Algorithm Design

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Chapter 6 Dynamic Programming
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they need to pay a fixed price P for delivery in addition to the cost of the
gas ordered. However, it costs c to store a gallon of gas for an extra day,
so ordering too much ahead increases the storage cost.
They are planning to close for a week in the winter, and they want
their tank to be empty by the time they close. Luckily, based on years of
experience, they have accurate projections for how much gas they will
need each day tmtfl this point in time. Assume that there are n days left
unti! they close, and they need gt gallons of gas for each of the days
i = 1 ..... n. Assume that the tank is empty at the end of day 0. Give an
algorithm to decide on which days they should place orders, and how
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much to order so as to minimize their total cost.
Recall the scheduling problem from Section 4.2 in which we sought to
~ze the maximum lateness. There are r~ jobs, each with a deadline
dt and a required processing time tt, and all jobs are available to be
scheduled starting at l~me s. For a job i to be done, it needs tO be assighed
a period from st >_ s to ft = st + tt, and different jobs should be assigned
nonoverlapping intervals. As usual, such an assignment of times will be
called a schedule.
In this problem, we consider the same setup, but want to optimize a
different objective. In particular, we consider the case in which each job
must either be done by its deadline or not at all. We’ll say that a subset J of
the jobs is schedulable if there is a schedule for the jobs in J so that each
of them finishes by its deadline. Your problem is to select a schedulable
subset of maximum possible size and give a schedule for this subset that
allows each job to finish by its deadline.
(a) Prove that there is an optimal solution J (i.e., a schedulable set of
(b)
maximum size) in which the jobs in J are scheduled in increasing
order of their deadlines.
Assume that a~ deadlines dt and required times tt are integers. Give
an algorithm to find an optimal solution. Your algorithm should
run in time polynomial in the number of jobs n, and the maximum
deadline D = rnaxt dr.
Let G : (V, E) be a graph with n nodes in which each pair of nodes is
joined by an edge. There is a positive weight tvq on each edge (i,j); and
we will assume these weights satisfy the triangle inequality tvtk < tvq + tvjk.
For a subset V’ ___ V, we will use G[V’] to denote the subgraph (with edge
weights) induced on the nodes in V’.
We are given a set X c V of k terminals that must be connected by
edges. We say that a Stein~r tree on X is a set Z so that X __ Z _ V, together
Notes and Further Reading
with a spanning subtree T of G[Z]. The weight of the Steiner tree is the
weight of the tree T.
Show that there is function f(.) and a polynomial function p(.) so that
the problem of finding a ~um-weight Steiner tree on X can be solved
in time O(f(k). p(n)).
Notes and Further Reading
Richard Bellman is credited with pioneering the systematic study of dynamic
programming (Bellman 1957); the algorithm in this chapter for segmented least
squares is based on Bellman’s work from this early period (Bellman 1961).
Dynamic programming has since grown into a technique that is widely used
across computer science, operations research, control theory, and a number
of other areas. Much of the recent work on this topic has been concerned with
stochastic dynamic programming: Whereas our problem formulations tended
to tacitly assume that al! input is known at the outset, many problems in
scheduling, production and inventory planning, and other domains involve
uncertainty, and dynamic programming algorithms for these problems encode
this uncertainty using a probabilistic formulation. The book by Ross (1983)
provides an introduction to stochastic dynamic programming.
Many extensions and variations of the Knapsack Problem have been
studied in the area of combinatorial optimization. As we discussed in the
chapter, the pseudo-polynomial bound arising from dynamic programming
can become prohibitive when the input numbers get large; in these cases,
dynamic programming is often combined with other heuristics to solve large
instances of Knapsack Problems in practice. The book by Martello and Toth
(1990) is devoted to computational approaches to versions of the Knapsack
Problem.
Dynamic programming emerged as a basic technique in computational biology
in the early 1970s, in a flurry of activity on the problem of sequence
comparison. Sankoff (2000) gives an interesting historical account of the early
work in this period. The books by Waterman (1995) and Gusfield (1997) provide
extensive coverage of sequence alignment algorithms (as well as many
related algorithms in computational biology); Mathews and Zuker (2004) discuss
further approaches to the problem of RNA secondary structure prediction.
The space-efficient algorithm for sequence alignment is due to Hirschberg
(1975).
The algorithm for the Shortest-Path Problem described in this chapter is
based originally on the work of Bellman (1958) and Ford (1956). Many optimizations,
motivated both by theoretical and experimental considerations,
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