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1

Main Index

Contents

- Chapter 16 - Graphs
- Graph Categories
- Example of Digraph
- Connectedness of Digraph
- Adjacency Matrix
- Adjacency Set
- vertexInfo Object
- Vertex Map and Vector vInfo
- VtxMap and Vinfo Example
- Breadth-First Search Algorithm
- Dfs()

- Strong Components
- Graph G and Its Transpose GT
- Shortest-Path Example
- Dijkstra Minimum-Path Algorithm
- Minimum Spanning Tree
- Minimum Spanning Tree: vertices A and B
- Completing the Minimum Spanning-Tree with Vertices C and D
- Summary Slides

Main Index

Contents

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

Main Index

Contents

Connectedness of Digraph- Strongly connected if there is a path from any
- vertex to any other vertex.
- 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).

Main Index

Contents

Adjacency Matrix- An m by m matrix, called an adjacency matrix, identifies the 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.

Neighbor object (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;

}

};

Main Index

Contents

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.

Main Index

Contents

Vertex Map and Vector vInfo- To store the vertices in a graph, we provide a map<T,int> container, called vtxMap, where a vertex name is the key of type T. The int field of a map object is an index into a vector of vertexInfo objects, called vInfo. The size of the vector is initially the number of vertices in the graph, and there is a 1-1 correspondence between an entry in the map and a vertexInfo entry in the vector

parent

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

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)

{}

};

Graph object data items

template <typename T>

class graph

{

public:

~~~~~ member functions ~~~~

private:

// vertex names and corresponding vInfo indices

typedef map<T,int> vertexMap;

vertexMap vtxMap;

vector<vertexInfo<T> > vInfo; // info on each vertex

int numVertices; // current # of vertices in graph

int numEdges; // current # of edges in graph

stack<int> availStack; // available (unused) vInfo indices

};

Find location for vertexInfo of vertex with name v

// uses vtxMap to obtain the index of v in vInfo

// 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;

}

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

template <typename T>

int graph<T>::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");

}

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");

}

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++;

}

}

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");

}

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]

Breadth First Search (traversal)

- Uses a queue to order search and a set to store visited vertices
- Start with all unvisited (white) vertices
- Push start vertex onto Q
- While Q is not empty
- Remove vertex V from Q
- Mark V as visited (black)
- Insert V into the visited set
- For each adjacent vertex (each neighbor) U
- If U is unvisited (white)
- Mark it seen (gray) and push it onto Q
- Return visited set (vertices reached from start)
- Running time – O(V + E)

Depth first search (traversal)

- Emulates a postorder traversal, backtracking search
- Visits occur while backing out
- DfsVisit(V, checkCycles)
- If V is unvisited (white)
- Mark V as seen (gray)
- For each neighbor U of V
- If U is unvisited (white)
- DfsVisit(U,checkCycles)
- Else if U is previously discovered (gray) && checkCycles
- Throw exception (found cycle)
- Mark V as visited (black)
- Push V onto FRONT of dfsList

Strong Components

- A strongly connected component of a graph G is a maximal set of vertices SC in G that are mutually accessible.

Graph G and Its Transpose GT

- The transpose has the same set of vertices V as graph G but a new edge set ET consisting of the edges of G but with the opposite direction.

Finding Strong Components

- Perform dfs() of graph G, creating dfsGList
- Create GT (transform of graph G)
- Color all vertices in GT white
- For each vertex V in GT
- If V is white
- Perform dfsVisit() of GT from V and create dfsGTList
- Append dfsGTList to component vector
- At end of process, the component vector contains a set of vertices for each strong component in the graph G
- Finding GT is O(V+E) and dfs() is O(V+E)
- So, finding strong components is also O(V+E)

G, GT and its Strong Components

- dfsGList: A B C E D G F
- dfsGTLists: {A C B} , {E} , {D F G}

Topological sort of acyclic graphs

- Important in determining precedence order in graphs representing scheduling of activities
- Dfs() produces a topological sort of the vertices in the graph, returning them in the dfsList
- Graph must be acyclic
- To show that dfs() performs a topological sort
- show that if a path exists from V to W then V always appears ahead of W in dfsList
- We examine the three colors W may have when first encountered in path …

Shortest-Path Example

- Shortest-path is a modified breadth-first search
- Path length is number of edges traversed and is stored in dataValue field of vertex at time of its discovery
- The parent field is set to the index of the parent at the same time
- Path is recovered in reverse, using parent fields

Shortest-Path Example(path from F to C)

- Start: visitQ = F:0:F format(vertex:dataValue:parent)
- Next: visitQ = D:1:F, E:1:F
- Next: visitQ = E:1:F, A:2:D
- Next: visitQ = A:2:D
- Next: visitQ = B:3:A, C:3:A
- Next: visitQ = C:3:A
- Finish: C found, path length = 3, path = F,D,A,C : parent( parent( parent(C) ) )

Minimum (weight) path – Dijkstra’s algorithm

- Uses priority queue containing identities of all fringe vertices and the length of the minimum path to each from the start
- Algorithm builds a tree of all minimum length paths from start
- Each vertex is either tree, fringe or unseen

At each step

The fringe vertex V with the minimum path is

removed from priorityQ and added to the tree

V’s non-tree neighbors U become fringe and

the minimum path length is computed from start,

thru V to U and is stored in U.dataValue,

V is saved as U.parent and

V:pathlength is added to priorityQ

- Process stops when queue is empty, or chosen destination vertex is found

Main Index

Contents

Dijkstra Minimum-Path Algorithm From A to D ExamplePriQ: (A,0) Tree (vertices & path weight)

(B,4) (C,11) (E,4) A,0

(E,4) (C,11) (C,10) (D,12) A,0 B,4

(C,10) (C,11) (D,12) A,0 B,4 C,10

empty A,0 B,4 C,10 D,12

Minimum Spanning TreePrim’s Algorithm

- Spanning tree for graph with minimum TOTAL weight
- Min Spanning Tree may not be unique, but total weight is same value for all
- All vertices are either tree, fringe, or unseen
- Priority queue is used to hold fringe vertices and the minimum weight edge connecting each to the tree

Put start vertex in priorityQ

While priorityQ not empty

The nearest vertex V is removed from the queue

and added to the tree

For each non-tree neighbor U of V

if the edge V,U weight < current U.dataValue

U.dataValue is set to weight of edge V,U

U.parent is set to V

push U:weight pair onto priority queue

Runtime Orders of Complexity

- Min Spanning Tree – O(V + E log2E)
- Min Path (Dijkstra) – O(V + E log2E)
- Strong Components – O(V + E)
- Dfs – O(V+E)
- BFS – O(V+E)

Main Index

Contents

Summary Slide 1§- Undirected and Directed Graph (digraph)

- Both types of graphs can be either weighted or nonweighted.

Main Index

Contents

Summary Slide 2§- Breadth-First, bfs()

- locates all vertices reachable from a starting vertex

- can be used to find the minimum distance from a starting vertex to an ending vertex in a graph.

Main Index

Contents

Summary Slide 3§- Depth-First search, dfs()

- produces a list of all graph vertices in the reverse order of their finishing times.

- supported by a recursive depth-first visit function, dfsVisit()

- an algorithm can check to see whether a graph is acyclic (has no cycles) and can perform a topological sort of a directed acyclic graph (DAG)

- forms the basis for an efficient algorithm that finds the strong components of a graph

Main Index

Contents

Summary Slide 4§-Dijkstra's algorithm

- if weights, uses a priority queue to determine a path from a starting to an ending vertex, of minimum weight

- This idea can be extended to Prim's algorithm, which computes the minimum spanning tree in an undirected, connected graph.

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