Capstone - 03 - Shortest Path Computation

Capstone - 03 - Shortest Path Computation - 1 2 Yet,...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2
Background image of page 2
Yet, another example is the problem of arranging guests attending a wedding around tables. Given a list of “irreconcilable differences” between invitees, you task is to find a seating arrangement that uses the least number of tables while avoiding to have two individuals with “irreconcilable differences” sitting around the same table. This is a graph coloring problem! Namely, we can solve the problem by (1) modeling each individual as a vertex in a graph, (2) representing the fact that two individuals have irreconcilable differences by drawing an edge between the vertices corresponding to th id ii d l d (3) ii ii th b f l (t bl ) d t l th these individuals, and (3) minimizing the number of colors (tables) used to color the vertices (seat the guests) such that no two vertices connected with an edge (two guests with irreconcilable differences) have the same color (are sitting on the same table). Along the same lines, one can use graph vertex coloring to solve other (quite important) problems, including assigning the minimum number of radio frequencies to various radio transmitters who may interfere with one another – here the radio stations would be the vertices, and if two stations interfere with one another (because of geographical proximity) then we draw an edge between them. Now coloring the graph is akin to assigning frequencies since we would not want to give the same frequency (color) to interfering stations (adjacent vertices). Yet, another famous application of graph coloring is “map coloring”. Here our job is to color adjacent states (vertices) on a planar map (e.g., map of the US) with different colors , but use the minimum number of colors. This fairly classical problem was settled by proving that one need no more than 4 colors for such (planar) graphs. 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The attempt on the left shows a coloring (assignment of guests to tables) that uses 4 colors (i.e., we need 4 tables). The one on the right shows a coloring that uses only 3 colors. One can show that for this graph, three is the minimum number of colors. Try to prove it! Accordingly, we would seat A, B, and E on the first (red) table; D, G, and H on the second (green) table; and C, and F on the third (blue) table. 4
Background image of page 4
So, how many colors do we need? Well, the answer is relatively easy for some special graphs: 1. For a complete graph with N nodes (a graph where each pair of vertices are connected with an edge), we would obviously need N colors. One can prove this very easily by contradiction – Assume that one can color a complete graph with less than N colors such that no two adjacent vertices have the same color. Since the h h N ti th it t b th th t t lt t ti h th graph has N vertices, then it must be the case that at least two vertices have the same color. But since there is an edge between any pair of vertices, it follows that two adjacent vertices have the same color, which is a contradiction.
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 38

Capstone - 03 - Shortest Path Computation - 1 2 Yet,...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online