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1 The most common error in simulation is using a poor model A simulation is only as good as the accuracy of its random input
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Both examples in this chapter are available online
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Josephuscpp Contains both implementations of j osephus and a main to test them Modemscpp Contains the code for the modem bank simulation
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141 If M = 0, who wins the Josephus game
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Show the operation of the Josephus algorithm in Figure 143 for the case of seven people with three passes Include the computation of r a n k and a picture that contains the remaining elements after each iteration Are there any values of M for which player 1 wins a 30-person Josephus game Show the state of the priority queue after each of the first 10 lines of the simulation depicted in Figure 144
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Let N = 2"or any integer k Prove that if M is 1, then player 1 always wins the Josephus game Let J(N) be the winner of an N-player Josephus game with M = 1 Show that a if N is even then J(h = 2 J ( N l 2 ) - I ) b if N is odd and J ( r ~ 1 2 1 # 1 then J(N) = 2J(rNl 21) - 3 c if N is odd and J ( r N 121) = I , then J ( N ) = N Use the results in Exercise 146 to write an algorithm that returns the winner of an N-player Josephus game with M = I What is the running time of your algorithm' Give a general formula for the winner of an N-player Josephus game with M = 2 Using the algorithm for N = 20, determine the order of insertion into
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the~inar~search~ree~ith~ank
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Suppose that the Josephus algorithm shown in Figure 142 is implemented with a v e c t o r instead of a 1i s t a If the change worked, what would be the running time b The change has a subtle error What is the problem and how can it be fixed In the Josephus algorithm shown in Figure 142, why can't we l replace lines 27 and 28 with the single assignment n e x t = i t r + Write a program that solves the historical version of the Josephus problem Give both the linked list and search tree algorithms Implement the Josephus algorithm with a queue Each pass of the potato is a dequeue followed by an enqueue
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1414 Rework the simulation so that the clock is represented as a double the time between dial-in attempts is modeled with a negative exponential distribution, and the connect time is modeled with a negative exponential distribution 1415 Rework the modem bank simulation so that Event is an abstract base class and DialInEvent and HangUpEvent are derived classes The Event class should store a pointer to a ModemSim object as an additional data member, which is initialized on construction It should also provide an abstract method named doEvent that is implemented in the derived classes and that can be called from runsim to process the event
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1416 Implement the Josephus algorithm with splay trees (see 22) and sequential insertion (The splay tree class is available online, but it will need a findKth method) Compare the performance with that in the text and with an algorithm that uses a linear-time, balanced tree-building algorithm 1417 Rewrite the Josephus algorithm shown in Figure 143 to use a median heap (see Exercise 719) Use a simple implementation of the median heap; the elements are maintained in sorted order Compare the running time of this algorithm with the time obtained by using the binary search tree 1418 Suppose that FIU has installed a system that queues phone calls when all modems are busy Rewrite the simulation routine to allow for queues of various sizes Make an allowance for an infinite queue 1419 Rewrite the modem bank simulation to gather statistics rather than output each event Then compare the speed of the simulation, assuming several hundred modems and a very long simulation, with some other possible priority queues (some of which are available online)-namely, the following a An asymptotically inefficient priority queue representation described in Exercise 714 b An asymptotically inefficient priority queue representation described in Exercise 715 c Splay trees (see 22) d Skew heaps (see 23) e Pairing heaps (see 23)
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