1. Refresh and Get Ready for More
Graphs
Refresh and Get Ready for More
Copyright © – Curt Hill

2. Copyright © 2001-2016 – Curt Hill
Introduction
We have seen some of this in the tree presentation
We will quickly review the definitions
Then some graph problems and algorithms
Another presentation will consider algorithms in more detail
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3. Mathematical definitions
Nodes or Vertices
Arcs or Edges
Indegree
Outdegree
Directed or digraph
Undirected
Network 2 1 3 6 4 5
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4. Copyright © 2001-2016 – Curt Hill
Undirected graphs
Two nodes are adjacent if an arc connects them
A path is a collection of arcs leading from one node to another
A cycle is a path of at least three arcs that starts and ends on the same node
A graph is connected if there is a path from any node to any other
A disconnected graph contains components
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5. Copyright © 2001-2016 – Curt Hill
Directed graphs
Also has the notion of paths and cycles
A strongly connected digraph has a path from every node to every other node
A weakly connected digraph has a path from every node to every other node, only if converted to an undirected graph
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6. Mathematical definitions
Two components
Strongly connected
Weakly connected 2 1 3 6 4 5 9 8 7
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7. Copyright © 2001-2016 – Curt Hill
Real Graphs
Airline/bus/train routes
Circuit board component connections
Molecule diagrams
People with a “works with” relationship
Rooms of a building
Flow graphs of programs
Moves in games or puzzles
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8. Copyright © 2001-2016 – Curt Hill
Flow Graphs
2: a = b * 2;
3: while(b < 4) {
4: c += a/2 + b--;
5: if(a<0)
6: a = -a*2 + 1;
7: }
8: a = 0; 2 3 4 5 6 7 8
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9. Copyright © 2001-2016 – Curt Hill
Set Representation
Set of destinations
Each node contains a set of those nodes that are connected by an arc
7: {8, 9}
8: {9}
Sets of ordered pairs
First is source, second is destination
{(7,8),(7,9),(8,9)} 9 8 7
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Adjacency Matrix
Matrix contains Booleans or weights
A sparse matrix can be a list of lists 7 8 9 1 2 2 9 8 1 7 2
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11. Traversing/Searching Graphs
The big danger is cycles
Three methods are related
Depth first
Breadth first
Best first
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12. Copyright © 2001-2016 – Curt Hill
Topological Sort
Suppose a directed graph with no cycles
A topological order of this graph has all nodes in sequential order such that no node is listed until all predecessors are listed
Not always unique
1,2,3,4,5,6
1,3,2,4,5,6
1,2,3,5,4,6
1,3,2,5,4,6
Among others 4 2 6 1 5 3
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13. Copyright © 2001-2016 – Curt Hill
Spanning Trees
A tree that starts at a particular node and contains each node of a graph but has only a subset of the arcs
Spanning trees are not usually unique
If the arcs have a cost function
A minimal spanning tree has the lowest sum of costs
Prim’s Algorithm will find the minimal spanning tree
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14. Traveling Salesman’s Problem
There are a number of cities that are connected by roads
A traveling salesman must visit them all
What is the shortest route?
This is a classic intractable problem
The solution takes O(en)time
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15. Copyright © 2001-2016 – Curt Hill
Finally
Graphs provide several problems
Although a tree is a graph it is very manageable
Graphs provide problems with representation
Each application may need a different
Sometimes the graph is not represented completely at all
The cycle issue provides problems for searching and traversal
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