 FS lgorithm The algorithm uses a mechanism for setting and getting labels of vertices and edges lgorithm FS(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) = UNXPLOR FS(G, v) lgorithm FS(G, s) L 0 new empty sequence L 0.addLast(s) setlabel(s, VISIT) i 0 while L i.ismpty() L i +1 new empty sequence for all v L i.elements() for all e G.incidentdges(v) if getlabel(e) = UNXPLOR w opposite(v,e) if getlabel(w) = UNXPLOR setlabel(e, ISOVRY) setlabel(w, VISIT) L i +1.addLast(w) else setlabel(e, ROSS) i i +1

36 xample unexplored vertex L 0 visited vertex unexplored edge L 1 discovery edge cross edge F L 0 L 0 L 1 L 1 F F

37 xample (cont.) L 0 L 0 L 1 L 1 F L 2 F L 0 L 0 L 1 L 1 L 2 F L 2 F

38 xample (cont.) L 0 L 0 L 1 L 1 L 2 F L 2 F L 0 L 1 L 2 F

39 Properties Notation G s : connected component of s Property 1 FS(G, s) visits all the vertices and edges of G s Property 2 The discovery edges labeled by FS(G, s) form a spanning tree T s of G s Property 3 For each vertex v in L i The path of T s from s to v has i edges very path from s to v in G s has at least i edges L 1 L 0 L 2 F F

40 nalysis Setting/getting a vertex/edge label takes O(1) time ach vertex is labeled twice once as UNXPLOR once as VISIT ach edge is labeled twice once as UNXPLOR once as ISOVRY or ROSS ach vertex is inserted once into a sequence L i Method incidentdges is called once for each vertex FS runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that Σ v deg(v) = 2m

41 pplications Using the template method pattern, we can specialize the FS traversal of a graph G to solve the following problems in O(n + m) time ompute the connected components of G ompute a spanning forest of G Find a simple cycle in G, or report that G is a forest Given two vertices of G, find a path in G between them with the minimum number of edges, or report that no such path exists

42 FS vs. FS pplications FS FS Spanning forest, connected components, paths, cycles Shortest paths iconnected components L 0 L 1 F L 2 F FS FS

43 FS vs. FS (cont.) ack edge (v,w) w is an ancestor of v in the tree of discovery edges ross edge (v,w) w is in the same level as v or in the next level L 0 L 1 F L 2 F FS FS

44 Graphs II igraphs, Strongly onnective omponent, Topological Sorting, and Minimum Spanning Tree

45 Java Graph Library No standard library JGraphT n open source library Supports most mentioned Graph functions You can simply download the file and use the library to create your graph

46 Schedule on Jan. 17 I II III IV V 9:00~10:00 10:00~11:00 11:00~12:00 12:00~1:00

47 Schedule on Jan. 17 1:00~2:00 儫 2:00~3:00 I II III IV V 喆 珉彣

48 Lab on Jan 14 Ts will offer an extra lab for you to answer questions on the project MIS P classroom 5F

49 Makeup exam on Jan. 17 Thursday 4:00-5:30. ollege of ommerce 313 Maximal 80 points ynamic programing on LS inary Search Tree (VL) Hash Table ycle etection