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column and thus confirms our analysis, whereas the increasing numbers for T/N suggest that O(N) is an underestimate and the quickly decreasing values for T/N2 suggest that O(M2is an overestimate Note in particular that we do not have definitive convergence One problem is that the clock that we used to time the program ticks only every 10 ms Note also that there is no great difference between O(N) and O(N log N) Certainly an O(N log N) algorithm is much closer to being linear than being quadratic Finally, note that the machine in this example has enough memory to store 640,000 objects (in the case of this experiment, integers) If your machine does not have this much memory, you will not be able to reproduce these results
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Limitations of Big-Oh Analysis
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Worst case is sometimes uncommon and can be safely ignored At other times, it is very common and cannot be ignored
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Average-case analysis is almost always much more difficult than worstcase analysis
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Big-Oh analysis is a very effective tool, but it does have limitations As already mentioned, its use is not appropriate for small amounts of input For small amounts of input, use the simplest algorithm Also, for a particular algorithm, the constant implied by the Big-Oh may be too large to be practical For example, if one algorithm's running time is governed by the formula 2N log Nand another has a running time of IOOON, the first algorithm would most likely be better, even though its growth rate is larger Large constants can come into play when an algorithm is excessively complex They also come into play because our analysis disregards constants and thus cannot differentiate between things like memory access (which is cheap) and disk access (which typically is many thousand times more expensive) Our analysis assumes infinite memory, but in applications involving large data sets, lack of sufficient memory can be a severe problem Sometimes, even when constants and lower order terms are considered, the analysis is shown empirically to be an overestimate In this case, the analysis needs to be tightened (usually by a clever observation) Or the averagecase running time bound may be significantly less than the worst-case running time bound, and so no improvement in the bound is possible For many complicated algorithms the worst-case bound is achievable by some bad input, but in practice it is usually an overestimate Two examples are the sorting algorithms Shellsort and quicksort (both described in 9) However, worst-case bounds are usually easier to obtain than their averagecase counterparts For example, a mathematical analysis of the average-case running time of Shellsort has not been obtained Sometimes, merely defining what average means is difficult We use a worst-case analysis because it is expedient and also because, in most instances, the worst-case analysis is very meaningful In the course of performing the analysis, we frequently can tell whether it will apply to the average case
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In this chapter we introduced algorithm analysis and showed that algorithmic decisions generally influence the running time of a program much more than programming tricks do We also showed the huge difference between the running times for quadratic and linear programs and illustrated that cubic algorithms are, for the most part, unsatisfactory We examined an algorithm that could be viewed as the basis for our first data structure The binary search efficiently supports static operations (ie, searching but not updating), thereby providing a logarithmic worst-case search Later in the text we examine dynamic data structures that efficiently support updates (both insertion and deletion) In 7 we discuss some of the data structures and algorithms included in C++'s STL We also look at some applications of data structures and discuss their efficiency
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