computer ten times faster yetthen the new computer ( times faster than the original computerwill only run problem of size if you had second program whose growth rate is and for which the original computer could run problem of size in an hourthan machine ten times faster can run problem only of size in an hourthusan exponential growth rate is radically different than the other growth rates shown in figure the significance of this difference is explored in instead of buying faster computerconsider what happens if you replace an algorithm whose running time is proportional to with new algorithm whose running time is proportional to log in the graph of figure fixed amount of time would appear as horizontal line if the line for the amount of time available to solve your problem is above the point at which the curves for the two growth rates in question meetthen the algorithm whose running time grows less quickly is faster an algorithm with running time (nn requires time steps for an input of size an algorithm with running time (nn log requires time steps for an input of size which is an improvement of much more than factor of ten when compared to the algorithm with running time (nn because log whenever if the typical problem size is larger than for this examplethen you would be much better off changing algorithms instead of buying computer ten times faster furthermorewhen you do buy faster computeran algorithm with slower growth rate provides greater benefit in terms of larger problem size that can run in certain time on the new computer asymptotic analysis despite the larger constant for the curve labeled in figure the curve labeled crosses it at the relatively small value of what if we double the value of the constant in front of the linear equationas shown in the graphthe curve labeled is surpassed by the curve labeled once the additional factor of two for the linear growth rate does not much matterit only doubles the -coordinate for the intersection point in generalchanges to constant factor in either equation only shift where the two curves crossnot whether the two curves cross when you buy faster computer or faster compilerthe new problem size that can be run in given amount of time for given growth rate is larger by the same factorregardless of the constant on the running-time equation the time curves for two algorithms with different growth rates still crossregardless of their running-time equation constants for these reasonswe usually ignore the constants