Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 462 — #470
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Chapter 12
Simple Graphs
Figure 12.13
A 7-node star graph.
12.6.3 Why coloring?
One reason coloring problems frequently arise in practice is because scheduling
conflicts are so common. For example, at the internet company Akamai, cofounded
by Tom Leighton, a new version of software is deployed over each of its servers
(200,000 servers in 2016) every few days. It would take more than twenty years to
update all these the servers one at a time, so the deployment must be carried out
for many servers simultaneouly. On the other hand, certain pairs of servers with
common critical functions cannot be updated simultaneouly, since a server needs
to be taken offline while being updated.
This problem gets solved by making a 200,000-node conflict graph and coloring
it with with a dozen or so colors—so only a dozen or so waves of installs are
needed!
Another example comes from the need to assign frequencies to radio stations. If
two stations have an overlap in their broadcast area, they can’t be given the same
frequency. Frequencies are precious and expensive, it is important to minimize the
number handed out. This amounts to finding the minimum coloring for a graph
whose vertices are the stations and whose edges connect stations with overlapping
areas.
Coloring also comes up in allocating registers for program variables. While a
variable is in use, its value needs to be saved in a register. Registers can be reused
for different variables, but two variables need different registers if they are referenced
during overlapping intervals of program execution. So register allocation is
the coloring problem for a graph whose vertices are the variables: vertices are adjacent
if their intervals overlap, and the colors are registers. Once again, the goal is
to minimize the number of colors needed to color the graph.
Finally, there’s the famous map coloring problem stated in Proposition 1.1.4. The
question is how many colors are needed to color a map so that adjacent territories
get different colors? This is the same as the number of colors needed to color a
graph that can be drawn in the plane without edges crossing. A proof that four