Beyond Comparison: Distribution. Necessary Choices

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1 Better than N lg N? atifallyoucandotokeysiscomparethem, thensorting (N lgn). there are N! possible ways the input data could be your program must be prepared to do N! different comdata-moving operations. there must be N! possible combinations of outcomes of sts in your program, since those determine what move where (we re assuming that comparisons are -way). Tree rting time abc b < c T acb a < c a < b cab F bac a < c bca b < c cba :: CSB: Lectures # Beyond Comparison: Distribution can do more than compare keys?, how can we sort a set of N integer keys whose values to kn, for some small constant k? ue: put the integers into N buckets, with an integer p ket p/k. eys per bucket, so catenate and use insertion sort, which ast. N = : ts: on sort is fast. Putting in buckets takes time Θ(N), and rt takes Θ(kN). When k is fixed (constant), we have me Θ(N). :: CSB: Lectures # Distribution Counting Example tems are between and as in this example: < < < < < < e gives # occurrences of each key. < Running sum m gives cumulative count of keys < each value... s us where to put each key: stance of key k goes into slot m, where m is the number nces that are < k. :: CSB: Lectures # CSB Lectures # s on sorting by comparison counting, radix sorts day: DS(IJ), Chapter ; Next topic: Chapter. Necessary Choices f-test goes two ways, number of possible different outif-tests is k. enough tests so that k N!, which means k Ω(lgN!). ng s approximation, N N πn +Θ, e N /(lgπ +lgn)+n lgn N lge+lg +Θ = Θ(N lgn) N that k, the worst-case number of tests needed to sort comparison sorting, is in Ω(N lgn): there must be cases eed (some multiple of) N lgn comparisons to sort N Distribution Counting hnique: count the number of items <, <, etc. ems with value < p, then in sorted order, the j th item must be item #M p +j. er linear-time algorithm. :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures #

2 Running sum of Running sum of Running sum of :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures # Running sum of Running sum of Running sum of :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures #

3 Running sum of Running sum of Running sum of :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures # Running sum of Running sum of Running sum of :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures #

4 Running sum of Running sum of Running sum of :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures # Running sum of Running sum of Running sum of :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures #

5 Running sum of MSD Radix Sort omplicated: must keep lists from each step separate p processing -element lists A posn, cat, cad, con, bat, can, be, let, bet, be, bet / cat, cad, con, can / let / set be, bet / cat, cad, con, can / let / set be / bet / cat, cad, con, can / let / set be / bet / cat, cad, can / con / let / set be / bet / cad / can / cat / con / let / set And Don t Forget Search Trees h tree is in sorted order, when read in inorder. e to really use for sorting [next topic]. e, same performance as heapsort: N insertions in time us Θ(N) to traverse, gives Θ(N +N lgn) = Θ(N lgn) :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures # Running sum of Radix Sort ys one character at a time. ribution counting for each digit. her right to left (LSD radix sort) or left to right (MSD ort is venerable: used for punched cards. e Initial: set, cat, cad, con, bat, can, be, let, bet cad can con bet let bat cat set d n t n, can, set, cat, bat, let, bet Pass (by char #) bet be bat con cat can cad Pass (by char #) let bat cat can cad b c l s bat, be, bet, cad, can, cat, con, let, set bet let set be con a e o cad, can, cat, bat, be, set, let, bet, con set Performance of Radix Sort akes Θ(B) time where B is total size of the key data. red other sorts as function of #records. pare? different records, must have keys at least Θ(lg N) long e, comparison actually takes time Θ(K) where K is size rst case [why?] mparisons really means N(lgN) operations. sort would take B = N lg N time with minimal-length er hand, must work to get good constant factors with :: CSB: Lectures # :: CSB: Lectures # :: CSB: Lectures #

6 Summary rt: Θ(Nk) comparisons and moves, where k is maximum is displaced from final position. small datasets or almost ordered data sets. Θ(N lgn) with good constant factor if data is not pathost case O(N ). Θ(N lg N) guaranteed. Good for external sorting. reesort with guaranteed balance: Θ(N lg N) guaranteed. distribution sort: Θ(B) (number of bytes). Also good for ting. :: CSB: Lectures #