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Unformatted text preview: graph where each vertex is connected to exactly one other vertex forming a connected cycle). One can prove that if N is even, then one needs two colors, and if N is odd, one needs 3 colors. 3. For a planar graph (a graph that can be drawn on a plane without having any of the graph’s edges intersect), as we mentioned before, it was shown that one needs at most 4 colors… How about arbitrary graphs? Unfortunately, one can show that finding out the minimum number of colors for arbitrary graphs cannot be done in an efficient way (unless the graph is relatively small). 5 Recall the problem of finding the shortest path from a starting vertex to all other vertices in a graph (modeling a real‐world artifact)… 6 Last time we discussed a “brute‐force” approach that consists of (1) enumerating all the paths from the starting point, (2) figuring out how much each such path costs, and then (3) for each destination selecting the path with the minimum cost. This approach is not really practical, in terms of how long it would take to do all this for graphs of sizes as small as (say) 20. Our goal in this lecture is to come up with an “efficient” algorithm… 7 The best way to do this is to work on a simple example (such as the one shown above) and see if we can come up with a procedure. To be able to carry out our procedure, we would need some scratch spa...
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This note was uploaded on 02/10/2014 for the course CS 109 taught by Professor Azerbestavros during the Spring '13 term at BU.
- Spring '13