Lecture15

# Lecture15 - Computational Optimization ISE 407 Lecture 15...

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Computational Optimization ISE 407 Lecture 15 Dr. Ted Ralphs

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ISE 407 Lecture 15 1 References for Today’s Lecture Sections 17.2–17.5, R. Sedgewick, Algorithms in C++, Part 5 . AMO Sections 2.3 CLRS Section 22.1
ISE 407 Lecture 15 2 Connectivity Relations So far, we have only considered sets of items that are related to each other through some kind of ordering (if at all). In other words, two items x and y are only related by their relative positions in the ordered list. We will now generalize this idea by considering additional connectivity relationships between items. To do so, we will specify that there is a direct link between certain pairs of items. This will allow us to ask questions such as the following. Is x connected “directly” to y ? Is x connected to y “indirectly,” i.e., through a sequence of direct connections ? What is the set of of all items connected to x , directly or indirectly ? What is the shortest number of connections needed to get from x to y ?

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ISE 407 Lecture 15 3 Graphs A graph is an abstract object used to model such connectivity relations. A graph consists of a list of items, along with a set of connections between the items. The study of such graphs and their properties, called graph theory , is hundreds of years old. Graphs can be visualized easily by creating a physical manifestation. There are several variations on this theme. The connections in the graph may or may not have an orientation or a direction . We may not allow more than one connection between a pair of items. We may not allow an item to be connected to itself. For now, we consider graphs that are undirected , i.e., the connections do not have an orientation, and simple , i.e., we allow only one connection between each pair of items and no connections from an item to itself.
ISE 407 Lecture 15 4 Applications of Graphs Maps Social Networks World Wide Web Circuits Scheduling Communication Networks Matching and Assignment

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ISE 407 Lecture 15 5 Graph Terminology and Notation In an undirected graph, the “items” are usually called vertices (sometimes also called nodes ). The set of vertices is denoted V and the vertices are indexed from 0 to n - 1 , where n = | V | . The connections between the vertices are unordered pairs called edges . The set of edges is denoted E and m = | E | ≤ n ( n - 1) / 2 . An undirected graph G = ( V, E ) is then composed of a set of vertices V and a set of edges E V × V . If e = { i, j } ∈ E , then i and j are called the endpoints of e, e is said to be incident to i and j , and i and j are said to be adjacent vertices.
ISE 407 Lecture 15 6 More Terminology Let G = ( V, E ) be an undirected graph. A subgraph of G is a graph composed of an edge set E E along with all incident vertices.

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• Fall '13
• TedRalphs

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