where ( the parameter is often referred to as the "size of the problem,and we can read this as " (nis the time it takes to solve problem of size nnamely steps in the summation functions given aboveit makes sense to use the number of terms in the summation to denote the size of the problem we can then say that the sum of the first integers is bigger instance of the summation problem than the sum of the first because of thisit might seem reasonable that the time required to solve the larger case would be greater than for the smaller case our goal then is to show how the algorithm' execution time changes with respect to the size of the problem computer scientists prefer to take this analysis technique one step further it turns out that the exact number of operations is not as important as determining the most dominant part of the (nfunction in other wordsas the problem gets largersome portion of the (nfunction tends to overpower the rest this dominant term is whatin the endis used for comparison the order of magnitude function describes the part of (nthat increases the fastest as the value of increases order of magnitude is often called big- notation (for "order"and written as ( ( )it provides useful approximation to the actual number of steps in the computation the function (nprovides simple representation of the dominant part of the original (nin the above examplet ( as gets largethe constant will become less and less significant to the final result if we are looking for an approximation for ( )then we can drop the and simply say that the running time is (nit is important to note that the is certainly significant for (nhoweveras gets largeour approximation will be just as accurate without it as another examplesuppose that for some algorithmthe exact number of steps is ( when is smallsay or the constant seems to be the dominant part of the function howeveras gets largerthe term becomes the most important in factwhen is really largethe other two terms become insignificant in the role that they play in determining the final result againto approximate (nas gets largewe can ignore the other terms and focus on in additionthe coefficient becomes insignificant as gets large we would say then that the function (nhas an order of magnitude (nn or simply that it is ( although we do not see this in the summation examplesometimes the performance of an algorithm depends on the exact values of the data rather than simply the size of the problem for these kinds of algorithms we need to characterize their performance in terms of best caseworst caseor average case performance the worst case performance refers to particular data set where the algorithm performs especially poorly whereas different data set for the exact same algorithm might have extraordinarily good performance howeverin most cases the algorithm performs somewhere in between these two extremes (average caseit is important for computer scientist to understand these distinctions so they are not misled by one particular case number of very common order of magnitude functions will come up over and over as you study algorithms these are shown in table in order to decide which of these functions is the dominant part of any (nfunctionwe must see how they compare with one another as gets large figure shows graphs of the common functions from table notice that when is small what is algorithm analysis