Mathematics for Computer Science
Mathematics for Computer Science
MIT 6.042J/18.062J
MIT 6.042J/18.062J
Team Problem 1
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.1
Airline Gate Allocation
October 5, 2005
Lec 5W.2
122
145
Flight 67
Numbers 257
306
99
Lec 5W.3
October 5, 2005
Copyright © Albert R. Meyer, 2005.
Lec 5W.4
Model as a Graph
257
145
306
122
145
67
306
99
Copyright © Albert R. Meyer, 2005.
October 5, 2005
time
Model as a Graph
Needs gate at same time
Copyright © Albert R. Meyer, 2005.
Airline Schedule
Given a set of airline flights needing gates
at overlapping times,
how many different gates do I need in
order to accommodate them?
Copyright © Albert R. Meyer, 2005.
Graph Coloring
99
October 5, 2005
Lec 5W.5
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.6
1
Color vertices
Coloring the Vertices
so that adjacent vertices have
different colors.
257
October 5, 2005
4 gates
Lec 5W.7
Better: 3 Colors
257
145
assign
67 Gates:
306
# colors = # gates needed
Copyright © Albert R. Meyer, 2005.
122
Copyright © Albert R. Meyer, 2005.
257, 67
122,145
99
306
99
October 5, 2005
Lec 5W.8
Final Exam Time Slots
122
Two subjects conflict if a student
is taking both. Assign conflicting
subjects to different time slots,
keeping exam period short.
145
67
306
99
so 3 airline gates will do
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.9
Model as a Graph
6.042
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.10
More Conflicting Allocation Problems
8.02
3.091
4 time slots
(best possible)
Copyright © Albert R. Meyer, 2005.
18.02
assign
times:
6.001
October 5, 2005
M 9am
M 1pm
T 9am
T 1pm
Lec 5W.11
• # separate habitats to house different
species of animals, some incompatible
with others?
• # different frequencies for radio stations
that interfere with each other?
• # different colors to color a map?
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.12
2
Map Coloring
Copyright © Albert R. Meyer, 2005.
Countries are the Vertices
October 5, 2005
Lec 5W.13
Four Color Theorem
October 5, 2005
Lec 5W.15
Trees are 2-colorable
χ(G) = Chromatic Number of G
:= minimum #colors for G
Copyright © Albert R. Meyer, 2005.
Lec 5W.16
χ (Ceven ) = 2
Pick any vertex as “root.”
If (unique) path from root to w is
even length: w
odd length: w
October 5, 2005
October 5, 2005
Simple Cycles
root
Copyright © Albert R. Meyer, 2005.
Lec 5W.14
Chromatic Number
Any planar map is 4-colorable.
False proof published 1850’s
(was correct for 5 colors).
Proof with computer calculations: 1970’s.
Much improved: 1990’s
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Copyright © Albert R. Meyer, 2005.
χ (Codd ) = 3
Lec 5W.17
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.18
3
The Wheel Wn
Complete Graph K5
W5
χ (Kn ) = n
Copyright © Albert R. Meyer, 2005.
October 5, 2005
χ (Wodd ) = 4
Lec 5W.19
If deg(v) ≤ k for all vertices v of G,
then
χ (G ) ≤ k +1
A simple recursive coloring
procedure achieves this.
October 5, 2005
October 5, 2005
Lec 5W.20
Arbitrary Graphs
Bounded Degree
Copyright © Albert R. Meyer, 2005.
Copyright © Albert R. Meyer, 2005.
χ (Weven ) = 3
Lec 5W.21
2-colorable? --easy to check
3-colorable? --hard to check
(even if planar)
χ (G ) ? --harder still
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.25
Team Problems
Problems 2 & 3
Copyright © Albert R. Meyer, 2005.
October 5, 2005
Lec 5W.26
4