!" !"#"! $%& % # of bitstime' ( %% #) )***********************+' ( %% #) )**** *********** **********+' ( %% #) )**** ********* *** *** ********+' ( %% #) )**** ******* *** *** * *** ********+' ( %% #) )**** ***** *** *** * *** * *** ********+' ( %% #) )**** * *** * *** * *** * *** * *** * *** * *** * *** * *** ****+* * + , % %%*** * *** * *** * *** * *** * *** * *** * *** * *** * *** +* * * *** T(n) = The amount of time grade school addition uses to add two n-bit numbersWhat do you mean by “time”?$%)- .../ ( - %,, , %% , 0– Processor speed– Instruction set– Disk speed– Brand of compiler & 1 ( %, 2 % % % ( - ) % %% ( *Hold on! You just admitted that it makes no sense to measure the time, T(n), taken by the method of grade school addition since the time depends on the implementation details. We will have to speak of the time taken by a particular implementation, as opposed to the time taken by the method in the abstract.Don’t jump to conclusions!Your objections are serious, but not insurmountable. There is a very nice sense in which we can analyze grade school addition without ever having to worry about implementation details.Here is how it works . . .On any reasonable computer adding 3 bits and writing down the two bit answer can be done in constant time. Pick any particular computer A and define c to be the time it takes to perform on that computer. Total time to add two n-bit numbers using grade school addition: cn[c time for each of n columns]Implemented on another computer B the running time will be c’n where c’ is the time it takes to perform on that computer. Total time to add two n-bit numbers using grade school addition: c’n[c’ time for each of n columns]The fact that we get a line is invariant under changes of implementations. Different machines result in different slopes, but time grows linearly as input size increases. # of bits in numberstimeMachine A: cnMachine B: c’nThus we arrive at an implementation independent insight: Grade School Addition is a linear timealgorithm.Determining the growth rate of the resource curve as the problem size increases is one of the fundamental ideas of computer science.I see! We can define away the details of the world that we do not wish to currently study,in order to recognize the similarities between seemingly different things.. AbstractionAbstraction: : Abstract away the inessential Abstract away the inessential features of a problem or solutionfeatures of a problem or solution=TIME vs INPUT SIZEFor any algorithm, defineINPUT SIZE = # of bits to specify inputs,DefineTIMEn= the worst-case amount of time used on inputs of size n.We Often Ask:What is the GROWTH RATE of Timen?' ( #) )*X* * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * * * * * * * * * *n2How to multiply 2 nHow to multiply 2 n--bit numbers.bit numbers.X* * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * * * * * * * * * *n2I get it! The total time is bounded by cn2.& % /%%0 & % 0 3 % 4 ( % %,, 5% ( % # of bits in numberstimeOk, so…How much time does it take to square the number n using grade school multiplication?& % 03 % 6 7 5 ) # of bits in numberstime % ) ( ( *# of bits used to describe inputtime' ( % -. !"#1
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