Question

12. Suppose you haven video streams that need to be sent, one after another, over a communication link. Streami consists of a total of b, bits that need to be sent, at a constant rate, over a period of t seconds. You cannot send two streams at the same time, so you need to determine a schedule for the streams: an order in which to send them. Whichever order you choose, there cannot be any delays between the end of one stream and the start of the next. Suppose your schedule starts at time 0 (and therefore ends at time t, whichever order you choose). We assume that all the values bi and are positive integers. Now, because you're just one user, the link does not want you taking up too much bandwidth, so it imposes the following constraint, using a fixed parameter r: () For each natural number t>0, the total number of bits you send over the time interval from 0 to t cannot exceed rt Note that this constraint is only imposed for time intervals that start at 0, not for time intervals that start at any other value. We say that a schedule is valid if it satisfies the constraint () imposed by the link. 194 Chapter 4 Greedy Algorithms The Problem. Given a set of n streams, each specified by its number of bits b, and its time duration t as well as the link parameter r, determine whether there exists a valid schedule. Example. Suppose we have n-3 streams, with (b (2000, 1), (b2, t2) (6000, 2), (b3, (3) (2000,1 and suppose the link's parameter is r 5000. Then the schedule that runs the streams in the order 1,2, 3, is valid, since the constraint () is satis fied: t = 1: the whole first stream! has been sent, and 2000 < 5000-1 t 2: half of the second stream has also been sent and 2000+30005000.2 Similar calculations hold for t 3 and t 4 (a) Consider the following claim: Claim: There exists a valid schedule if and only if each stream i satisfies Decide whether you think the claim is true or false, and give a proof of either the claim or its negation. Give an algorithm that takes a set of n streams, each specified by its number of bits b, and its time duration t, as well as the link parameter r, and determines whether there exists a valid schedule. The running time of your algorithm should be polynomial in n. (b)

Answer 1

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