the amount of time an algorithm takes to run andrelated to thatthe amount of space an algorithm uses while running typicallycomputer scientists will talk about space/time tradeoff in algorithms sometimes we can achieve faster running time by using more memory butif we use too much memory we can slow down the computer and other running programs the space that is referred to is the amount of ram needed to solve problem the time we are concerned with is measure of how the number of operations grow as the size of the data grows consider function (nthat is description of the running time of an algorithmwhere is the size of the data given to the algorithm as computer scientists we want to study the asymptotic behavior of this function in other wordswe want to study how (nincreases as the value of is natural number representing possible sizes of input data the natural numbers are the set of non-negative integers the definition in sect is re-statement of the big-oh notation definition presented earlier in this big-oh asymptotic upper bound ( ( ) (nd and we write that (nis ( ( ) (( ( ))and we say that is big-oh of the definition of big-oh says that we can find an upper bound for the time it will take for an algorithm to run consider the plot of time versus data size given in fig data sizeor is the axiswhile time is the axis imagine that the green line represents the observed behavior of some algorithm the blue line clearly is an upper bound to the green line after about this is what the definition of big-oh means for whilethe upper bounding function may not be an upper boundbut eventually it becomes an upper bound and stays that way all the way to the limit as approaches infinity butdoes the blue line represent tight bound on the complexity of the algorithm whose running time is depicted by the green linewe' like to know that when we describe the complexity of an algorithm it is truly representational of the actual running time saying that the algorithm runs in ( is accurate even if the algorithm runs in time proportional to because big-oh notation only describes an upper bound if we truly want to say what the algorithm' running time is proportional tothen we need little more power this leads us to our next definition in sect this copy belongs to 'acha