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Under these circumstances, we say that the code (or program or algorithm) runs in time O(6N + 5) This is usually read as big-O of 6N + 5 We need not know what the constant c will be In fact, it will undoubtedly be different for different computers, but we must know that there is one such c for any reasonable computer system If the computer is very fast, the c might be less than 1 say, 0001 If the computer is very slow, the c might be very large say, 1,000 Moreover, since changing the units (say from nanosecond to second) only involves a constant multiple, there is no need to give any units of time Be sure to notice that a big-O estimate is an upper-bound estimate We always approximate by taking numbers on the high side rather than the low side of the true count Also notice that when performing a big-O estimate, we need not determine an exact count of the number of operations performed We only need an estimate that is correct up to a constant multiple If our estimate is twice as large as the true number, that is good enough An order-of-magnitude estimate, such as the previous 6N + 5, contains a parameter for the size of the task solved by the algorithm (or program or piece of code) In our
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sample case, this parameter N was the number of array elements to be searched Not surprisingly, it takes longer to search a larger number of array elements than it does to search a smaller number of array elements Big-O running-time estimates are always expressed as a function of the size of the problem In this chapter, all our algorithms will involve a range of values in some container In all cases N will be the number of elements in that range The following is an alternative, pragmatic way to think about big-O estimates: Only look at the term with the highest exponent and do not pay attention to constant multiples For example, all of the following are O(N 2): N 2 + 2N + 1, All of the following are O(N 3): N 3 + 5N 2 + N + 1, 8N 3 + 7, 100N 3 + 4N + 1 3N 2 + 7, 100N 2 + N
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These big-O running-time estimates are admittedly crude, but they do contain some information They will not distinguish between a running time of 5N + 5 and a running time of 100N, but they do let us distinguish between some running times and so determine that some algorithms are faster than others Look at the graphs in Display 1915 and notice that all the graphs for functions that are O(N) eventually fall below the graph for the function 05N 2 The result is inevitable: An O(N) algorithm will always run faster than any O(N 2) algorithm, provided we use large enough values of N Although an O(N 2) algorithm could be faster than an O(N) algorithm for the problem size you are handling, programmers have found that, in practice, O(N) algorithms perform better than O(N) algorithms for most practical applications that are intuitively large Similar remarks apply to any other two different big-O running times Some terminology will help with our descriptions of generic algorithm running times Linear running time means a running time of T(N) = aN + b A linear running time is always an O(N) running time Quadratic running time means a running time with a highest term of N 2 A quadratic running time is always an O(N 2) running time We will also occasionally have logarithms in running-time formulas Those normally are given without any base, since changing the base is just a constant multiple If you see log N, think log base 2 of N, but it would not be wrong to think log base 10 of N Logarithms are very slow growing functions So, a O(log N) running time is very fast In many cases, our running-time estimates will be better than big-O estimates In particular, when we specify a linear running time, that is a tight upper bound and you can think of the running time as being exactly T(N) = cN, although the c is still not speci ed
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