Graphs

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Some Examples and Terminology

A graph is a set of vertices (nodes) and a set of edges (arcs) such that each edge is associated with exactly two vertices• Edges can be undirected or directed

(digraph)

A subgraph is a portion of a graph that itself is a graph

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A portion of a road map treated as a graph

NodesNodes

EdgesEdges

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A directed graph representing part of a city street map

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(a) A maze; (b) its representation as a graph

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The prerequisite structure for a selection of courses as a directed graph without cycles.

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PathsA sequence of edges that connect two vertices in a graphIn a directed graph the direction of the edges must be considered• Called a directed path

A cycle is a path that begins and ends at same vertex and does not pass through any vertex more than onceA graph with no cycles is acyclic

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Weights

A weighted graph has values on its edges• Weights, costs, etc.

A path in a weighted graph also has weight or cost• The sum of the edge weights

Examples of weights• Miles between nodes on a map• Driving time between nodes• Taxi cost between node locations

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A weighted graph

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Connected Graphs

A connected graph• Has a path between every pair of

distinct vertices

A complete graph• Has an edge between every pair of

distinct vertices

A disconnected graph• Not connected

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Connected Graphs

Undirected graphs

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Adjacent Vertices

Two vertices are adjacent in an undirected graph if they are joined by an edge

Sometimes adjacent vertices are called neighbors

Vertex A is adjacent to B, but B is not adjacent to A.

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Note the graph with two subgraphs • Each subgraph connected• Entire graph disconnected

Airline routes

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TreesAll trees are graphs• But not all graphs are trees

A tree is a connected graph without cyclesTraversals• Preorder (and, technically, inorder and

postorder) traversals are examples of depth-first traversal

• Level-order traversal of a tree is an example of breadth-first traversal

Visit a node• Process the node’s data and/or mark the node

as visited

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Trees

Visitation order of two traversals; (a) depth first; (b) breadth first.

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Depth-First Traversal

Visits a vertex, then• A neighbor of the vertex, • A neighbor of the neighbor,• Etc.

Advance as far as possible from the original vertex

Then back up by one vertex• Considers the next neighbor

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Algorithm depthFirstTraversal (originVertex)traversalOrder = a new queue for the resulting traversal ordervertexStack = a new stack to hold vertices as they are visitedMark originVertex as visitedtraversalOrder.enqueue (originVertex)vertexStack.push (originVertex)while (!vertexStack.isEmpty ()) {

topVertex = vertexStack.peek ()if (topVertex has an unvisited neighbor) {

nextNeighbor = next unvisited neighbor of topVertexMark nextNeighbor as visitedtraversalOrder.enqueue (nextNeighbor)vertexStack.push (nextNeighbor)

}else // all neighbors are visited

vertexStack.pop ()}return traversalOrder

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Depth-First Traversal

Trace of a depth-first traversal

beginning at vertex A.

Assumes that children are placed on the stack in alphabetic (or numeric order).

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Breadth-First TraversalAlgorithm for breadth-first traversal of nonempty graph beginning at a given vertex

Algorithm breadthFirstTraversal(originVertex)vertexQueue = a new queue to hold neighborstraversalOrder = a new queue for the resulting traversal orderMark originVertex as visitedtraversalOrder.enqueue(originVertex)vertexQueue.enqueue(originVertex)while (!vertexQueue.isEmpty()) {

frontVertex = vertexQueue.dequeue()while (frontVertex has an unvisited neighbor) {

nextNeighbor = next unvisited neighbor of frontVertexMark nextNeighbor as visitedtraversalOrder.enqueue(nextNeighbor)vertexQueue.enqueue(nextNeighbor)

}}return traversalOrder

A breadth-first traversal visits a vertex and then each of the

vertex's neighbors before advancing

A breadth-first traversal visits a vertex and then each of the

vertex's neighbors before advancing

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Breadth-First TraversalA trace of a breadth-first

traversal for a directed graph,

beginning at vertex A.

Assumes that children are placed in the queue in alphabetic (or numeric order).

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Implementations of the ADT Graph

A directed graph and implementations using adjacency lists and an adjacency matrix.

A B C D

A 0 1 1 1

B 0 0 0 0

C 0 0 0 0

D 1 0 1 0

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Topological Order

Given a directed graph without cycles

In a topological order • Vertex a precedes vertex b whenever

a directed edge exists from a to b

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Topological Order

Three topological orders for the indicated graph.

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Topological Order

An impossible prerequisite structure for three courses as a directed graph with a cycle.

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Topological Order

Algorithm for a topological sort

Algorithm getTopologicalSort()vertexStack = a new stack to hold vertices as they are visitedn = number of vertices in the graphfor (counter = 1 to n) {

nextVertex = an unvisited vertex having no unvisited successorsMark nextVertex as visitedstack.push(nextVertex)

}return stack

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Topological Sorting

Algorithm getTopologicalSort()vertexStack = a new stack to hold vertices as they are visitedn = number of vertices in the graphfor (counter = 1 to n) {

nextVertex = an unvisited vertex having no unvisited successorsMark nextVertex as visitedstack.push(nextVertex)

}return stack

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Topological Order

Finding a topological order

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Shortest Path in an Unweighted Graph

(a) an unweighted graph and (b) the possible paths from vertex A to vertex H.

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Shortest Path in an Unweighted Graph

The previous graph after the shortest-path algorithm has traversed from vertex A to vertex H

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Shortest Path in an Unweighted Graph

Finding the shortest path from vertex A to vertex H.

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Shortest Path in an Weighted Graph

(a) A weighted graph and (b) the possible paths from vertex A to vertex H.

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Shortest Path in an Weighted Graph

Shortest path between two given vertices• Smallest edge-weight sum

Algorithm based on breadth-first traversal

Several paths in a weighted graph might have same minimum edge-weight sum• Algorithm given by text finds only one of these

paths

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Shortest Path in an Weighted Graph

Finding the shortest path from vertex A

to vertex H

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Shortest Path in an Weighted Graph

The previous graph after finding the shortest path from vertex A to vertex H.

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Java Interfaces for the ADT Graph

Methods in the BasicGraphInterface• addVertex• addEdge• hasEdge• isEmpty• getNumberOfVertices• getNumberOfEdges• clear