Graphs concept in data structures and algorithms.ppt

Graphs
Data Structures
Prepared By K.Vimala
M.Sc.,M.Phil.,MBA.,B.Ed.,

What is a graph?
• A data structure that consists of a set of nodes
(vertices) and a set of edges that relate the nodes
to each other
• The set of edges describes relationships among the
vertices

Formal definition of graphs
• A graph G is defined as follows:
G=(V,E)
V(G): a finite, nonempty set of
vertices
E(G): a set of edges (pairs of vertices)

Directed vs. undirected graphs
• When the edges in a graph have no
direction, the graph is called undirected

• When the edges in a graph have a direction,
the graph is called directed (or digraph)
Directed vs. undirected graphs
(cont.)
E(Graph2) = {(1,3) (3,1) (5,9) (9,11)
(5,7)
Warning: if the graph is
directed, the order of the
vertices in each edge is
important !!

• Trees are special cases of graphs!!
Trees vs graphs

Graph terminology
• Adjacent nodes: two nodes are adjacent if they are
connected by an edge
• Path: a sequence of vertices that connect two nodes in
a graph
• Complete graph: a graph in which every vertex is
directly connected to every other vertex
5 is adjacent to 7
7 is adjacent from 5

• What is the number of edges in a complete
directed graph with N vertices?
N * (N-1)
Graph terminology (cont.)
2
( )
O N

• What is the number of edges in a complete
undirected graph with N vertices?
N * (N-1) / 2
2
( )
O N

• Weighted graph: a graph in which each edge
carries a value

Graph implementation
• Array-based implementation
– A 1D array is used to represent the vertices
– A 2D array (adjacency matrix) is used to
represent the edges

Array-based implementation

Graph implementation (cont.)
• Linked-list implementation
– A 1D array is used to represent the vertices
– A list is used for each vertex v which contains the
vertices which are adjacent from v (adjacency list)

Linked-list implementation

Adjacency matrix vs. adjacency list
representation
• Adjacency matrix
– Good for dense graphs --|E|~O(|V|2
)
– Memory requirements: O(|V| + |E|) = O(|V|2
)
– Connectivity between two vertices can be tested
quickly
• Adjacency list
– Good for sparse graphs -- |E|~O(|V|)
– Memory requirements: O(|V| + |E|)=O(|V|)
– Vertices adjacent to another vertex can be found
quickly Prepared By K.Vimala

Graph specification based on
adjacency matrix representation
const int NULL_EDGE = 0;
template<class VertexType>
class GraphType {
public:
GraphType(int);
~GraphType();
void MakeEmpty();
bool IsEmpty() const;
bool IsFull() const;
void AddVertex(VertexType);
void AddEdge(VertexType, VertexType, int);
int WeightIs(VertexType, VertexType);
void GetToVertices(VertexType, QueType<VertexType>&);
void ClearMarks();
void MarkVertex(VertexType);
bool IsMarked(VertexType) const;
private:
int numVertices;
int maxVertices;
VertexType* vertices;
int **edges;
bool* marks;
};
(continues)

GraphType<VertexType>::GraphType(int maxV)
{
numVertices = 0;
maxVertices = maxV;
vertices = new VertexType[maxV];
edges = new int[maxV];
for(int i = 0; i < maxV; i++)
edges[i] = new int[maxV];
marks = new bool[maxV];
}
GraphType<VertexType>::~GraphType()
{
delete [] vertices;
for(int i = 0; i < maxVertices; i++)
delete [] edges[i];
delete [] edges;
delete [] marks;
}
(continues)

void GraphType<VertexType>::AddVertex(VertexType vertex)
{
vertices[numVertices] = vertex;
for(int index = 0; index < numVertices; index++) {
edges[numVertices][index] = NULL_EDGE;
edges[index][numVertices] = NULL_EDGE;
}
numVertices++;
}
void GraphType<VertexType>::AddEdge(VertexType fromVertex,
VertexType toVertex, int weight)
{
int row;
int column;
row = IndexIs(vertices, fromVertex);
col = IndexIs(vertices, toVertex);
edges[row][col] = weight;
}
(continues)

int GraphType<VertexType>::WeightIs(VertexType fromVertex,
VertexType toVertex)
{
int row;
int column;
row = IndexIs(vertices, fromVertex);
col = IndexIs(vertices, toVertex);
return edges[row][col];
}

Graph searching
• Problem: find a path between two nodes of
the graph (e.g., Austin and Washington)
• Methods: Depth-First-Search (DFS) or
Breadth-First-Search (BFS)

Depth-First-Search (DFS)
• What is the idea behind DFS?
– Travel as far as you can down a path
– Back up as little as possible when you reach a
"dead end" (i.e., next vertex has been "marked"
or there is no next vertex)
• DFS can be implemented efficiently using a
stack

Set found to false
stack.Push(startVertex)
DO
stack.Pop(vertex)
IF vertex == endVertex
Set found to true
ELSE
Push all adjacent vertices onto stack
WHILE !stack.IsEmpty() AND !found
IF(!found)
Write "Path does not exist"
Depth-First-Search (DFS) (cont.)

start end
(initialization)

template <class ItemType>
void DepthFirstSearch(GraphType<VertexType> graph, VertexType
startVertex, VertexType endVertex)
{
StackType<VertexType> stack;
QueType<VertexType> vertexQ;
bool found = false;
VertexType vertex;
VertexType item;
graph.ClearMarks();
stack.Push(startVertex);
do {
stack.Pop(vertex);
if(vertex == endVertex)
found = true;
(continues)

else {
if(!graph.IsMarked(vertex)) {
graph.MarkVertex(vertex);
graph.GetToVertices(vertex, vertexQ);
while(!vertexQ.IsEmpty()) {
vertexQ.Dequeue(item);
if(!graph.IsMarked(item))
stack.Push(item);
}
}
} while(!stack.IsEmpty() && !found);
if(!found)
cout << "Path not found" << endl;
}
(continues)

void GraphType<VertexType>::GetToVertices(VertexType vertex,
QueTye<VertexType>& adjvertexQ)
{
int fromIndex;
int toIndex;
fromIndex = IndexIs(vertices, vertex);
for(toIndex = 0; toIndex < numVertices; toIndex++)
if(edges[fromIndex][toIndex] != NULL_EDGE)
adjvertexQ.Enqueue(vertices[toIndex]);
}

Breadth-First-Searching (BFS)
• What is the idea behind BFS?
– Look at all possible paths at the same depth
before you go at a deeper level
– Back up as far as possible when you reach a
"dead end" (i.e., next vertex has been "marked"
or there is no next vertex)

• BFS can be implemented efficiently using a queue
Set found to false
queue.Enqueue(startVertex)
DO
queue.Dequeue(vertex)
IF vertex == endVertex
Set found to true
ELSE
Enqueue all adjacent vertices onto queue
WHILE !queue.IsEmpty() AND !found
• Should we mark a vertex when it is enqueued or
when it is dequeued ?
Breadth-First-Searching (BFS) (cont.)
IF(!found)
Write "Path does not exist"

void BreadthFirtsSearch(GraphType<VertexType> graph,
VertexType startVertex, VertexType endVertex);
{
QueType<VertexType> queue;
QueType<VertexType> vertexQ;//
bool found = false;
VertexType vertex;
VertexType item;
graph.ClearMarks();
queue.Enqueue(startVertex);
do {
queue.Dequeue(vertex);
if(vertex == endVertex)
found = true;
(continues)

else {
if(!graph.IsMarked(vertex)) {
graph.MarkVertex(vertex);
graph.GetToVertices(vertex, vertexQ);
while(!vertxQ.IsEmpty()) {
vertexQ.Dequeue(item);
if(!graph.IsMarked(item))
queue.Enqueue(item);
}
}
}
} while (!queue.IsEmpty() && !found);
if(!found)
cout << "Path not found" << endl;
}

Single-source shortest-path problem
• There are multiple paths from a source vertex
to a destination vertex
• Shortest path: the path whose total weight
(i.e., sum of edge weights) is minimum
• Examples:
– Austin->Houston->Atlanta->Washington:
1560 miles
– Austin->Dallas->Denver->Atlanta->Washington:
2980 miles

• Common algorithms: Dijkstra's algorithm,
Bellman-Ford algorithm
• BFS can be used to solve the shortest graph
problem when the graph is weightless
weightless or all
the weights are the same
(mark vertices before Enqueue)
Single-source shortest-path problem
(cont.)

Exercises
• 24-37

Graphs concept in data structures and algorithms.ppt

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