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can essentially ignore it The computer is much more likely to break Your job is secure Instead of using a shuffling technique, you can achieve the same result by choosing the pivot randomly instead of deterministically Take a random item in the array and swap it with the item in position l o w Take another random item and swap it with the item in position high Take a third random item and swap it with the item in the middle position Then continue as usual As before, degenerate partitions are always possible, but they now happen as a result of bad random numbers, not bad inputs Let us look at the differences between randomized and nonrandomized algorithms So far we have concentrated on nonrandomized algorithms When calculating their average running times, we assume that all inputs are equally likely This assumption does not hold, however, because nearly sorted input, for instance, occurs much more often than is statistically expected This situation can cause problems for some algorithms, such as quicksort But when we use a randomized algorithm, the particular input is no longer important The random numbers are important, and we get an expected running time, in which we average over all possible random numbers for any particular input Using quickselect with random pivots (or a shuffle preprocessing step) gives an O(N) expected time algorithm That is, for any input, including already sorted input, the running time is expected to be O(N), based on the statistics of random numbers On the one hand an expected time bound is somewhat stronger than an average-case time bound because the assumptions used to generate it are weaker (random numbers versus random input) but it is weaker than the corresponding worst-case time bound On the other hand, in many instances solutions that have good worst-case bounds frequently have extra overhead built in to assure that the worst case does not occur The O(N)worst-case algorithm for selection, for example, is a marvelous theoretical result but is not practical Randomized algorithms come in two basic forms The first, as already shown, always gives a correct answer but it could take a long time, depending on the luck of the random numbers The second type is what we examine in the remainder of this chapter Some randomized algorithms work in a fixed amount of time but randomly make mistakes (presumably with low probability) called false positives or false negatives This technique is commonly accepted in medicine False positives and false negatives for most tests are actually fairly common, and some tests have surprisingly high error rates Furthermore, for some tests the errors depend on the individual, not random numbers, so repeating the test is certain to produce another false result In randomized algorithms we can rerun the test on the same input using different random numbers If we run a randomized algorithm 10 times and get 10 positives-and if a single false positive is an unlikely occurrence.

Some randomized algorithms work in a fixed amount of time but randomly make mistakes(presumab1y with low probability) These mistakes are false positives or false negatives

What is the process for getting recommended changes implemented on your site Beyond knowing the people involved,

(say, 1 chance in 100)-the probability of 10 consecutive false positives ( 1 chance in 1001 or one hundred billion billion) is essentially zero

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skill in timber framing, like the large Dutch barns found in many German communities. In chapter 15, we dispel the common belief that the Town lattice is simpler to build than a Burr arch-truss. Louis Wernwag (1770 1843), who was born in Germany, and Timothy Palmer (1751 1821) of Newburyport, Massachusetts, built many great timber bridges. Their bridges varied in design and construction, each being designed for a speci c span, using a one-of-a-kind trussed arch. Because of their complexity, a pedestrian bridge built after the designs of Wernwag and Palmer was judged impractical for this project. For these reasons, it was decided that the Burr arch-truss was the logical choice for a covered bridge at Old Salem.

Trial division is the simplest algorithm primality testing It is fast for small (32-bit) numbers but cannot be used for larger numbers

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