comparing growth rates to sum uptable showsin ordereach of the seven common functions used in algorithm analysis constant logarithm log linear -log- log quadratic cubic exponential an table classes of functions here we assume that is constant (nideallywe would like data structure operations to run in times proportional to the constant or logarithm functionand we would like our algorithms to run in linear or -log- time algorithms with quadratic or cubic running times are less practicaland algorithms with exponential running times are infeasible for all but the smallest sized inputs plots of the seven functions are shown in figure exponential cubic quadratic -log- linear logarithmic constant figure growth rates for the seven fundamental functions used in algorithm analysis we use base for the exponential function the functions are plotted on log-log chartto compare the growth rates primarily as slopes even sothe exponential function grows too fast to display all its values on the chart the ceiling and floor functions one additional comment concerning the functions above is in order when discussing logarithmswe noted that the value is generally not an integeryet the running time of an algorithm is usually expressed by means of an integer quantitysuch as the number of operations performed thusthe analysis of an algorithm may sometimes involve the use of the floor function and ceiling functionwhich are defined respectively as followsxthe largest integer less than or equal to xthe smallest integer greater than or equal to