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9104302 - CS 473(ug Combinatorial Algorithms Fall 2005...

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CS 473 (ug): Combinatorial Algorithms, Fall 2005 Homework 1 Solutions This homework was due on Sep 9, with an automatic extension to Sep 12.
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CS 473 (ug) Homework 1 Solutions Fall 2005 1. Problem 4.12: a) The statement is false. As a counterexample, consider the streams ( b 1 , t 1 ) = (1000 , 1) and ( b 2 , t 2 ) = (100 , 100) with r = 500. The streams can be scheduled (do stream 2 first), but b 1 > rt 1 . b) Notice that the question does not ask for an algorithm that produces a schedule if one exists; you only need an algorithm that outputs whether a schedule exists or not. A simple linear-time solution is to calculate B = n i =1 b i and T = n i =1 t i . If B > rT , no schedule exists. (This follows because after all streams have been scheduled, the total bandwidth used will be B and the time elapsed will be T , and since B > rT , the constraint is violated.) If B rT , we claim that a valid schedule exists. Proof: Consider the schedule in which streams are sorted in increasing order of b/t . After stream i has completed, the total number of bits sent so far is i j =1 b j (call this B i ), and the total time elapsed is i j =1 t j (call this T i ). We can see that B 1 T 1 B 2 T 2 . . . B n T n = B T r , because B i T i is simply the average bit-rate of the first i streams, and since the streams are sorted in increasing order of bit-rates, the average of the first i + 1 streams must be greater than that of the first i streams. But the average after all the streams is r , and so for all i , B i T i r . We show that B i +1 T i +1 B i T i more formally below: B i +1 T i +1 = B i + b i +1 T i + t i +1 B i + t i +1 B i T i T i + t i +1 Because b i +1 t i +1 B i T i = B i T i + t i +1 T i T i + t i +1 = B i T i We have shown that if B rT (we do not violate the constraint at the end), then there is a schedule such that the constraint is not violated after each stream is completed. What about the time while a stream is being sent? If the average bit-rate is r at the beginning and end of the stream, then can it exceed r in the middle of the stream? It’s easy to show that it cannot; here’s an interesting idea that can be converted into a proof with very little effort. Suppose stream i runs from t 1 to t 2 , and at both times t 1 and t 2 , the average bit-rate is r . To show that the bit-rate is r at any time t , t 1 < t < t 2 , simply break stream i into two streams, one called i 1 of length t - t 1 and the other, i 2 , of length t 2 - t . Now we have a new schedule with 1 extra stream; by the argument above we know that the average bit-rate is r after the end of every stream. In particular, at time t , after stream i 1 has been sent, the average bit-rate is r . Another solution to this problem is to sort the streams in increasing order of b/t , and show that if any schedule does not violate the constraint, then this one does not. The argument is essentially similar to the one above; the main difference is that the first method just gives a 1
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CS 473 (ug) Homework 1 Solutions Fall 2005 condition for a schedule to exist, while the second method actually outputs a feasible sched- ule.
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