What You Will Learn • Trees • Spanning Trees • Kruskal’s Algorithm
Tree • A tree is a connected graph in which each edge is a bridge.
Examples Not Trees • Trees
Spanning Tree • A spanning tree is a tree that is created from another graph by removing edges while still maintaining a path to each vertex.
Example 3: A Spanning Tree Problem • Schoolcraft College is considering adding awnings above its sidewalks to help shelter students from the snow and rain while they walk between some of the buildings on campus. A diagram of the buildings and the connecting sidewalks where the awnings are to be added is on the next slide.
Example 3: A Spanning Tree Problem • Originally, the president of the college wished to have awnings placed over all the sidewalks shown, but that was found to be too costly. Instead, the president has proposed to place just enough awnings over a select number of sidewalks so that, by moving from building to building, students would still
Example 3: A Spanning Tree Problem • be able to reach any location shown without being exposed to the elements. • a) Represent all the buildings and sidewalks shown with a graph. • b) Create three different spanning trees from this graph that would satisfy the president’s proposal.
Example 3: A Spanning Tree Problem • Solution • a) Using letters to represent the building names, vertices to represent the buildings, andedges to representthe sidewalksbetween buildings, wegenerate this graph.
Example 3: A Spanning Tree Problem • Solution • b) To create a spanning tree we remove nonbridge edges until a tree is created.
Example 3: A Spanning Tree Problem • Solution • b) Here are two more possibilities.
Minimum-cost spanning tree • A minimum cost spanning tree is the least expensive spanning tree of all spanning trees under consideration.
Kruskal’s Algorithm • To construct the minimum-cost spanning tree from a weighted graph: • 1. Select the lowest-cost edge on the graph. • 2. Select the next lowest-cost edge that does not form a circuit with the first edge. • 3. Select the next lowest-cost edge that does not form a circuit with the previously selected edges.
Kruskal’s Algorithm 4. Continue selecting the lowest-cost edges that do not form circuits with the previously selected edges. 5. When a spanning tree is complete, you have the minimum-cost spanning tree.
Example 7 • Schools in Budville, Fairplay, Happy Corners, Kieler, Louisburg, and Sinsinawa, Wisconsin, all wish to establish a fiber-optic computer network to share information and to obtain Internet access. The most efficient method of establishing such a network would be to install fiber-optic cable along roadsides.
Example 7 • The weighted graph shows the distance in miles between schools along existing roads.
Example 7 • a) Determine the shortest distance to link these six schools. • b) The cost to install fiber-optic cable is $1257 per mile. What is the minimum cost to install the fiber-optic cable along the roadsides determined in part (a)?
Example 7 • Solution • a) We are seeking the minimum-cost spanning tree. • Use Kruskal’s algorithm. • Select edge HB, 1 mi; • edge BL, 1.5 mi; • edge FS, 2 mi
Example 7 Solution Selecting HL, 2.5 mi creates a circuit between H, B, L; so we must select LF, 3 mi. Finally, select edge KL, 3.5 mi.
Example 7 • Solution • According to Kruskal’s algorithm, this figure shows the minimum-cost spanning tree. • Place the fiber-optic cable along this path.
Example 7 • Solution • b) From the figure, there are • 1 + 1.5 + 3.5 + 3 + 2 = 11miles of fiber-opticcable needed. • At $1257 per mile,the cost is • $1257 × 11 = $13,827.