Graphs
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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Nodes
Edges
A portion of a road map treated as a graph
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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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Paths
A sequence of edges that connect two
vertices in a graph
In 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 once
A 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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Trees
All trees are graphs
• But not all graphs are trees
A tree is a connected graph without cycles
Traversals
• Preorder (and, technically, inorder and
postorder) traversals are examples of depthfirst 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 order
vertexStack = a new stack to hold vertices as they are visited
Mark originVertex as visited
traversalOrder.enqueue (originVertex)
vertexStack.push (originVertex)
while (!vertexStack.isEmpty ()) {
topVertex = vertexStack.peek ()
if (topVertex has an unvisited neighbor) {
nextNeighbor = next unvisited neighbor of topVertex
Mark nextNeighbor as visited
traversalOrder.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 Traversal
Algorithm for breadth-first traversal of nonempty
graph beginning at a given vertex
Algorithm breadthFirstTraversal(originVertex)
vertexQueue = a new queue to hold neighbors
traversalOrder = a new queue for the resulting traversal order
Mark originVertex as visited
A breadth-first traversal visits
traversalOrder.enqueue(originVertex)
a vertex and then each of the
vertexQueue.enqueue(originVertex)
vertex's neighbors before
while (!vertexQueue.isEmpty()) {
advancing
frontVertex = vertexQueue.dequeue()
while (frontVertex has an unvisited neighbor) {
nextNeighbor = next unvisited neighbor of frontVertex
Mark nextNeighbor as visited
traversalOrder.enqueue(nextNeighbor)
vertexQueue.enqueue(nextNeighbor)
}
}
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return traversalOrder
Breadth-First Traversal
A 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 B C D
A 0
1
1
1
B 0
0
0
0
C 0
0
0
0
D 1
0
1
0
A directed graph and implementations using
adjacency lists and an adjacency matrix.
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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 visited
n = number of vertices in the graph
for (counter = 1 to n) {
nextVertex = an unvisited vertex having no unvisited successors
Mark nextVertex as visited
stack.push(nextVertex)
}
return stack
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Topological Sorting
Algorithm getTopologicalSort()
vertexStack = a new stack to hold vertices as they are visited
n = number of vertices in the graph
for (counter = 1 to n) {
nextVertex = an unvisited vertex having no unvisited successors
Mark nextVertex as visited
stack.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
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