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1 Graphs & Characteristics Graph Representations A Representation in C++ (Ford & Topp) CSE 30331 Lectures 18 – Graphs

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2 What is a graph? Formally, a graph G(V,E) is A set of vertices V A set of edges E, such that each edge e i,j connects two vertices v i and v j in V V and E may be empty

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3 Graph Categories A graph is connected if each pair of vertices have a path between them A complete graph is a connected graph in which each pair of vertices are linked by an edge

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4 Example of Digraph Graphs with ordered edges are called directed graphs or digraphs

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5 Strength of Connectedness (digraphs only) Strongly connected if there is a path from each vertex to every other vertex. A C B E D (a) Not Strongly or Weakly Connected (No pathE to D or D toE) C A ED B (b) Strongly Connected

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6 Strength of Connectedness (digraphs only) Weakly connected if, for each pair of vertices v i and v j, there is either a path P(v i, v j ) or a path P(v i, v j ). A C B E D A D C B (a) (c) Not Strongly or Weakly Connected (No pathE to D or D toE) Weakly Connected (No path from D to a vertex)

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7 Representing Graphs Adjacency Matrix Edges are represented in a 2-D matrix Adjacency Set (or Adjacency List) Each vertex has an associated set or list of edges leaving Edge List The entire edge set for the graph is in one list Mentioned in discrete math (probably)

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8 Adjacency Matrix An m x m matrix, called an adjacency matrix, identifies graph edges. An entry in row i and column j corresponds to the edge e = (v i, v j ). Its value is the weight of the edge, or -1 if the edge does not exist. D A C E B

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9 Adjacency Set (or List) An adjacency set or adjacency list represents the edges in a graph by using … An m element map or vector of vertices Where each vertex has a set or list of neighbors Each neighbor is at the end of an out edge with a given weight (a) D A C E B A E D C B Vertices Set of Neighbors B1 C1 C1 B1 D1B1

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10 Adjacency Matrix and Adjacency Set (side-by-side) D B A E C 4 2 7 3 6 4 1 2

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11 Building a graph class Neighbor Identifies adjacent vertex and edge weight VertexInfo Contains all characteristics of a given vertex, either directly or through links VertexMap Contains names of vertices and links to the associated VertexInfo objects

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12 vertexInfo Object A vertexInfo object consists of seven data members. The first two members, called vtxMapLoc and edges, identify the vertex in the map and its adjacency set.

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13 vertexInfo object vtxMapLoc – iterator to vertex (name) in map edges – set of vInfo index / edge weight pairs Each is an OUT edge to an adjacent vertex vInfo[index] is vertexInfo object for adjacent vertex inDegree – # of edges coming into this vertex outDegree is simply edges.size() occupied – true (this vertex is in the graph), false (this vertex was removed from graph) color – (white, gray, black) status of vertex during search dataValue – value computed during search (distance from start, etc) parent – parent vertex in tree generated by search

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14 A Neighbor class (edges to adjacent vertices) class neighbor { public: int dest; // index of destination vertex in vInfo vector int weight; // weight of this edge // constructor neighbor(int d=0, int c=0) : dest(d), weight(c) {} // operators to compare destination vertices friend bool operator<(const neighbor& lhs, const neighbor& rhs) { return lhs.dest < rhs.dest; } friend bool operator==(const neighbor& lhs, const neighbor& rhs) { return lhs.dest == rhs.dest; } };

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15 vertexInfo object (items in vInfo vector) template class vertexInfo { public: enum vertexColor { WHITE, GRAY, BLACK }; map ::iterator vtxMapLoc; // to pair in map set edges; // edges to adjacent vertices int inDegree; // # of edges coming into vertex bool occupied; // currently used by vertex or not vertexColor color; // vertex status during search int dataValue; // relevant data values during search int parent; // parent in tree built by search // default constructor vertexInfo(): inDegree(0), occupied(true) {} // constructor with iterator pointing to vertex in map vertexInfo(map ::iterator iter) : vtxMapLoc(iter), inDegree(0), occupied(true) {} };

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16 A graph using a vertexMap and vertexInfo vector Graph vertices are stored in a map, called vtxMap Each entry is a key, value pair The initial size of the vertexInfo vector is the number of vertices in the graph There is a 1-1 correspondence between an entry in the map and a vertexInfo entry in the vector vertex mIter (iterator location) index... index vtxMapLoc edges inDegree occupied color dataValue vtxMap parent vInfo

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17 A graph class (just the private members) private: typedef map vertexMap; vertexMap vtxMap; // store vertex in a map with its name as the key // and the index of the corresponding vertexInfo // object in the vInfo vector as the value vector > vInfo; // list of vertexInfo objects for the vertices int numVertices; int numEdges; // current size (vertices and edges) of the graph stack availStack; // availability stack, stores unused vInfo indices

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18 VtxMap and Vinfo Example A DC B

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19 Find the location for vertexInfo of vertex with name v // uses vtxMap to obtain the index of v in vInfo. // this is a private helper function template int graph ::getvInfoIndex(const T& v) const { vertexMap::const_iterator iter; int pos; // find the vertex : the map entry with key v iter = vtxMap.find(v); if (iter == vtxMap.end()) pos = -1; // wasn’t in the map else pos = (*iter).second; // the index into vInfo return pos; }

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20 Find in and out degree of v // return the number of edges entering v template int graph ::inDegree(const T& v) const { int pos=getvInfoIndex(v); if (pos != -1) return vInfo[pos].inDegree; else throw graphError("graph inDegree(): v not in the graph"); } // return the number of edges leaving v template int graph ::outDegree(const T& v) const { int pos=getvInfoIndex(v); if (pos != -1) return vInfo[pos].edges.size(); else throw graphError("graph outDegree(): v not in the graph"); }

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21 Insert a vertex into graph template void graph ::insertVertex(const T& v) { int index; // attempt insertion, set vInfo index to 0 for now pair result = vtxMap.insert(vertexMap::value_type(v,0)); if (result.second) { // insertion into map succeeded if (!availStack.empty()) { // there is an available index index = availStack.top(); availStack.pop(); vInfo[index] = vertexInfo (result.first); } else { // vInfo is full, increase its size vInfo.push_back(vertexInfo (result.first)); index = numVertices; } (*result.first).second = index; // set map value to index numVertices++; // update size info } else throw graphError("graph insertVertex(): v in graph"); }

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22 Insert an edge into graph // add the edge (v1,v2) with specified weight to the graph template void graph ::insertEdge(const T& v1, const T& v2, int w) { int pos1=getvInfoIndex(v1), pos2=getvInfoIndex(v2); if (pos1 == -1 || pos2 == -1) throw graphError("graph insertEdge(): v not in the graph"); else if (pos1 == pos2) throw graphError("graph insertEdge(): loops not allowed"); // insert edge (pos2,w) into edge set of vertex pos1 pair ::iterator, bool> result = vInfo[pos1].edges.insert(neighbor(pos2,w)); if (result.second) // it wasn’t already there { // update counts numEdges++; vInfo[pos2].inDegree++; }

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23 Erase an edge from graph // erase edge (v1,v2) from the graph template void graph ::eraseEdge(const T& v1, const T& v2) { int pos1=getvInfoIndex(v1), pos2=getvInfoIndex(v2); if (pos1 == -1 || pos2 == -1) throw graphError("graph eraseEdge(): v not in the graph"); // find the edge to pos2 in the list of pos1 neighbors set ::iterator setIter; setIter = vInfo[pos1].edges.find(neighbor(pos2)); if (setIter != edgeSet.end()) { // found edge in set, so remove it & update counts vInfo[pos1].edges.erase(setIter); vInfo[pos2].inDegree--; numEdges--; } else throw graphError("graph eraseEdge(): edge not in graph"); }

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24 Erase a vertex from graph (algorithm) Find index of vertex v in vInfo vector Remove vertex v from map Set vInfo[index].occupied to false Push index onto availableStack For every occupied vertex in vInfo Scan neighbor set for edge pointing back to v If edge found, erase it For each neighbor of v, decrease its inDegree by 1 Erase the edge set for vInfo[index]

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