DataStruct 13. Graph Algorithms

Transcription

1 DataStruct 13. Graph Algorithms Michael T. Goodrich, et. al, Data Structures and Algorithms in C++, 2 nd Ed., John Wiley & Sons, Inc., December 10, 2013 Advanced Networking Technology Lab. (YU-ANTL) Dept. of Information & Comm. Eng, College of Engineering, Yeungnam University, KOREA (Tel : ; Fax : ytkim@yu.ac.kr)

2 Graphs Data Structures for Graphs Graph Traversals Directed Graphs Shortest Paths Minimum Spanning Trees Outline DS 13-2

3 13.1 Graphs Graphs Definition Applications Terminology Properties ADT SFO ORD LAX DFW DS 13-3

4 Graph A graph is a pair (V, E), where V is a set of nodes, called vertices E is a collection of pairs of vertices, called edges Vertices and edges are positions and store elements Example: A vertex represents an airport and stores the three-letter airport code An edge represents a flight route between two airports and stores the mileage of the route SFO ORD LGA PVD HNL LAX DFW MIA DS 13-4

5 Edge Types Directed edge ordered pair of vertices (u,v) first vertex u is the origin second vertex v is the destination e.g., a flight Undirected edge unordered pair of vertices (u,v) e.g., a flight route Directed graph all the edges are directed e.g., route network Undirected graph all the edges are undirected e.g., flight network ORD ORD 849 miles flight AA 1206 PVD PVD DS 13-5

6 Applications Graphs Electronic circuits Printed circuit board Integrated circuit Transportation networks Highway network Flight network Computer networks Local area network Internet Web Databases Entity-relationship diagram cslab1a cslab1b math.brown.edu cs.brown.edu brown.edu qwest.net att.net cox.net John Paul David DS 13-6

7 Terminology Graphs End vertices (or endpoints) of an edge U and V are the endpoints of a Edges incident on a vertex a, d, and b are incident on V Adjacent vertices U and V are adjacent Degree of a vertex X has degree 5 Parallel edges h and i are parallel edges Self-loop j is a self-loop U a c V W d f b e X Y g h i Z j DS 13-7

8 Terminology (cont.) Graphs 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 Examples 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 U a c d P 2 V W f e b P 1 X g Y h Z DS 13-8

9 Terminology (cont.) Graphs Cycle circular sequence of alternating vertices and edges a V b each edge is preceded and followed by its endpoints Simple cycle cycle such that all its vertices and edges are distinct U c d C 2 W e X g C 1 h Z Examples f C 1 =(V,b,X,g,Y,f,W,c,U,a, ) is a simple cycle Y C 2 =(U,c,W,e,X,g,Y,f,W,d,V,a, ) is a cycle that is not simple DS 13-9

10 Properties Graphs Property 1 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) Notation - Example n m deg(v) n = 4 m = 6 deg(v) = 3 number of vertices number of edges degree of vertex v What is the bound for a directed graph? DS 13-10

11 Main Methods of the Graph ADT Graphs Vertices and edges are positions store elements Accessor methods avertex() incidentedges(v) endvertices(e) isdirected(e) origin(e) destination(e) opposite(v, e) areadjacent(v, w) Update methods insertvertex(o) insertedge(v, w, o) insertdirectededge(v, w, o) removevertex(v) removeedge(e) Generic methods numvertices() numedges() vertices() edges() DS 13-11

12 3.2 Data Structures for Graphs Data structures for graphs Edge list structure Adjacency list structure Adjacency matrix structure DS 13-12

13 Vertex object element reference to position in vertex sequence Edge object element origin vertex object destination vertex object reference to position in edge sequence Vertex sequence sequence of vertex objects Edge sequence sequence of edge objects Edge List Structure v a u b c u v w z a w d b c d z Graphs DS 13-13

14 Adjacency List Structure Graphs Edge list structure Incidence sequence for each vertex sequence of references to edge objects of incident edges Augmented edge objects references to associated positions in incidence sequences of end vertices a v b u w u v w a b DS 13-14

15 Adjacency Matrix Structure Edge list structure Augmented vertex objects Integer key (index) associated with vertex 2D-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 u a 0 u 1 v 2 w a v b w Graphs b DS 13-15

16 Asymptotic Performance n vertices, m edges no parallel edges no self-loops Bounds are big-oh Edge List Adjacency List Adjacency Matrix Space n + m n + m n 2 incidentedges(v) m deg(v) n areadjacent (v, w) m min(deg(v), deg(w)) 1 insertvertex(o) 1 1 n 2 insertedge(v, w, o) removevertex(v) m deg(v) n 2 removeedge(e) DS 13-16

17 13.3 Graph Traversals Definitions Subgraph Connectivity Spanning trees and forests Depth-first search Algorithm Example A Properties Analysis B D E Applications of DFS Path finding C Cycle finding DS 13-17

18 Subgraphs A 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 A spanning subgraph of G is a subgraph that contains all the vertices of G Subgraph Depth-First Search Spanning subgraph DS 13-18

19 Connectivity A graph is connected if there is a path between every pair of vertices A connected component of a graph G is a maximal connected subgraph of G Connected graph Depth-First Search Non connected graph with two connected components DS 13-19

20 Trees and Forests A (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 A forest is an undirected graph without cycles The connected components of a forest are trees Tree Depth-First Search Forest DS 13-20

21 Spanning Trees and Forests A spanning tree of a connected graph is a spanning subgraph that is a tree A spanning tree is not unique unless the graph is a tree Spanning trees have applications to the design of communication networks A spanning forest of a graph is a spanning subgraph that is a forest Graph Spanning tree Depth-First Search DS 13-21

22 Depth-First Search (DFS) Depth-first search (DFS) is a general technique for traversing a graph A DFS traversal of a graph G Visits all the vertices and edges of G Determines whether G is connected Computes the connected components of G Computes a spanning forest of G DFS on a graph with n vertices and m edges takes O(n + m ) time DFS can be further extended to solve other graph problems Find and report a path between two given vertices Find a cycle in the graph Depth-first search is to graphs what Euler tour is to binary trees DS 13-22

23 DFS Algorithm The algorithm uses a mechanism for setting and getting labels of vertices and edges Algorithm DFS(G) Input graph G Output labeling of the edges of G as discovery edges and back edges for all u G.vertices() setlabel(u, UNEXPLORED) for all e G.edges() setlabel(e, UNEXPLORED) for all v G.vertices() if getlabel(v) = UNEXPLORED DFS(G, v) Algorithm DFS(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, VISITED) for all e G.incidentEdges(v) if getlabel(e) = UNEXPLORED w opposite(v,e) if getlabel(w) = UNEXPLORED setlabel(e, DISCOVERY) DFS(G, w) else setlabel(e, BACK) DS 13-23

24 Example A unexplored vertex A A visited vertex unexplored edge B D E discovery edge back edge C A A B D E B D E C C DS 13-24

25 Example (cont.) A A B D E B D E C C A A B D E B D E C C DS 13-25

26 DFS and Maze Traversal The DFS 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) DS 13-26

27 Property 1 DFS(G, v) visits all the vertices and edges in the connected component of v Property 2 The discovery edges labeled by DFS(G, v) form a spanning tree of the connected component of v Properties of DFS B A C D E DS 13-27

28 Analysis of DFS Setting/getting a vertex/edge label takes O(1) time Each vertex is labeled twice once as UNEXPLORED once as VISITED Each edge is labeled twice once as UNEXPLORED once as DISCOVERY or BACK Method incidentedges is called once for each vertex DFS runs in O(n + m) time provided the graph is represented by the adjacency list structure Recall that S v deg(v) = 2m DS 13-28

29 Path Finding We can specialize the DFS algorithm to find a path between two given vertices v and z using the template method pattern We call DFS(G, v,z) with v as the start vertex We use a stack S to keep track of the path between the start vertex and the current vertex As soon as destination vertex z is encountered, we return the path as the contents of the stack Algorithm pathdfs(g, v, z) setlabel(v, VISITED) S.push(v) if v = z return S.elements() for all e G.incidentEdges(v) if getlabel(e) = UNEXPLORED w opposite(v,e) if getlabel(w) = UNEXPLORED setlabel(e, DISCOVERY) S.push(e) pathdfs(g, w, z) S.pop(e) else setlabel(e, BACK) S.pop(v) DS 13-29

30 Cycle Finding We can specialize the DFS 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 As 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 Algorithm cycledfs(g, v) setlabel(v, VISITED) S.push(v) for all e G.incidentEdges(v) if getlabel(e) = UNEXPLORED w opposite(v,e) S.push(e) if getlabel(w) = UNEXPLORED setlabel(e, DISCOVERY) if ((U=cycleDFS(G, w)=empty) S.pop(e) else return U else T new empty stack repeat o S.pop() T.push(o) until o = w return T.elements() S.pop(v) DS 13-30

31 Generic DFC Implementation in C++ Graph.h (1) /** Graph.h (1) */ #ifndef GRAPH_H #define GRAPH_H #include <list> #include <vector> #include <iostream> #include <iomanip> #include <limits> #include <string> #define PLUS_INF INT_MAX/2 enum VertexStatus UN_VISITED, VISITED, VRTX_NOT_FOUND; enum EdgeStatus DISCOVERY, BACK, CROSS, EDGE_UN_VISITED, EDGE_VISITED, EDGE_NOT_FOUND; using namespace std; class Graph // Graph based on Adjacency Matrix public: class Vertex; class Edge; DS 13-31

32 Graph.h (2) /** Graph.h (2) */ public: class Vertex friend ostream& operator<<(ostream& fout, Vertex v) fout << v.getname(); return fout; public: Vertex() : name(), ID(-1) Vertex(string n, int id, VertexStatus vs) : name(n), ID(id), vtxstatus(vs) Vertex(char *pn, int id, VertexStatus vs) : name(string(pn)), ID(id), vtxstatus(vs) Vertex(int id) : ID(id) string getname() return name; int getid() return ID; void setname(string c_name) name = c_name; void setid(int id) ID = id; bool operator==(vertex v) return ((ID == v.getid()) && (name == v.getname())); bool operator!=(vertex v) return ((ID!= v.getid()) (name!= v.getname())); void setvtxstatus(vertexstatus vs) vtxstatus = vs; VertexStatus getvtxstatus() return vtxstatus; DS 13-32

33 Graph.h (3) /** Graph.h (3) */ private: string name; int ID; VertexStatus vtxstatus; ; // end class Vertex public: typedef std::list<graph::vertex> VtxList; public: class Edge friend ostream& operator<<(ostream& fout, Edge& e) fout << "Edge(" << e.getvertex_1() << ", " << e.getvertex_2(); fout << ", d(" << setw(2) << e.getdistance() << "))"; return fout; public: Edge() : pvrtx_1(null), pvrtx_2(null), distance(plus_inf) Edge(Vertex v1, Vertex v2, int d) :vrtx_1(v1), vrtx_2(v2), distance(d), pvrtx_1(null), pvrtx_2(null), edgestatus(edge_un_visited DS 13-33

34 Graph.h (4) /** Graph.h (4) */ void endvertices(vtxlist& vtxlist) vtxlist.push_back(*pvrtx_1); vtxlist.push_back(*pvrtx_2); Vertex opposite(vertex v) if (v == *pvrtx_1) return *pvrtx_2; else if (v == *pvrtx_2) return *pvrtx_1; else //cout << "Error in opposite()" << endl; return Vertex(NULL); Vertex getvertex_1() return vrtx_1; Vertex getvertex_2() return vrtx_2; Vertex* getpvrtx_1() return pvrtx_1; Vertex* getpvrtx_2() return pvrtx_2; int getdistance() return distance; void setpvrtx_1(vertex* pv) pvrtx_1 = pv; void setpvrtx_2(vertex* pv) pvrtx_2 = pv; void setdistance(int d) distance = d; bool operator!=(edge e) return ((*pvrtx_1!= e.getvertex_1()) (*pvrtx_2!= e.getvertex_2())); bool operator==(edge e) return ((*pvrtx_1 == e.getvertex_1()) && (*pvrtx_2 == e.getvertex_2())); void setedgestatus(edgestatus es) edgestatus = es; EdgeStatus getedgestatus() return edgestatus; DS 13-34

35 Graph.h (5) /** Graph.h (5) */ private: Vertex vrtx_1; Vertex vrtx_2; Vertex* pvrtx_1; Vertex* pvrtx_2; int distance; EdgeStatus edgestatus; ; public: typedef std::list<graph::edge> EdgeList; public: typedef std::list<vertex> VtxList; typedef std::list<edge> EdgeList; typedef std::list<vertex>::iterator VtxItor; typedef std::list<edge>::iterator EdgeItor; Graph() : pvrtxarray(null), padjlstarray(null) // default constructor Graph(int num_nodes) : pvrtxarray(null), padjlstarray(null) typedef Edge* EdgePtr; num_vertices = num_nodes; pvrtxarray = new Graph::Vertex[num_vertices]; for (int i=0; i<num_nodes; i++) pvrtxarray[i] = NULL; DS 13-35

36 Graph.h (6) /** Graph.h (6) */ padjlstarray = new EdgeList[num_vertices]; for (int i=0; i<num_vertices; i++) padjlstarray[i].clear(); void vertices(vtxlist& vtxlst); void edges(edgelist&); bool isadjacentto(vertex v, Vertex w); void insertvertex(vertex& v); void insertedge(edge& e); void eraseedge(edge e); void erasevertex(vertex v); int getnumvertices() return num_vertices; void incidentedges(vertex v, EdgeList& edges); Vertex* getpvrtxarray() return pvrtxarray; EdgeList* getpadjlstarray() return padjlstarray; void printgraph(); private: Vertex* pvrtxarray; EdgeList* padjlstarray; int num_vertices; ; #endif DS 13-36

37 Graph.cpp (1) /** Graph.cpp (1) */ #include "Graph.h" using namespace std; typedef std::vector<graph::vertex> VtxList; typedef std::list<graph::edge> EdgeList; typedef std::vector<graph::vertex>::iterator VtxItor; typedef std::list<graph::edge>::iterator EdgeItor; void Graph::insertVertex(Vertex& v) int vtx_id; vtx_id = v.getid(); if (pvrtxarray[vtx_id] == NULL) pvrtxarray[vtx_id] = v; void Graph::vertices(VtxList& vtxlst) vtxlst.clear(); for (int i=0; i<getnumvertices(); i++) vtxlst.push_back(pvrtxarray[i]); DS 13-37

38 Graph.cpp (2) /** Graph.cpp (2) */ void Graph::insertEdge(Edge& e) Vertex vtx_1, vtx_2; Vertex* pvtx; int vtx_1_id, vtx_2_id; vtx_1 = e.getvertex_1(); vtx_2 = e.getvertex_2(); vtx_1_id = vtx_1.getid(); vtx_2_id = vtx_2.getid(); if (pvrtxarray[vtx_1_id] == NULL) pvrtxarray[vtx_1_id] = vtx_1; if (pvrtxarray[vtx_2_id] == NULL) pvrtxarray[vtx_2_id] = vtx_2; e.setpvrtx_1(&pvrtxarray[vtx_1_id]); e.setpvrtx_2(&pvrtxarray[vtx_2_id]); //edgelist.push_back(e); padjlstarray[vtx_1_id].push_back(e); DS 13-38

39 Graph.cpp (3) /** Graph.cpp (3) */ void Graph::edges(EdgeList& edges) EdgeItor eitor; edges.clear(); for (int i=0; i<getnumvertices(); i++) eitor = padjlstarray[i].begin(); while (eitor!= padjlstarray[i].end()) edges.push_back(*eitor); eitor++; DS 13-39

40 Graph.cpp (4) /** Graph.cpp (4) */ void Graph::incidentEdges(Vertex v, EdgeList& edgelst) Graph::Edge e; EdgeItor eitor; int vtx_id = v.getid(); eitor = padjlstarray[vtx_id].begin(); while (eitor!= padjlstarray[vtx_id].end()) edgelst.push_back(*eitor); eitor++; void Graph::printGraph() int i, j; EdgeItor eitor; int numoutgoingedges; for (i=0; i<num_vertices; i++) cout << setw(2) << (char)(i+'a') << " "; numoutgoingedges = padjlstarray[i].size(); eitor = padjlstarray[i].begin(); DS 13-40

41 Graph.cpp (3) /** Graph.cpp (3) */ void Graph::incidentEdges(Vertex v, EdgeList& edgelst) Graph::Edge e; EdgeItor eitor; int vtx_id = v.getid(); eitor = padjlstarray[vtx_id].begin(); while (eitor!= padjlstarray[vtx_id].end()) edgelst.push_back(*eitor); eitor++; DS 13-41

42 Graph.cpp (4) /** Graph.cpp (4) */ void Graph::printGraph() int i, j; EdgeItor eitor; int numoutgoingedges; for (i=0; i<num_vertices; i++) cout << setw(2) << (char)(i+'a') << " "; numoutgoingedges = padjlstarray[i].size(); eitor = padjlstarray[i].begin(); while (eitor!= padjlstarray[i].end()) cout << *eitor << " "; eitor++; cout << endl; DS 13-42

43 DepthFirstSearch.h (1) /** DepthFirstSearch.h (1)*/ #ifndef DEPTHFIRSTSEARCH_H #define DEPTHFIRSTSEARCH_H #include <iostream> #include "Graph.h" using namespace std; class DepthFirstSearch protected: typedef Graph::Vertex Vertex; typedef Graph::Edge Edge; typedef std::list<graph::vertex> VertexList; typedef std::list<graph::edge> EdgeList; protected: Graph& graph; Vertex start; bool done; // flag of search done protected: void initialize(); void dfstraversal(vertex& v, Vertex& target, VertexList& path); virtual void traversediscovery(const Edge& e, const Vertex& from) virtual void traverseback(const Edge& e, const Vertex& from) virtual void finishvisit(const Vertex& v) virtual bool isdone() const return done; DS 13-43

44 DepthFirstSearch.h (2) /** DepthFirstSearch.h (2)*/ // marking utilities void visit(vertex& v); void visit(edge& e); void unvisit(vertex& v); void unvisit(edge& e); bool isvisited(vertex& v); bool isvisited(edge& e); void setedgestatus(edge& e, EdgeStatus es); EdgeStatus getedgestatus(edge& e); public: DepthFirstSearch(Graph& g); void findpath(vertex& s, Vertex& t, VertexList& path); Graph& getgraph() return graph; void showconnectivity(); private: VertexStatus* pvrtxstatus; EdgeStatus** ppedgestatus; int** ppconnectivity; ; #endif DS 13-44

45 DepthFirstSearch.cpp (1) /** DepthFirstSearch.cpp (1)*/ #include <iostream> #include <list> #include <algorithm> #include "Graph.h" #include "DepthFirstSearch.h" using namespace std; typedef Graph::Vertex Vertex; typedef Graph::Edge Edge; typedef std::list<graph::vertex> VertexList; typedef std::list<graph::vertex>::iterator VertexItor; typedef std::list<graph::edge> EdgeList; typedef std::list<graph::edge>::iterator EdgeItor; DepthFirstSearch::DepthFirstSearch(Graph& g) :graph(g) int num_nodes = graph.getnumvertices(); pvrtxstatus = new VertexStatus[num_nodes]; for (int i=0; i<num_nodes; i++) pvrtxstatus[i] = UN_VISITED; ppedgestatus = new EdgeStatus*[num_nodes]; for (int i=0; i<num_nodes; i++) ppedgestatus[i] = new EdgeStatus[num_nodes]; for (int i=0; i<num_nodes; i++) for (int j=0; j<num_nodes; j++) ppedgestatus[i][j] = EDGE_UN_VISITED; DS 13-45

46 DepthFirstSearch.cpp (2) /** DepthFirstSearch.cpp (2)*/ ppconnectivity = new int*[num_nodes]; for (int i=0; i<num_nodes; i++) ppconnectivity[i] = new int[num_nodes]; for (int i=0; i<num_nodes; i++) for (int j=0; j<num_nodes; j++) ppconnectivity[i][j] = PLUS_INF; // initially not connected Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; EdgeList edges; edges.clear(); graph.edges(edges); for (EdgeItor pe = edges.begin(); pe!= edges.end(); ++pe) vtx_1 = (*pe).getvertex_1(); vtx_1_id = vtx_1.getid(); vtx_2 = (*pe).getvertex_2(); vtx_2_id = vtx_2.getid(); ppconnectivity[vtx_1_id][vtx_2_id] = (*pe).getdistance(); for (int i=0; i<num_nodes; i++) ppconnectivity[i][i] = 0; // distance of same node DS 13-46

47 DepthFirstSearch.cpp (3) /** DepthFirstSearch.cpp (3)*/ void DepthFirstSearch::initialize() int num_nodes = graph.getnumvertices(); VertexList verts; graph.vertices(verts); Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; done = false; for (int i=0; i<num_nodes; i++) pvrtxstatus[i] = UN_VISITED; for (int i=0; i<num_nodes; i++) for (int j=0; j<num_nodes; j++) ppedgestatus[i][j] = EDGE_UN_VISITED; DS 13-47

48 DepthFirstSearch.cpp (4) /** DepthFirstSearch.cpp (4)*/ void DepthFirstSearch::showConnectivity() int num_nodes = graph.getnumvertices(); int dist; cout << "Connectivity of graph: " << endl; cout << " "; for (int i=0; i<num_nodes; i++) cout << setw(5) << (char)(i + 'A'); cout << endl; cout << "-----+"; for (int i=0; i<num_nodes; i++) cout << "-----"; cout << endl; for (int i=0; i<num_nodes; i++) cout << " " << (char)(i + 'A') << " "; for (int j=0; j<num_nodes; j++) dist = ppconnectivity[i][j]; if (dist == PLUS_INF) cout << " +oo"; else cout << setw(5) << dist; cout << endl; DS 13-48

49 DepthFirstSearch.cpp (5) /** DepthFirstSearch.cpp (5)*/ void DepthFirstSearch::dfsTraversal(Vertex& v, Vertex& target, VertexList& path) visit(v); if (v == target) done = true; return; EdgeList incidentedges; incidentedges.clear(); graph.incidentedges(v, incidentedges); EdgeItor pe = incidentedges.begin(); while (!isdone() && pe!= incidentedges.end()) Edge e = *pe++; EdgeStatus estat = getedgestatus(e); if (estat == EDGE_UN_VISITED) visit(e); Vertex w = e.opposite(v); if (!isvisited(w)) //traversediscovery(e, v); path.push_back(w); setedgestatus(e, DISCOVERY); if (!isdone()) dfstraversal(w, target, path); if (!isdone()) //traverseback(e, v); // erase some possible cycle Vertex last_pushed = path.back(); // for debugging path.pop_back(); DS 13-49

50 DepthFirstSearch.cpp (6) /** DepthFirstSearch.cpp (6)*/ else setedgestatus(e, BACK); // end of processing of all incident edges //template<typename G> void DepthFirstSearch::findPath(Vertex &start, Vertex &target, VertexList& path) initialize(); path.clear(); path.push_back(start); dfstraversal(start, target, path); void DepthFirstSearch::visit(Vertex& v) Graph::Vertex* pvtx; int numnodes = getgraph().getnumvertices(); int vtx_id = v.getid(); if (vtx_id >= 0 && vtx_id < numnodes) pvrtxstatus[vtx_id] = VISITED; else cout << "Vertex (" << v << ") ID is out-of-range (" ; cout << numnodes << ")" << endl; Yeungnam University (YU-ANTL) DS 13-50

51 DepthFirstSearch.cpp (7) /** DepthFirstSearch.cpp (7)*/ void DepthFirstSearch::visit(Edge& e) Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; int numnodes = getgraph().getnumvertices(); vtx_1 = e.getvertex_1(); vtx_1_id = vtx_1.getid(); vtx_2 = e.getvertex_2(); vtx_2_id = vtx_2.getid(); if ((vtx_1_id >= 0 && vtx_1_id < numnodes) && (vtx_2_id >= 0 && vtx_2_id < numnodes)) ppedgestatus[vtx_1_id][vtx_2_id] = EDGE_VISITED; else cout << "Vertex IDs (" << vtx_1_id << ", " << vtx_2_id; cout << ") of Edge (" << e << ") is out-of-range (" << numnodes << ")" << endl; void DepthFirstSearch::unvisit(Vertex& v) Graph::Vertex* pvtx; int numnodes = getgraph().getnumvertices(); int vtx_id = v.getid(); if (vtx_id >= 0 && vtx_id < numnodes) pvrtxstatus[vtx_id] = UN_VISITED; else cout << "Vertex (" << v << ") ID is out-of-range (" << numnodes << ")" << endl; DS 13-51

52 DepthFirstSearch.cpp (8) /** DepthFirstSearch.cpp (8)*/ void DepthFirstSearch::unvisit(Edge& e) Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; int numnodes = getgraph().getnumvertices(); vtx_1 = e.getvertex_1(); vtx_1_id = vtx_1.getid(); vtx_2 = e.getvertex_2(); vtx_2_id = vtx_2.getid(); if ((vtx_1_id >= 0 && vtx_1_id < numnodes) && (vtx_2_id >= 0 && vtx_2_id < numnodes)) ppedgestatus[vtx_1_id][vtx_2_id] = EDGE_UN_VISITED; else cout << "Vertex IDs (" << vtx_1_id << ", " << vtx_2_id << ") of Edge ( ; cout << e << ") is out-of-range (" << numnodes << ")" << endl; DS 13-52

53 DepthFirstSearch.cpp (9) /** DepthFirstSearch.cpp (9)*/ bool DepthFirstSearch::isVisited(Vertex& v) Graph::Vertex* pvtx; int numnodes = getgraph().getnumvertices(); int vtx_id = v.getid(); if (vtx_id >= 0 && vtx_id < numnodes) return (pvrtxstatus[vtx_id] == VISITED); else cout << "Vertex (" << v << ") ID is out-of-range (" << numnodes << ")" << endl; return false; DS 13-53

54 DepthFirstSearch.cpp (10) /** DepthFirstSearch.cpp (10)*/ bool DepthFirstSearch::isVisited(Edge& e) Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; EdgeStatus estat; int numnodes = getgraph().getnumvertices(); vtx_1 = e.getvertex_1(); vtx_1_id = vtx_1.getid(); vtx_2 = e.getvertex_2(); vtx_2_id = vtx_2.getid(); if ((vtx_1_id >= 0 && vtx_1_id < numnodes) && (vtx_2_id >= 0 && vtx_2_id < numnodes)) estat = ppedgestatus[vtx_1_id][vtx_2_id]; if ((estat == EDGE_VISITED) (estat == DISCOVERY) (estat == BACK)) return true; else return false; else cout << "Vertex IDs (" << vtx_1_id << ", " << vtx_2_id << ") of Edge ( ; cout << e << ") is out-of-range (" << numnodes << ")" << endl; return false; DS 13-54

55 DepthFirstSearch.cpp (11) /** DepthFirstSearch.cpp (11)*/ void DepthFirstSearch::setEdgeStatus(Edge& e, EdgeStatus es) Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; int numnodes = getgraph().getnumvertices(); vtx_1 = e.getvertex_1(); vtx_1_id = vtx_1.getid(); vtx_2 = e.getvertex_2(); vtx_2_id = vtx_2.getid(); if ((vtx_1_id >= 0 && vtx_1_id < numnodes) && (vtx_2_id >= 0 && vtx_2_id < numnodes)) ppedgestatus[vtx_1_id][vtx_2_id] = es; else cout << "Vertex IDs (" << vtx_1_id << ", " << vtx_2_id << ") of Edge ( ; cout << e << ") is out-of-range (" << numnodes << ")" << endl; DS 13-55

56 DepthFirstSearch.cpp (12) /** DepthFirstSearch.cpp (12)*/ EdgeStatus DepthFirstSearch::getEdgeStatus(Edge& e) Vertex vtx_1, vtx_2; int vtx_1_id, vtx_2_id; int numnodes = getgraph().getnumvertices(); EdgeStatus estat; vtx_1 = e.getvertex_1(); vtx_1_id = vtx_1.getid(); vtx_2 = e.getvertex_2(); vtx_2_id = vtx_2.getid(); if ((vtx_1_id >= 0 && vtx_1_id < numnodes) && (vtx_2_id >= 0 && vtx_2_id < numnodes)) estat = ppedgestatus[vtx_1_id][vtx_2_id]; return estat; else cout << "Edge (" << e << ") was not found from AdjacencyList" << endl; return EDGE_NOT_FOUND; DS 13-56

57 Fig Example of depth-first search traversal on a graph starting at vertex A. Discovery edges are shown with solid lines and back edges are shown with dashed lines: DS 13-57

58 main.cpp (1) /** main.cpp (1) */ #include <iostream> #include <string> //#include "Vertex.h" //#include "Edge.h" #include "Graph.h" #include "DepthFirstSearch.h" #include "BreadthFirstSearch.h" #define NUM_NODES 16 #define NUM_EDGES 50 typedef Graph::Vertex Vertex; typedef Graph::Edge Edge; typedef std::list<graph::vertex> VtxList; typedef std::list<graph::edge> EdgeList; typedef std::list<graph::vertex>::iterator VtxItor; typedef std::list<graph::edge>::iterator EdgeItor; void main() Vertex v[num_nodes] = // Topology of Figure 13.6 (p. 608) Vertex("A", 0, UN_VISITED), Vertex("B", 1, UN_VISITED), Vertex("C", 2, UN_VISITED), Vertex("D", 3, UN_VISITED), Vertex("E", 4, UN_VISITED), Vertex("F", 5, UN_VISITED), Vertex("G", 6, UN_VISITED), Vertex("H", 7, UN_VISITED), Vertex("I", 8, UN_VISITED), Vertex("J", 9, UN_VISITED), Vertex("K", 10, UN_VISITED), Vertex("L", 11, UN_VISITED), Vertex("M", 12, UN_VISITED), Vertex("N", 13, UN_VISITED), Vertex("0", 14, UN_VISITED), Vertex("P", 15, UN_VISITED) ; DS 13-58

59 main.cpp (2) /** main.cpp (2) */ Graph::Edge edges[num_edges] = Edge(v[0], v[1], 10), Edge(v[1], v[0], 10), Edge(v[0], v[4], 10), Edge(v[4], v[0], 10), Edge(v[0], v[5], 15), Edge(v[5], v[0], 15), Edge(v[1], v[2], 10), Edge(v[2], v[1], 10), Edge(v[1], v[5], 10), Edge(v[5], v[1], 10), Edge(v[2], v[3], 10), Edge(v[3], v[2], 10), Edge(v[2], v[6], 10), Edge(v[6], v[2], 10), Edge(v[3], v[6], 15), Edge(v[6], v[3], 15), Edge(v[3], v[7], 10), Edge(v[7], v[3], 10), Edge(v[4], v[5], 10), Edge(v[5], v[4], 10), Edge(v[4], v[8], 10), Edge(v[8], v[4], 10), Edge(v[5], v[8], 15), Edge(v[8], v[5], 15), Edge(v[6], v[9], 15), Edge(v[9], v[6], 15), Edge(v[6], v[10], 10), Edge(v[10], v[6], 10), Edge(v[6], v[11], 15), Edge(v[11], v[6], 15), Edge(v[7], v[11], 10), Edge(v[11], v[7], 10), Edge(v[8], v[9], 10), Edge(v[9], v[8], 10), Edge(v[8], v[12], 10), Edge(v[12], v[8], 10), Edge(v[8], v[13], 15), Edge(v[13], v[8], 15), Edge(v[9], v[10], 10), Edge(v[10], v[9], 10), Edge(v[10], v[13], 15), Edge(v[13], v[10], 15), Edge(v[10], v[14], 10), Edge(v[14], v[10], 10), Edge(v[11], v[15], 10), Edge(v[15], v[11], 10), Edge(v[12], v[13], 10), Edge(v[13], v[12], 10), Edge(v[14], v[15], 10), Edge(v[15], v[14], 10) ; Graph simplegraph(num_nodes); DS 13-59

60 main.cpp (3) /** main.cpp (3) */ Graph simplegraph(num_nodes); cout << "Inserting vertices.." << endl; for (int i=0; i<num_nodes; i++) simplegraph.insertvertex(v[i]); VtxList vtxlst; simplegraph.vertices(vtxlst); int count = 0; cout << "Inserted vertices: "; for (VtxItor vitor = vtxlst.begin(); vitor!= vtxlst.end(); ++vitor) cout << *vitor << " "; cout << endl; cout << "Inserting edges.." << endl; for (int i=0; i<num_edges; i++) simplegraph.insertedge(edges[i]); cout << "Inserted edges: " << endl; count = 0; EdgeList eglst; simplegraph.edges(eglst); DS 13-60

61 main.cpp (4) /** main.cpp (4) */ for (EdgeItor p = eglst.begin(); p!= eglst.end(); ++p) count++; cout << *p << ", "; if (count % 5 == 0) cout << endl; cout << endl; cout << "Print out Graph based on Adjacency List.." << endl; simplegraph.printgraph(); cout << "Testing dfsgraph..." << endl; DepthFirstSearch dfsgraph(simplegraph); VtxList path; dfsgraph.findpath(v[0], v[15], path); cout << "Path (" << v[0] << " => " << v[15] << ) : "; for (VtxItor vitor = path.begin(); vitor!= path.end(); ++vitor) cout << *vitor << " "; cout << endl; dfsgraph.findpath(v[15], v[0], path); cout << "Path (" << v[15] << " => " << v[0] << ) : "; for (VtxItor vitor = path.begin(); vitor!= path.end(); ++vitor) cout << *vitor << " "; cout << endl; DS 13-61

62 Example output (1) DS 13-62

63 Example output (2) DS 13-63