algorithm analysis function (nindicates the rate of growth at which the run time of an algorithm increases as the input sizenincreases to specify the time-complexity of an algorithmwhich runs on the order of ( )we use the notation of (nconsider the two versions of our algorithm from earlier for version onethe time was computed to be ( if we let then < for result of ( for version twowe computed time of (nn againif we let then < for result of ( in this casethe choice of comes from the observation that when > we have < and < which satisfies the equation in the definition of big- the function (nn is not the only choice for satisfying the condition ( <cf (nwe could have said the algorithms had run time of ( or ( since the objectivehoweveris to find function (*that provides the tightest (lowestupper bound or limit for the run time of an algorithm the big- notation is intended to indicate an algorithm' efficiency for large values of there is usually little difference in the execution times of algorithms when is small constant of proportionality the constant of proportionality is only crucial when two algorithms have the same (nit usually makes no difference when comparing algorithms whose growth rates are of different magnitudes suppose we have two algorithmsl and with run times equal to and respectively has time-complexity of ( with and has time of (nwith even though has smaller constant of proportionalityl is still slower andin fact an order of magnitude slowerfor large values of thusf (ndominates the expression cf (nand the run time performance of the algorithm the differences between the run times of these two algorithms is shown numerically in table and graphically in figure constructing (ninstead of counting the number of logical comparisons or arithmetic operationswe evaluate an algorithm by considering every operation for simplicitywe assume that each basic operation or statementat the abstract leveltakes the same amount of time andthuseach is assumed to cost constant time the total number of