DEFINITION:
The routing problem are concerned with
finding ways to route the delivery of goods and/or services to an
assortment of destinations. Examples of goods in question are
packages, mail, newspapers; examples of services are police
protection, Internet access, garbage collection; examples of
destinations are houses, computer terminals, towns etc. In
addition, proper routes must satisfy what we will call the routes
of the road:
i)
if there is a “direction of traffic” (as in one way
streets, pipeline flows, communication protocols),
then the direction of traffic must be followed, and
ii) if there is not direct way to get from destination X to
destination Y, then a proper route cannot go directly
from point X to point Y.
OBS: Two fundamental questions can come up in a routing
problem:
1) Is there a proper route for the particular problem?
2) If there is more than one possible route, which one is
the best? (Where best is a function of some
predetermined variable such as cost, distance, time etc)
Example 1 (page 159): The Walking Patrolman
A private security guard is hired to patrol on foot the streets of
the small neighborhood shown in the figure below. He is being paid for
just one walk-through and is anxious to get the job done and go home.
He has two questions he would like answered: (1) is there any route that
allows him to walk through every block exactly once (starting and
ending at the corner S where he parked his car)? (2) if not, what is the
most efficient possible way to walk the neighborhood (starting and
ending at S)? Here, efficiency is measured in total number of blocks
walked.
Example 2 (page 159): The Walking Mail Carrier
Consider now the problem of a mail carrier, who has exactly the
same neighborhood as the security guard (see the above figure) as his
designated mail delivery area. The big difference is that for those blocks
in which there are homes on both sides of the street the mail carrier
must walk through the block twice (he does each side of the street
separately). The carrier asks two similar questions: (1) starting at P.O.,
can he cover every sidewalk along which there are homes once and only
once, ending his walk back at P.O.? (2) if this cannot be done, what is
the most efficient way to deliver the mail throughout the neighborhood?
Example 3 (page 160): The Seven Bridges of Königsberg
Our story begins more than 250 years ago in the medieval town of
Königsberg, in Eastern Europe. Königsberg was divided by the river
Pregel into four separate land areas which were connected to each other
by seven bridges (see the figure from below). A prize (7 gold coins) is
offered to the first person who can
find a way to walk across each one of
the 7 bridges of Königsberg without
recrossing any and return to the
original point. A smaller prize (5 gold
coins) is offered to the first person
who can across each of the 7 bridges
exactly once without necessarily
returning to the original starting
point. So far, no one has collected on
either prize. How come?
Example 4 (page 160): The Bridges of Madison County
This is a modern version of the previous example. A beautiful
river runs through Madison County as shown in the picture from below.
A photographer is hired to take picture of each of the 11 bridges for a
national magazine. He needs to drive across each bridge once for a
photo shoot, and since there is a $25 toll (the locals call it “maintenance
fee”) every time an out-of-town visitor drives across a bridge, the
photographer wants to minimize the total cost of his trip and to recross
bridges only if it is absolutely necessary. What is the best (cheapest)
route for him to follow?
Example 5 (page 161): Unicursal Tracings
Can we trace each of the drawing shown below without lifting the
pencil or retracing any of the lines, and end in the same place we
started? What if we are not required to end back at the starting place?