time span they describe hence the following table gives time spans rather than instruction countsbased on the assumption that we have computer which can operate at speed of mipwhere one mip million instructions per secondf ( log log log nlog = usec usec usec usec usec usec usec usec usec usec usec usec usec usec msec msec usec usec usec usec msec msec sec yr usec usec usec msec msec sec min yr usec usec usec sec sec wk yr yr it is clear thatas the sizes of the problems get really bigthere can be huge differences in the time it takes to run algorithms from different complexity classes for algorithms with exponential complexityo( )even modest sized problems have run times that are greater than the age of the universe (about yr)and current computers rarely run uninterrupted for more than few years this is why complexity classes are so important they tell us how feasible it is likely to be to run program with particular large number of data items typicallypeople do not worry much about complexity for sizes below or maybe but the above numbers make it clear why it is worth thinking about complexity classes where bigger applications are concerned another useful way of thinking about growth classes involves considering how the compute time will vary if the problem size doubles the following table shows what happens for the various complexity classesf ( log log log nlog if the size of the problem doubles then (nwill be the samef ( nf (nalmost the samelog (log ( )log (log ( more by log ( nf ( twice as big as beforef ( (na bit more than twice as big as before nlog ( (nlog four times as big as beforef ( (neight times as big as beforef ( (nthe square of what it was beforef ( ( ( )) this kind of information can be very useful in practice we can test our program on problem that is half or quarter or one eighth of the full sizeand have good idea of how long we will have to wait for the full size problem to finish moreoverthat estimate won' be affected by any constant factors ignored in computing the growth classor the speed of the particular computer it is run on the following graph plots some of the complexity class functions from the table note that although these functions are only defined on natural numbersthey are drawn as though they were defined for all real numbersbecause that makes it easier to take in the information presented