Algorithm Design

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Chapter 6 Dynamic Programming
/3 cannot appear in consecutive minutes. For example, if your job is on
machine A in minute i, and you want to switch to mach~e B, then your
choice for minute i + 1 must be move, and then your choice for minute i + 2
canbe B. The value of a plan is the total number of steps that you manage
to execute over the n minutes: so it’s the sum of ai over all minutes in
which the job is on A, plus the sum of bi over all minutes in which the job
is on B.
bn, find a plan of
maximum value. (Such a strategy will be called optgmal.) Note that your
plan can start with either of the machines A or B in minute 1.
Example. Suppose n = 4, and the values of a~ and bi are given by the
following table.
Minute 1 Minute 2 Minute 3 Minute 4
A 10 1 1 10
B 5 1 20 20
Then the plan of maximum value would be to choose A for:minute 1,
then move for minute 2, and then B for minutes 3 and 4. The value of this
plan would be 10 + 0 + 2O + 20 = 5O.
(a) Show that the following algorithm does not correctly solve this
problem, by giving an instance on which it does not return the correct
In minute I, choose the machine achieving the larger of a I, bl
Set
~hile i < n
What was the choice in minute i--I?
If A:
If hi+ l >ai+ai+l then
Choose moue in minute i and B in minute i+ 1
Proceed to iteration i+ 2
Else
Choose A in minute
Proceed to iteration i+
Endif
If B: behave as above with roles of A and B reversed
EndWhile
11.
12.
Exercises
In your example, say what the correct answer is and also what the
algorithm above finds.
b n and returns the value of an optimal plan.
an and
Suppose you’re consulting for a company that manufactures PC equipment
and ships it to distributors all over the country. For each of the
next n weeks, they have a projected supply s~ of equipment (measured in
pounds), whi4h has to be shipped by an air freight carrier.
Each week’s supply can be carried by one of two air freight companies,
AorB.
Company A charges a fixed rate r per pound (so it costs r- s~ to ship
a week’s supply si).
Company B makes contracts for a fixed amount c per week, independent
of the weight. However, contracts with company B must be made
In blocks of four consecutive weeks at a time.
A schedule, for the PC company, is a choice of air freight Company
(A or B) for each of the n weeks, with the restriction that company B,
whenever it is chosen, must be chosen for blocks of four contiguous
weeks at a 0me. The cost of the schedule is the total amount paid to
company A and B, according to the description above.
Give a polynomial-time algorithm that takes a sequence of supply
Sn and returns a schedule of minimum cost.
Example. Suppose r = 1, c = 10, and the sequence of values is
11, 9, 9, 12, 12, 12, 12, 9, 9, 11.
Then the optimal schedule would be to choose company A for the first
three weeks, then company B for a block of four consecutive weeks, and
then company A for the fInal three weeks.
Suppose we want to replicate a file over a collection of n servers, labeled
Sn. TO place a copy of the file at server Si results in a placement
cost of q, for an integer q > 0.
Now, if a user requests the file from server Si, and no copy of the file is
present at S,, then the servers S~+l, S~+2, S,+3... are searched In order until
a copy of the file is fInally found, say at server S i, where j > i. This results
In an access cost ofj - i. (Note that the lower-indexed servers S~_> S~_2 ....
are not consulted In this search.) The access cost is 0 if S i holds a copy of
the file. We will require that a copy of the file be placed at server Sn, so
that all such searches ~ terminate, at the latest, at
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