Spring 2015 Mathematics in Management Science Network Problems Networks & Trees Minimum Networks Spanning Trees Minimum Spanning Trees

Recap Have looked at apps where need to efficiently: traverse collection of edges (Euler circuits) visit specific points (Hamiltonian circuits) Next study routing epitomized by wiring problems (electrical, phone, network, etc.) as well as plumbing and even “phone trees”.

Network Routing Next study routing epitomized by wiring problems (electrical, phone, network, etc.) as well as plumbing and even “phone trees”. Here want “connections” between every pair of vertices, but do not need direct connections—indirect is fine; to go from A to B ok to go via C,D,…).

Internet Example To your friend, do not need a wire directly connecting your device to theirs—your communication can be routed. Cost to route traffic is minimal, but physical connections are expensive. So, want to minimize # of physical connections, keeping everyone connected to everyone else.

Other Examples Electrical wiring. Water supply. These are examples of distribution systems: do not need direct connection from source to every location. (In wiring these direct connections are called “home runs”.) In all these examples need a spanning tree.

Data Lines Example Must install new data lines connecting the buildings on a university campus. All buildings must connect to network, but not necessarily to each other. Want to do this efficiently; that is, with minimum cost.

Graph Model Construct a graph with the properties: one vertex for each building an edge links two buildings if it is possible to install a line between them add weights to each edge for the cost of laying this line

Example: 12 bldgs and costs of laying lines between them

Kruskal’s Algorithm Use the edges, in order of increasing cost, so that all vertices on some added edge (when done) no circuits get formed connected (when done)

Example

Example

Networks & Trees A network is a connected graph. A tree is a network which contains no circuits. Given a network, we’ll search for subgraphs which are spanning trees.

Trees?

Trees?

Trees A tree is a connected graph with no circuits. A graph is a tree if and only if there is a unique path connecting any two vertices.

Tree Equivalences These are equivalent notions: There is a unique path between each pair of vertices in G. Every edge of G is a bridge. G is a network and # of edges = # vtxs -1. G is a tree.

Trees & Spanning Trees A tree is a network (a connected graph) which contains no circuits. We’ll look for subgraphs of a given network which are trees. Such a tree which contains all of the vertices of original network is called a spanning tree.

Spanning Trees A spanning tree of a graph: is a subgraph (uses only edges & vertices of the original graph) is a tree (no loops) includes all original vtxs (it spans)

Examples of Spanning Trees Two different spanning trees (green) for the same graph:

Spanning Trees A spanning tree gives minimal connectivity: Adding any edge will create a circuit (introducing redundancy). Removing any edge will destroy the connectivity.

Why want Spanning Trees? Since spanning, it includes all vertices. Since connected, can get from any vtx A to any other vtx B. No circuits (part of being a tree) means no duplicate paths (only one way to get from A to B), so no excess cost.

Examples of subgraphs (blue wiggly lines) of graph (a)

(b) is a tree but not spanning; (c) not a tree- --it is disconnected; (d) not a tree---contains a circuit; (e) is the only spanning tree

Minimal Cost Spanning Trees Of all possible spanning trees, the one with the smallest total weight is called a minimum spanning tree. First Network Problem Given weighted network, find a minimum spanning tree.

Example of an MST

Minimal Cost Spanning Trees Of all possible spanning trees, the one with the smallest total weight is called a minimum spanning tree. First Network Problem Given weighted network, find a minimum spanning tree.

Kruskal’s Algorithm Use the edges, in order of increasing cost, so that no circuits get formed all vertices on some added edge (when done) connected (when done)

Example Example Apply KA to wtd graph blo

Application of Kruskal’s Algorithm

Optimality of Kruskal’s Algorithm Optimality of Kruskal’s Algorithm Kruskal’s Algorithm always yields an optimal solution to the Minimum Spanning Tree Problem. In contrast, the similar Sorted Edges Algorithm (used to attack the Traveling Salesman Problem) does not always yield the optimal solution.

Counting Spanning Trees The network in (a) has 8 edges, 5 of which are bridges. There are 3 different spanning trees pictured in (b), (c), and (d).