Searching Algorithm Efficiencies


 Imogen Kelley
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1 Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay (search) algorithm. Sice a more efficiet algorithm would take less time to execute, oe approach would be to write a program for each of the algorithms to be compared, ad execute them, measurig the time each takes to fiish. However, a better metric would allow algorithms to be evaluated before implemetig them. Oe such metric is the umber of mai steps the algorithm will require to fiish. Of course, the exact umber of steps depeds o the iput data. For the liear search algorithm, the umber of steps depeds o whether the target is i the list, ad if so, where i the list, as well as o the legth of the list. For search algorithms, the mai steps are checkig a particular item to see if it is what we are lookig for. Doig a compariso (Is this item the same as the umber I am lookig for?) takes a certai amout of time. Coutig the umber of comparisos eeded i the best case, the worst case, ad the average case produces the followig table. Model Best Case (fewest comparisos) (most comparisos) Average Case (average umber of comparisos) (for = 00,000) (target is first item) 00,000 (target is last item) 50,000 (target is middle item) as a fuctio of /2 The best case aalysis does't tell us much. If the first elemet checked happes to be the target, ay algorithm will take oly oe compariso. The worst ad average case aalyses give a better idicatio of algorithm efficiecy. The worst case tells us the logest possible time this algorithm ca take ad average case tells us what to expect i geeral. Below is a chart showig average ad worst case comparisos required for searchig lists of various sizes: Items () Average Case Worst Case
2 Here are those poits plotted i a graph: Average Case List Legth Whether we look at average or worst case, it is clear the patter is a straight lie. Both of these expressios represet liear fuctios: comparisos = comparisos = / 2 This algorithm is called liear search because the time required ca be expressed as a liear fuctio: the umber of comparisos to fid a target icreases liearly as the size of the list.
3 Efficiecy of Biary Search Is biary search more efficiet tha liear search? If so, how much more efficiet is it? To evaluate biary search, we will cout the umber of comparisos i the best case ad worst case. This aalysis omits the average case, which is a bit more difficult. The best case occurs if the middle item happes to be the target. The oly oe compariso is eeded to fid it sice we always start there for biary search. As before, the best case aalysis does ot reveal much. Whe does the worst case occur? Well, if the item we are lookig for is the last possible thig we check (or if it is ot i the list at all). I either of these cases we cotiue dividig the list ito half util there is othig left to check. If we start with a list of 000 items, ad elimiate half of them with each check this is the patter we get Items Left to Search So Far For a list size of 000, there are at most 0 comparisos required to check the etire list to decide if a particular value is i it or ot. 9 checks gets us dow to oe last possibility, ad oe last check decides if it is actually the oe we wat. The way we ca express the cocept "How may times do I have to divide this umber by 2 i order to reach?" is with a logarithm. Logarithms are the opposite of expoets 2 says "multiply 2's together". Log 2 says " is equal to this may 2's all multiplied". Log 2 6 = 4 because 6 = 2 4 you have to divide 6 by 2 four times to reach Log 2 52 = 9 because 52 = 2 9 you have to divide 256 by 2 ie times to reach For our example above, we ca calculate Log the easiest way to do this is by searchig google for "log2(000)"
4 Well, we ca't do 9.96 checks so we will roud that up to 0 the same umber we got dividig by 2 util we ra out of thigs to check! Usig this method we ca quickly predict how much work to: Check a list of 5000 items: Log 2 (5000) = = oly 3 comparisos (do't roud dow we have to do more tha 2 comparisos of work, so that is 3) Check a list of,000,000 items: Log 2 (000000) = = 20 comparisos Compared to our list of 000 items, it oly takes 3 more comparisos to search a list of 5000 thigs. It oly takes 0 additioal comparisos to search a millio thigs! I geeral, if is the size of the list to be searched ad C is the umber of comparisos to do so i the worst case, C = log 2. Thus, the efficiecy of biary search ca be expressed as a logarithmic fuctio, i which the umber of comparisos required to fid a target icreases oly logarithmically with the size of the list. The followig table summarizes the aalysis for biary search. Model Best Case (fewest comparisos) (most comparisos) (for = 00000) (target is middle item) 6 (target ot i list) as a fuctio of log 2
5 Efficiecy Compariso Here is a graph of the umber of comparisos required for lists of a give legth give the worst case for the two strategies Liear Biary List Legth Pretty astoudig differece! If we kept goig, the worst case compariso would be eve more extreme. The followig table shows how the maximum umber of comparisos icreases for biary search ad liear search. List Size Liear Search Biary Search 00,000 00, , , , , , ,000 20,600,000,600,000 2 Check here to see the differece i actio: Searchig: Summary. A commo operatio o lists of records is to search the list to retrieve a particular record. 2. There are alterative algorithms for searchig a list. The umber of comparisos required is a useful metric for their efficiecy. 3. Liear search examies each list item i tur util the target is located or the ed of the list is reached. 4. Liear search efficiecy for the worst case data model is a liear fuctio of the umber of items i the list. 5. If the list is sorted, a biary search strategy may be used. 6. Biary search uses the result of each compariso to elimiate half of the list from further searchig. 7. Biary search efficiecy for the worst case is a logarithmic fuctio of list size.
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