Coloring
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Coloring 3/16/12 1
description
Coloring. Flight Gates. flights need gates, but times overlap. how many gates needed?. time. Airline Schedule. 122 145 67 257 306 99. Flights. Needs gate at same time. Conflicts Among 3 Flights. 145. 306. 99. 257. 122. 145. 67. 306. 99. - PowerPoint PPT Presentation
Transcript of Coloring
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Coloring
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Flight Gates
flights need gates, but times overlap. how many gates needed?
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Airline Schedule
122145 67257306 99
Flights
time
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Conflicts Among 3 Flights
99
145
306
Needs gate at same time
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Model all Conflicts with a Graph
257
67
99
145
306
122
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Color vertices so that adjacent
vertices have different colors.
min # distinct colors needed =
min # gates needed
Color the vertices
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Coloring the Vertices
257, 67122,14599306
4 colors4 gates
assigngates:
257
67
99
145
306
122
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Better coloring
3 colors3 gates
257
67
99
145
306
122
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Final Exams
Courses conflict if student takes both, so need different time slots.
How short an exam period?
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Harvard’s SolutionDifferent “exam group” for every teaching hour. Exams for different groups at different times.
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But This May be Suboptimal• Suppose course A and course B meet
at different times• If no student in course A is also in
course B, then their exams could be simultaneous
• Maybe exam period can be compressed!
• (Assuming no simultaneous enrollment)
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Model as a Graph
CS 20
Psych 1201
Celtic 101
Music 127r
AM 21b
M 9amM 2pmT 9amT 2pm4 time slots
(best possible)
A BMeans A and B have at least one student in common
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Map Coloring
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Planar Four Coloring
any planar map is 4-colorable.1850’s: false proof published (was correct for 5 colors).
1970’s: proof with computer1990’s: much improved
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Chromatic Numbermin #colors for G ischromatic number, χ(G)
lemma:
χ(tree) = 23/16/12
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Pick any vertex as “root.”if (unique) path from root iseven length: odd length:
Trees are 2-colorable
root
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Simple Cycles
χ(Ceven) = 2
χ(Codd) = 3
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Bounded Degreeall degrees ≤ k, implies
very simple algorithm…
χ(G) ≤ k+1
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“Greedy” Coloring…color vertices in any order. next vertex gets a color different from its neighbors. ≤ k neighbors, so k+1 colors always work3/16/12
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coloring arbitrary graphs
2-colorable? --easy to check3-colorable? --hard to check (even if planar)find χ(G)? --theoretically no harder than 3-color, but harder in practice
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Finis
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