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Capstone - 03 - Shortest Path Computation

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

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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 i di id l d (3) i i i i 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 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

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