Lower Bounds for Sorting

Transcription

1 Liear Sortig Topics Covered: Lower Bouds for Sortig Coutig Sort Radix Sort Bucket Sort Lower Bouds for Sortig Compariso vs. o-compariso sortig Decisio tree model Worst case lower boud Compariso Sortig Relies solely o relative orderig of elemets Geeralizatio: oly operatio allowed betwee ay two elemets Examples of compariso sorts: quicksort, heapsort, bubblesort, mergesort, selectio sort, etc. Decisio Tree Model p. Iteral ode = compariso betwee two elemets Leaf ode = correct orderig (sort complete) Decisio Tree Model p.2 Path = comparisos made to arrive at a orderig Every possible orderig must appear as a leaf Worst Case Lower Boud The legth of logest path from root to leaf is umber of comparisos i worst case Problem reduces to: what is mi height of decisio tree i which every possible permutatio is a leaf? Let h = height of tree, = size of set. The aswer = mi{ h:! 2 h } or mi { h: log(!) h }, that is h = log (! ) = Θ( log( ) = Θ( log ) = Ω( log )

2 Proof of Lower Boud Stirlig s Approximatio for!! = 2π + O e ( ) = ( π ) + ( ) ( ) + ( + ( )) log! log 2 log log e log O 2 ( ) O( ) ( log ) = log + =Θ Coutig Sort Tallies each umber s occurrece withi the array The algorithm assumes each elemet i the array is a iteger, i the rage to k Needs a orary array for workig space with legth k Also eeds aother array to hold the output Coutig Sort Algorithm Origial Array out Output Array Temporary array for i to k do [i] for j to legth[ ] do [ [ j ] ] [ [ j ] ] + Coutig Sort Made a secod array with legth equal to the ial array s legth Two s Two 2 Oe 3 Three 4s Two s Coutig Sort Calculatig the umber of elemets less tha or equal to each slot umber (Stable) for i to k do [i] [i] + [i -] before for j legth[ ] - to do [ [ j ] ] [ [ j ] ] - out[ [ [ j ] ] ] [ j ] out

3 Coutig Sort Coutig Sort Two s Two 2 Oe 3 Three 4s Two s Fial Result Alterative versio: Goig through the array, use the iformatio to fill the ial array with sorted umbers (Ustable): Advatage: Do t eed output array Coutig Sort Advatages Fast Stable Duplicates maitai order Disadvatages Requires additioal memory Radix sort or Old School Defiitio is the algorithm used by the card-sortig machies A multiple pass sort algorithm that distributes each item to a bucket accordig to part of the item's key begiig with the least sigificat part of the key. After each pass, items are collected from the buckets, keepig the items i order, the redistributed accordig to the ext most sigificat part of the key How it works The cards are orgaized ito 8 colums i each colum a hole ca be puched i oe of 2 places The sorter mechaically examies a give colum of each card i a deck ad distributes the card ito oe of 2 bis depedig o which place has bee puched 3

4 A operator gathers the cards bi by bi, so that cards with the first place puched are o top of cards with the secod place puched, ad so o. For decimal digits, oly places are used i each colum. The other two places are used for ecodig oumeric characters A d-digit umber would the occupy a field of d colums Ufortuately, sice the cards i 9 of the bis must be put aside to sort each of the bis, this procedure geerates may itermediate piles of cards that must be kept track of. It sorts o the least sigificat digit first. LDS2/radixsort.html Bucket Sort Bucket sort rus i liear time whe the iput is draw from a uiform distributio. Like coutig sort, bucket sort is fast because it assumes somethig about the iput. Whereas coutig sort assumes that the iput cosists of itegers i a small rage, bucket sort assumes that the iput is geerated by a radom process that distributes elemets uiformly over the iterval [, ). Idea divide the iterval [, ) ito equal-sized subitervals, or buckets distribute the iput umbers ito the buckets Assumptio: Sice the iputs are uiformly distributed over [, ), we do't expect may umbers to fall ito each bucket sort the umbers i each bucket go through the buckets i order, listig the elemets i each. 4

5 Algorithm Let be S be a sequece of (key, elemet) items with keys i the rage [, N - ] Bucket-sort uses the keys as idices ito a auxiliary array B of sequeces (buckets) Phase : Empty sequece S by movig each item (k, o) ito its bucket B[k] Phase 2: For i =,, N -, move the items of bucket B[i] to the ed of sequece S Aalysis Phase takes O() time Phase 2 takes O( + N) time Bucket-sort takes O( + N) time Example y/code/bucket.html Refereces Thomas H. Corme et al. Itroductio to Algorithms, 2d Editio. MIT Press 2 Coutig Sort Algorithm. Worcester Polytechical Istitute < tig_sort.htm> 3/8/23 CS2 Lab 7. Rochester Istitute of Techology < 3/8/23 Refereces part 2 ml DS2/radixsort.html code/bucket.html