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Random numbers are used in many applications In this section we discuss a few of the most common ones One important application of random numbers is in program testing Suppose, for example, that we want to test whether a sorting algorithm written in 9 actually works Of course, we could provide some small amount of input, but if we want to test the algorithms for the large data sets they were designed for, we need lots of input Providing sorted data as input tests one case, but more convincing tests would be preferable For instance, we would want to test the program by perhaps running 5000 sorts for inputs of size 1000 To do so requires writing a routine to generate the test data, which in turn requires the use of random numbers Once we have the random number inputs, how do we know whether the sorting algorithm works One test is to determine whether the sort arranged the array in nondecreasing order Clearly, we can run this test in a linear-time sequential scan But how do we know that the items present after the sort are



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the same as those prior to the sort One method is to fix the items in an arrangement of 1 2, N In other words, we start with a random pernzutation of the first N integers A permutation of 1 , 2, N is a sequence of N integers that includes each of 1 , 2, , N exactly once Then, no matter what permutation we start with, the result of the sort will be the sequence 1 2, , N, which is also easily tested In addition to helping us generate test data to verify program correctness, random numbers are useful in comparing the performance of various algorithms The reason is that, once again, they can be used to provide a large number of inputs Another use of random numbers is in simulations If we want to know the average time required for a service system (eg, teller service in a bank) to process a sequence of requests, we can model the system on a computer In this computer simulation we generate the request sequence with random numbers Still another use of random numbers is in the general technique called the randonzi:ed algoritlznz, wherein a random number is used to determine the next step performed in the algorithm The most common type of randomized algorithm involves selecting from among several possible alternatives that are more or less indistinguishable For instance, in a commercial computer chess program, the computer generally chooses its first move randomly rather than playing deterministically (ie, rather than always playing the same move) In this chapter we look at several problems that can be solved more efficiently by using a randomized algorithm





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Even publishing a restricted form with a blank access path can cause con icts InfoPath always infers an access path even when one doesn t exist in the template The inferred path is the physical location of the restricted form being opened So it s recommended that the access path on restricted form templates be properly set to a shared location that is accessible by all of your users

Pseudorandom wmbers many have properties of random numbers Good random-number generators are hard to find

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How are random numbers generated True randomness is impossible to achieve on a computer because any numbers obtained depend on the algorithm used to generate them and thus cannot possibly be random Generally, it is sufficient to produce pseudorandom numbers, or numbers that appear to be random because they satisfy many of the properties of random numbers Producing them is much easier said than done Suppose that we need to simulate a coin flip One way to d o so is to examine the system clock Presumably, the system clock maintains the number of seconds as part of the current time If this number is even, we can return 0 (for heads): if it is odd, we can return 1 (for tails) The problem is that this strategy does not work well if we need a sequence of random numbers One second is a long time, and the clock might not change at all while the program is running, generating all 0s or all Is, which is hardly a random sequence Even if the time were recorded in units of microseconds (or

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