Algorithm Design

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Chapter 6 Dynamic Programming
13.
We’d like to place copies of the fries at the servers so as to minimize
the sum of placement and access costs. Formally, we say that a configu-
ration is a choice, for each server Si with i = 1, 2 ..... n - 1, of whether to
place a copy of the file at Si or not. (Recall that a copy is always placed at
Sn.) The total cost of a configuration is the sum of all placement costs for
servers with a copy of the file, plus the sum of all access costs associated
14,
with all n servers.
Give a p olynomial-time algorithm to find a configuration of minimum
total cost.
The problem of searching for cycles in graphs arises naturally in financial
trading applications. Consider a firm that trades shares in n different
companies. For each pair i ~j, they maintain a trade ratio rq, meaning
that one share of i trades for rq shares ofj. Here we allow the rate r to be
fractional; that is, rq = ~ means that you can trade ~ee shares of i to get
two shares of j.
A trading cycle for a sequence of shares ~1, iz ..... ~k consists of
successively trading shares in company il for shares in company ~z, then
shares in company iz for shares i3, and so on, finally trading shares in ik
back to shares in company ~. After such a sequence of trades, one’ends up
with shares in the same company i~ that one starts with. Trading around a
cycle is usually a bad idea, as you tend to end up with fewer shares than
you started with. ]But occasionally, for short periods of time, there are
opportunities to increase shares. We will call such a cycle an opportunity
cycle, if trading along the cycle increases the number of shares. This
happens exactly if the product of the ratios along the cycle is above 1. In
analyzing the state of the market, a firm engaged in trading would like
to know if there are any opportunity cycles.
Give a polynomial-time algorithm that finds such an opportunity
cycle, if one exists.
A large collection of mobile wireless devices can naturally form a network
in which the devices are the nodes, and two devices x and y are connected
by an edge if they are able to directly communicate with each other (e.g.,
by a short-range radio link). Such a network of wireless devices is a highly
dynamlc object, in which edges can appear and disappear over time as
the devices move around. For instance, an edge (x, y) might disappear as x
and y move far apart from each other and lose the ability to communicate
directly.
In a network that changes over time, it is natural to look for efficient
ways of maintaining a path between certain designated nodes. There are
Exercises
two opposing concerns in maintaining such a path: we want paths that are
short, but we also do not want to have to change the path frequently as the
network structure changes. (That is, we’d like a single path to continue
working, if possible, even as the network gains and loses edges.) Here is
a way we might model this problem.
Suppose we have a set of mobile nodes v, and at a particular point in
time there is a set E 0 of edges among these nodes. As the nodes move, the
set of edges changes from E 0 to E~, then to E2, then to E3, and so on, to an
edge set E b. Fir i = 0, 1, 2 ..... b, let G~ denote the graph (V, E~). So if we were
to watch the structure of the network on the nodes V as a "time lapse," it
would look precisely like the sequence of graphs Go, G~, G2 ..... Gb_~, G~.
We will assume that each of these graphs G~ is connected.
Now consider two particular nodes s, t ~ V. For an s-t path P in one
of the graphs Gi, we define the length of P to be simply the number of
edges in P, and we denote this g(P). Our goal is to produce a sequence of
paths P0, P~ ..... P~ so that for each i, Pg is an s-t path in G~. We want the
paths to be relatively short. We also do not want there to be too many
changes--points at which the identity of the path switches. Formally, we
define changes(Po, P~ .....P~) to be the number of indices i (0 < i < b - 1)
for which Pi # P~+I"
Fix a constant K > 0. We define the cost of the sequence of paths
PO, P1 ..... Pb tO be
b
COSt(Po, PI ..... Pb) = ~ f-(Pi) + K. changes(Po, P~ ..... Pb).
(a) Suppose it is possible to choose a single path P that is an s-t path in
each of the graphs Go, G~ .....Gb. Give a polynomial-time algorithm
to find the shortest such path.
(b) Give a polynomial-time algorithm to find a sequence of paths
P0, P~ .....P~ of minimum cost, where P~ is an s-t path in G~ for
i=0,1 .....b.
15. On most clear days, a group of your friends in the Astronomy Department
gets together to plan out the astronomical events ~they’re going to try
observing that night. We’ll make the following assumptions about the
events.
o There are n events, which for simplicity we’ll assume occur in sequence
separated by exactly one minute each. Thus event j occurs
at minute j; if they don’t observe this event at exactly minute j, then
they miss out on it.
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