coloring 3/16/121. flight gates flights need gates, but times overlap. how many gates needed?...

1

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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113/16/12

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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 B

Means 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 Number

min #colors for G is

chromatic 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 Degree

all 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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