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Graphs
Chapter 28
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Contents
Some Examples and Terminology
Road Maps
Airline Routes
Mazes
Course Prerequisites
Trees
Traversals
Breadth-First Traversal
Depth-First Traversal
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Contents
Topological Order
Paths
Finding a Path
The Shortest Path in an Unweighted Graph
The Shortest Path in a Weighted Graph
Java Interfaces for the ADT Graph
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Objectives
Describe characteristics of a graph, including vertices, edges, paths
Give examples of graphs, undirected, directed, unweighted, weighted
Give examples of vertices
Adjacent and not adjacent
For both directed and undirected graphs
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Objectives
Give examples of paths, simple paths, cycles, simple cycles
Give examples of connected graphs, disconnected graphs, complete graphs
Perform depth-first traversal, breadth-first traversal on a given graph
List topological order for vertices of a directed graph without cycles
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Objectives
Detect whether path exists between two given vertices of a graph
Find path with fewest edges that joins one vertex to another
Find path with lowest cost that joins one vertex to another in a weighted graph
Describe operations for ADT graph
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Examples
Not the graphs we study
Bar graphs, pie charts, etc.
Graphs we study include
Trees
Networks of nodes connected by paths
Could include a road map
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8. Figure 28-1 A portion of a road map
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Figure 28-2 A directed graph representing a portion of a city’s street map
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Terminology
Nodes connected by edges
Edges
Undirected
Directed (digraph)
Path between two vertices
Sequence of edges
Weights
Shortest, fastest, cheapest, costs
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11. Figure 28-3 A weighted graph
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12. Figure 28-4 Undirected graphs
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Figure 28-5 Vertex A is adjacent to vertex B, but B is not adjacent to A
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14. Figure 28-6 Airline routes
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15. Figure 28-7 (a) A maze; (b) its representation as a graph
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Figure 28-8 The prerequisite structure for a selection of courses as a directed graph without cycles
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17. Figure 28-9 The visitation order of two traversals: (a) depth first;
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18. Figure 28-9 The visitation order of two traversals: (b) breadth first
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19. Breadth First Traversal
Algorithm
getBreadthFirstTraversal (originVertex) {
traversalOrder =
a new queue (or string)
VQ = a new queue for visited vertices
Mark originVertex as visited
and output it
VQ.enqueue(originVertex)
While (! VQ.isEmpty()) {
FrontV = VQ.dequeue()
while (FrontV has neighbor)
{ nextN = next neighbor of
FrontV
if (nextN is NOT Visited)
{ Mark nextN as visited
and output it
VQ.enqueue(nextN) } }…
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FIGURE A trace of a breadth-first traversal beginning at vertex A of a directed graph
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Depth First Traversal
Algorithm
getDepthFirstTraversal (originVertex) {
traversalOrder =
a new queue (or string)
VS = a new stack for visited vertices
Mark originVertex as visited
and output it
VS.push(originVertex)
While (! VS.isEmpty()) {
TopV = VS.peek()
if (TopV has UNVISITED neighbor)
{ nextN = next unvisited neighbor
of TopV
Mark nextN as visited
and output it
VS.push(nextN) }
else
VS.pop() …
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Comparison
BREADTH FIRST  QUEUE While (! VQ.isEmpty()) { FrontV = VQ.dequeue() while (FrontV has neighbor) { nextN = next neighbor of FrontV if (nextN is NOT Visited) { Mark nextN as visited and output it VQ.enqueue(nextN) } }…
DEPTH FIRST  STACK
While (! VS.isEmpty()) {
TopV = VS.peek()
if (TopV has UNVISITED neighbor)
{ nextN = next unvisited neighbor
of TopV
Mark nextN as visited
and output it
VS.push(nextN) }
else
VS.pop() …
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Figure A trace of a depth-first traversal beginning at vertex A of a directed graph
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Figure A trace of a depth-first traversal beginning at vertex A of a directed graph
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25. Figure 28-12 Three topological orders for the graph in Figure 28-8
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Figure An impossible prerequisite structure for three courses, as a directed graph with a cycle
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27. Figure 28-14 Finding a topological order for the graph in Figure 28-8
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28. Figure 28-14 Finding a topological order for the graph in Figure 28-8
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29. Figure 28-14 Finding a topological order for the graph in Figure 28-8
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30. Figure 28-14 Finding a topological order for the graph in Figure 28-8
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31. Figure 28-14 Finding a topological order for the graph in Figure 28-8
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Figure (a) An unweighted graph and (b) the possible paths from vertex A to vertex H
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Figure (a) The graph in Figure 28-15a after the shortest-path algorithm has traversed from vertex A to vertex H; (b) the data in a vertex
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34. Shortest Path Algorithm
Algorithm getShortestPath (originV, endV, path) {
done = False
VQ = a new queue - visited Vertices
Mark originV as visited
(output it)
VQ.enqueue(originV)
While (!done && ! VQ.isEmpty()) {
FrontV = VQ.dequeue()
while ((!done &&
FrontV has neighbor)
{ nextN = next neighbor of FrontV
if (nextN is NOT Visited)
{ Mark nextN as visited
and adjust length
and set predecessor
VQ.enqueue(nextN) }
if (nextN == endV) done = true
}… // construct the path …
NOTE: In light blue is what is different from the Breadth First Traversal
Path.push (endV)
Vertex = endV
While (Vertex has a predecessor)
{ Vertex = predecessor of Vertex
path.push(vertex) }…
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NOTE: For printing you may want to remove the blue boxes
Figure A trace of the traversal in the algorithm to find the shortest path from vertex A to vertex H in an unweighted graph
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NOTE: For printing you may want to remove the blue boxes
Figure A trace of the traversal in the algorithm to find the shortest path from vertex A to vertex H in an unweighted graph
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Figure (a) A weighted graph and (b) the possible paths from vertex A to vertex H, with their weights
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38. Cheapest Path Algorithm
Algorithm getCheapestPath (originV, endV, path) {
done = False
PQ = a new priority queue
Mark originV as visited
(output it)
PQ.add(originV)
While (!done && ! PQ.isEmpty())
{ FrontV = PQ.remove()
if (FrontV is NOT Visited)
{ Mark nextN as visited and
set cost and predecessor
if (FrontV == endV) done = true
else
while (FrontV has a neighbor)
{ nextN = next neighbor of FrontV
if (nextN is NOT visited)
{ adjust cost and predecessor
PQ.add(nextN with new values)
} } // if // while
} } // if // while
}… // construct the path …
Path.push (endV)
Vertex = endV
While (Vertex has a predecessor)
{ Vertex = predecessor of Vertex
path.push(vertex) }…
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NOTE: For printing you may want to remove the blue boxes
Figure A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph
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NOTE: For printing you may want to remove the blue boxes
Figure A trace of the traversal in the algorithm to find the cheapest path from vertex A to vertex H in a weighted graph
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FIGURE The graph in Figure 28-18a after finding the cheapest path from vertex A to vertex H
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42. Java Interfaces for the ADT Graph
ADT graph different from other ADTs
Once instance created, we do not add, remove, or retrieve components
Interface to create the graph and to obtain basic information
Listing 28-1
Note: Code listing files
must be in same folder
as PowerPoint files
for links to work
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43. Java Interfaces for the ADT Graph
Interface to specify operations such as traversals and path searches
Listing 28-2
For convenience, a third interface to combine first two
Listing 28-3
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44. Figure 28-21 A portion of the flight map in Figure 28-6
The following statements create the graph shown
Figure A portion of the flight map in Figure 28-6
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End
Chapter 28
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