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Unformatted text preview: ine j with no task conﬂicting with task i then
schedule task i on machine j
else
m ← m+1
{add a new machine}
schedule task i on machine m
Algorithm 5.3: A greedy algorithm for the task scheduling problem. 262 Chapter 5. Fundamental Techniques Correctness of Greedy Task Scheduling
In the algorithm TaskSchedule, we begin with no machines and we consider the
tasks in a greedy fashion, ordered by their start times. For each task i, if we have a
machine that can handle task i, then we schedule i on that machine. Otherwise, we
allocate a new machine, schedule i on it, and repeat this greedy selection process
until we have considered all the tasks in T .
The fact that the above TaskSchedule algorithm works correctly is established
by the following theorem. Theorem 5.2: Given a set of n tasks speciﬁed by their start and ﬁnish times, Algorithm TaskSchedule produces a schedule of the tasks with the minimum number
of machines in O(n log n) time. Proof: We can show that the above simple greedy algorithm, TaskSchedule, ﬁnds
an optimal schedule on the minimum number of machines by a simple contradiction
argument.
So, suppose the algorithm does not work. That is, suppose the algorithm ﬁnds
a nonconﬂicting schedule using k machines but there is a nonconﬂicting schedule
that uses only k − 1 machines. Let k be the last machine allocated by our algorithm,
and let i be the ﬁrst task scheduled on k. By the structure of the algorithm, when
we scheduled i, each of the machines 1 through k − 1 contained tasks that conﬂict
with i. Since they conﬂict with i and because we consider tasks ordered by their
start times, all the tasks currently conﬂicting with task i must have start times less
than or equal to si , the start time of i, and have ﬁnish times after si . In other words,
these tasks not only conﬂict with task i, they all conﬂict with each other. But this
means we have k tasks in our set T that conﬂict with each other, which implies
it is impossible for us to schedule all the tasks in T using only k − 1 machines.
Therefore, k is the minimum number of machines needed to schedule all the tasks
in T .
We leave as a simple exercise (R5.2) the job of showing how to implement the
Algorithm TaskSchedule in O(n log n) time.
We consider several other applications of the greedy method in this book, including two problems in string compression (Section 9.3), where the greedy approach gives rise to a construction known as Huffman coding, and graph algorithms
(Section 7.3), where the greedy approach is used to solve shortest path and minimum spanning tree problems.
The next technique we discuss is the divideandconquer technique, which is a
general methodology for using recursion to design efﬁcient algorithms. 5.2. DivideandConquer 5.2 263 DivideandConquer
The divideandconquer technique involves solving a particular computational problem by dividing it into one or more subproblems of smaller size, recursively solving
each subproblem, and then “merging” or “marrying” the solutions to the subproblem(s) to produce a solution to the original problem.
We can model the divideandconquer approach by using a parameter n to denote the size of the original problem, and let S(n) denote this problem. We solve
the problem S(n) by solving a collection of k subproblems S(n1 ), S(n2 ), . . ., S(nk ),
where ni < n for i = 1, . . . , k, and then merging the solutions to these subproblems.
For example, in the classic mergesort algorithm (Section 4.1), S(n) denotes the
problem of sorting a sequence of n numbers. Mergesort solves problem S(n) by
dividing it into two subproblems S( n/2 ) and S( n/2 ), recursively solving these
two subproblems, and then merging the resulting sorted sequences into a single
sorted sequence that yields a solution to S(n). The merging step takes O(n) time.
This, the total running time of the mergesort algorithm is O(n log n).
As with the mergesort algorithm, the general divideandconquer technique
can be used to build algorithms that have fast running times. 5.2.1 DivideandConquer Recurrence Equations
To analyze the running time of a divideandconquer algorithm we utilize a recurrence equation (Section 1.1.4). That is, we let a function T (n) denote the running
time of the algorithm on an input of size n, and characterize T (n) using an equation
that relates T (n) to values of the function T for problem sizes smaller than n. In
the case of the mergesort algorithm, we get the recurrence equation
T (n) = b
if n < 2
2T (n/2) + bn if n ≥ 2, for some constant b > 0, taking the simplifying assumption that n is a power of 2.
In fact, throughout this section, we take the simplifying assumption that n is an
appropriate power, so that we can avoid using ﬂoor and ceiling functions. Every
asymptotic statement we make about recurrence equations will still be true, even if
we relax this assumption, but justifying this fact formally involves long and boring
proofs. As we observed above, we can show that T (n) is O(n log n) in this case. In
general, however, we will possibly get a...
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