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

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 ei,j connects two vertices vi and vj in V
V and E may be empty

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

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

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

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

7. Representing Graphs Adjacency Matrix Adjacency Set (or Adjacency List)
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)

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 = (vi, vj). Its value is the weight of the edge, or -1 if the edge does not exist. D A C E B

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
Vertices
Set of Neighbors 1

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

11. Building a graph class Neighbor VertexInfo VertexMap
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

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.

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

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; };

15. vertexInfo object (items in vInfo vector)
template <typename T>
class vertexInfo {
public:
enum vertexColor { WHITE, GRAY, BLACK };
map<T,int>::iterator vtxMapLoc; // to pair<T,int> in map
set<neighbor> 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<T,int>::iterator iter)
: vtxMapLoc(iter), inDegree(0), occupied(true) };

16. A graph using a vertexMap and vertexInfo vector
Graph vertices are stored in a map<T,int>, called vtxMap
Each entry is a <vertex name, vertexInfo index> 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
. . .
vtxMapLoc
edges
inDegree
occupied
color
dataValue
vtxMap
parent
vInfo

17. A graph class (just the private members)
typedef map<T,int> 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<vertexInfo<T> > vInfo;
// list of vertexInfo objects for the vertices
int numVertices;
int numEdges;
// current size (vertices and edges) of the graph
stack<int> availStack;
// availability stack, stores unused vInfo indices

18. VtxMap and Vinfo ExampleC B

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 <typename T>
int graph<T>::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; }

20. Find in and out degree of v
// return the number of edges entering v
template <typename T>
int graph<T>::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
int graph<T>::outDegree(const T& v) const
return vInfo[pos].edges.size();
throw graphError("graph outDegree(): v not in the graph");

21. Insert a vertex into graph
template <typename T>
void graph<T>::insertVertex(const T& v) {
int index;
// attempt insertion, set vInfo index to 0 for now
pair<vertexMap::iterator, bool> 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<T>(result.first);
} else { // vInfo is full, increase its size
vInfo.push_back(vertexInfo<T>(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");

22. Insert an edge into graph
// add the edge (v1,v2) with specified weight to the graph
template <typename T>
void graph<T>::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<set<neighbor>::iterator, bool> result =
vInfo[pos1].edges.insert(neighbor(pos2,w));
if (result.second) // it wasn’t already there
// update counts
numEdges++;
vInfo[pos2].inDegree++; }

23. Erase an edge from graph
// erase edge (v1,v2) from the graph
template <typename T>
void graph<T>::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<neighbor>::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");

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]