CSED233: Data Structures (2018F) Lecture13: Sorting and Selection

Transcription

1 (018F) Lecture13: Sortig ad Selectio Daiji Kim CSE, POSECH

2 Divide-ad-Coquer Divide-ad coquer a geeral algorithm desig paradigm: Divide: divide the iput data S i two djoit susets S 1 ad S Recur: solve the suprolems associated with S 1 ad S Coquer: comie the solutios or S 1 ad S ito a solutio or S he ase case or the recursio are suprolems o size 0 or 1 Merge-sort a sortig algorithm ased o the divide-ad-coquer paradigm Like heap-sort It has O( ) ruig time Ulike heap-sort It does ot use a auxiliary priority queue It accesses data i a sequetial maer (suitale to sort data o a dk) By Pro. Daiji Kim, Fall 018

3 Merge-Sort Merge-sort o a iput sequece S with elemets costs o three steps: Divide: partitio S ito two sequeces S 1 ad S o aout elemets each Recur: recursively sort S 1 ad S Coquer: merge S 1 ad S ito a uique sorted sequece Algorithm mergesort(s) Iput sequece S with elemets Output sequece S sorted accordig to C S.size() > 1 (S 1, S ) partitio(s, ) mergesort(s 1 ) mergesort(s ) S merge(s 1, S ) 3 By Pro. Daiji Kim, Fall 018

4 Mergig wo Sorted Sequeces he coquer step o merge-sort costs o mergig two sorted sequeces A ad B ito a sorted sequece S cotaiig the uio o the elemets o A ad B Mergig two sorted sequeces, each with elemets ad implemeted y meas o a douly liked lt, takes O() time Algorithm merge(a, B) Iput sequeces A ad B with elemets each Output sorted sequece o A B S empty sequece while A.Empty() B.Empty() A.irst().elemet() < B.irst().elemet() S.addLast(A.remove(A.irst())) else S.addLast(B.remove(B.irst())) while A.Empty() S.addLast(A.remove (A.irst())) while B.Empty() S.addLast(B.remove (B.irst())) retur S 4 By Pro. Daiji Kim, Fall 018

5 Java Merge Implemetatio 5 By Pro. Daiji Kim, Fall 018

6 Java Merge-Sort Implemetatio 6 By Pro. Daiji Kim, Fall 018

7 Merge-Sort ree A executio o merge-sort depicted y a iary tree each ode represets a recursive call o merge-sort ad stores usorted sequece eore the executio ad its partitio sorted sequece at the ed o the executio the root the iitial call the leaves are calls o susequeces o size 0 or By Pro. Daiji Kim, Fall 018

8 Executio Example Partitio By Pro. Daiji Kim, Fall 018

9 Executio Example (cot.) Recursive call, partitio By Pro. Daiji Kim, Fall 018

10 Executio Example (cot.) Recursive call, partitio By Pro. Daiji Kim, Fall 018

11 Executio Example (cot.) Recursive call, ase case By Pro. Daiji Kim, Fall 018

12 Executio Example (cot.) Recursive call, ase case By Pro. Daiji Kim, Fall 018

13 Executio Example (cot.) Merge By Pro. Daiji Kim, Fall 018

14 Executio Example (cot.) Recursive call,, ase case, merge By Pro. Daiji Kim, Fall 018

15 Merge Executio Example (cot.) By Pro. Daiji Kim, Fall 018

16 Executio Example (cot.) Recursive call,, merge, merge By Pro. Daiji Kim, Fall 018

17 Merge Executio Example (cot.) By Pro. Daiji Kim, Fall 018

18 Aalys o Merge-Sort he height h o the merge-sort tree O( ) at each recursive call we divide i hal the sequece, he overall amout or work doe at the odes o depth i O() we partitio ad merge i sequeces o size i we make i1 recursive calls hus, the total ruig time o merge-sort O( ) depth #seqs size i i i 18 By Pro. Daiji Kim, Fall 018

19 Summary o Sortig Algorithms Algorithm ime Notes selectio-sort O( ) ertio-sort O( ) heap-sort O( ) merge-sort O( ) slow i-place or small data sets (< 1K) slow i-place or small data sets (< 1K) ast i-place or large data sets (1K 1M) ast sequetial data access or huge data sets (> 1M) 19 By Pro. Daiji Kim, Fall 018

20 Divide-ad-Coquer Divide-ad coquer a geeral algorithm desig paradigm: Divide: divide the iput data S i two or more djoit susets S 1, S, Coquer: solve the suprolems recursively Comie: comie the solutios or S 1, S,, ito a solutio or S he ase case or the recursio are suprolems o costat size Aalys ca e doe usig recurrece equatios 0 By Pro. Daiji Kim, Fall 018

21 Merge-Sort Review Merge-sort o a iput sequece S with elemets costs o three steps: Divide: partitio S ito two sequeces S 1 ad S o aout elemets each Coquer: recursively sort S 1 ad S Comie: merge S 1 ad S ito a uique sorted sequece Algorithm mergesort(s) Iput sequece S with elemets Output sequece S sorted accordig to C S.size() > 1 (S 1, S ) partitio(s, ) mergesort(s 1 ) mergesort(s ) S merge(s 1, S ) 1 By Pro. Daiji Kim, Fall 018

22 Recurrece Equatio Aalys he coquer step o merge-sort costs o mergig two sorted sequeces, each with elemets ad implemeted y meas o a douly liked lt, takes at most steps, or some costat. Likewe, the as case ( < ) will take at most steps. hereore, we let () deote the ruig time o merge-sort: ( ) ( ) We ca thereore aalyze the ruig time o merge-sort y idig a closed orm solutio to the aove equatio. hat, a solutio that has () oly o the let-had side. By Pro. Daiji Kim, Fall 018

23 Iterative Sustitutio I the iterative sustitutio, or plug-ad-chug, techique, we iteratively apply the recurrece equatio to itsel ad see we ca id a patter: ( ) ( ) ( ( )) ( )) ( ) 3 ( 3 ) 3 4 ( 4 ) 4... Note that ase, ()=, case occurs whe i =. hat, i =. So, hus, () O( ). ( ) i ( i ) i 3 By Pro. Daiji Kim, Fall 018

24 he Recursio ree Draw the recursio tree or the recurrece relatio ad look or a patter: ( ) ( ) depth s size i i i time otal time = (last level plus all previous levels) 4 By Pro. Daiji Kim, Fall 018

25 Guess-ad-est Method I the guess-ad-test method, we guess a closed orm solutio a d the try to prove it true y iductio: ( ) ( Guess: () < c. ) ( ) ( ) ( c ( ) ( )) c ( ) c c Wrog: we caot make th last lie e less tha c 5 By Pro. Daiji Kim, Fall 018

26 Guess-ad-est Method, (cot.) Recall the recurrece equatio: ( ) ( ) Guess #: () < c. ( ) ( ) ( c ( ) ( )) c ( ) c c c c >. c So, () O( ). I geeral, to use th method, you eed to have a good guess ad you eed to e good at iductio proos. 6 By Pro. Daiji Kim, Fall 018

27 Master Method (Appedix) May divide-ad-coquer recurrece equatios have the orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. 7 By Pro. Daiji Kim, Fall 018

28 Master Method, Example 1 he orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) 4 ( ) Solutio: a=, so case 1 says () O( ). 8 By Pro. Daiji Kim, Fall 018

29 Master Method, Example he orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) ( ) Solutio: a=1, so case says () O( ). 9 By Pro. Daiji Kim, Fall 018

30 Master Method, Example 3 he orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) ( 3) Solutio: a=0, so case 3 says () O( ). 30 By Pro. Daiji Kim, Fall 018

31 Master Method, Example 4 he orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) 8 ( ) Solutio: a=3, so case 1 says () O( 3 ). 31 By Pro. Daiji Kim, Fall 018

32 Master Method, Example 5 he orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) 9 ( 3) 3 Solutio: a=, so case 3 says () O( 3 ). 3 By Pro. Daiji Kim, Fall 018

33 Master Method, Example 6 he orm: ( ) a c ( ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) ( ) 1 (iary search) Solutio: a=0, so case says () O( ). 33 By Pro. Daiji Kim, Fall 018

34 Master Method, Example 7 he orm: ( ) a ( c ) ( ) d d he Master heorem: 1. ( ) O ( a ), the ( ) ( a ). ( ) ( a k ), the ( ) ( a k 1 ) 3. ( ) ( a ), the ( ) ( ( )), provided a ( ) ( ) or some 1. Example: ( ) ( ) (heap costructio) Solutio: a=1, so case 1 says () O(). 34 By Pro. Daiji Kim, Fall 018

35 By Pro. Daiji Kim, Fall 018 Iterative Proo o the Master heorem Usig iterative sustitutio, let us see we ca id a patter: We the dtiguh the three cases as he irst term domiat Each part o the summatio equally domiat he summatio a geometric series 35 1 ) ( 0 1 ) ( ) ( (1) ) ( (1)... ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( )) ( ( ) ( ) ( ) ( i i i a i i i a a a a a a a a a a a

36 By Pro. Daiji Kim, Fall 018 Iteger Multiplicatio Algorithm: Multiply two -it itegers I ad J. Divide step: Split I ad J ito high-order ad low-order its We ca the deie I*J y multiplyig the parts ad addig: So, () = 4(), which implies () O( ). But that o etter tha the algorithm we leared i grade s chool. 36 l h l h J J J I I I l l h l l h h h l h l h J I J I J I J I J J I I J I ) ( * ) ( *

37 By Pro. Daiji Kim, Fall 018 A Improved Iteger Multiplicatio Algorithm Algorithm: Multiply two -it itegers I ad J. Divide step: Split I ad J ito high-order ad low-order its Oserve that there a deret way to multiply parts: So, () = 3(), which implies () O( 3 ), y the Master heorem. hus, () O( ). 37 l h l h J J J I I I l l h l l h h h l l l l h h h l h h l l l h h h l l l l h h h l l h h h J I J I J I J I J I J I J I J I J I J I J I J I J I J I J I J J I I J I J I ) ( ] ) [( ] ) )( [( *

38 Quick-Sort Quick-sort a radomized sortig algorithm ased o the divide-ad-coquer paradigm: Divide: pick a radom elemet x (called pivot) ad partitio S ito L elemets less tha x E elemets equal x G elemets greater tha x Recur: sort L ad G Coquer: joi L, E ad G L E x x x G 38 By Pro. Daiji Kim, Fall 018

39 Partitio We partitio a iput sequece as ollows: We remove, i tur, each elemet y rom S ad We ert y ito L, E or G, depedig o the result o the comparo with the pivot x Each ertio ad removal at the egiig or at the ed o a sequece, ad hece takes O(1) time hus, the partitio step o quick-sort takes O() time Algorithm partitio(s, p) Iput sequece S, positio p o pivot Output susequeces L, E, G o the elemets o S less tha, equal to, or greater tha the pivot, resp. L, E, G empty sequeces x S.remove(p) while S.Empty() y S.remove(S.irst()) y < x L.addLast(y) else y = x E.addLast(y) else { y > x } G.addLast(y) retur L, E, G 39 By Pro. Daiji Kim, Fall 018

40 Java Implemetatio 40 By Pro. Daiji Kim, Fall 018

41 Quick-Sort ree A executio o quick-sort depicted y a iary tree Each ode represets a recursive call o quick-sort ad stores Usorted sequece eore the executio ad its pivot Sorted sequece at the ed o the executio he root the iitial call he leaves are calls o susequeces o size 0 or By Pro. Daiji Kim, Fall 018

42 Executio Example Pivot selectio By Pro. Daiji Kim, Fall 018

43 Executio Example (cot.) Partitio, recursive call, pivot selectio By Pro. Daiji Kim, Fall 018

44 Executio Example (cot.) Partitio, recursive call, ase case By Pro. Daiji Kim, Fall 018

45 Executio Example (cot.) Recursive call,, ase case, joi By Pro. Daiji Kim, Fall 018

46 Executio Example (cot.) Recursive call, pivot selectio By Pro. Daiji Kim, Fall 018

47 Executio Example (cot.) Partitio,, recursive call, ase case By Pro. Daiji Kim, Fall 018

48 Joi, joi Executio Example (cot.) By Pro. Daiji Kim, Fall 018

49 Worst-case Ruig ime he worst case or quick-sort occurs whe the pivot the uique miimum or maximum elemet Oe o L ad G has size 1 ad the other has size 0 he ruig time proportioal to the sum ( 1) 1 hus, the worst-case ruig time o quick-sort O( ) depth time By Pro. Daiji Kim, Fall 018

50 Expected Ruig ime Cosider a recursive call o quick-sort o a sequece o size s Good call: the sizes o L ad G are each less tha 3s4 Bad call: oe o L ad G has size greater tha 3s Good call A call good with proaility 1 1 o the possile pivots cause good calls: Bad call Bad pivots Good pivots Bad pivots 50 By Pro. Daiji Kim, Fall 018

51 Expected Ruig ime, Part Proailtic Fact: he expected umer o coi tosses required i order to get k heads k For a ode o depth i, we expect i acestors are good calls he size o the iput sequece or the curret call at most (34) i hereore, we have For a ode o depth 43, the expected iput size oe he expected height o the quick-sort tree O( ) he amout or work doe at the odes o the same depth O() hus, the expected ruig time o quick-sort O( ) expected height O( ) s(a) s(r) s() s(c) s(d) s(e) s() time per level O() O() O() total expected time: O( ) 51 By Pro. Daiji Kim, Fall 018

52 I-Place Quick-Sort Quick-sort ca e implemeted to ru i-place I the partitio step, we use replace operatios to rearrage the elemets o the iput sequece such that the elemets less tha the pivot have rak less tha h the elemets equal to the pivot have rak etwee h ad k the elemets greater tha the pivot have rak greater tha k he recursive calls cosider elemets with rak less tha h elemets with rak greater tha k Algorithm iplacequicksort(s, l, r) Iput sequece S, raks l ad r Output sequece S with the elemets o rak etwee l ad r rearraged i icreasig order l r retur i a radom iteger etwee l ad r x S.elemAtRak(i) (h, k) iplacepartitio(x) iplacequicksort(s, l, h 1) iplacequicksort(s, k 1, r) 5 By Pro. Daiji Kim, Fall 018

53 I-Place Partitioig Perorm the partitio usig two idices to split S ito L ad E U G (a similar method ca split E U G ito E ad G j k Repeat util j ad k cross: Sca j to the right util idig a elemet > x. Sca k to the let util idig a elemet < x. Swap elemets at idices j ad k (pivot = 6) j k By Pro. Daiji Kim, Fall 018

54 Java Implemetatio 54 By Pro. Daiji Kim, Fall 018

55 Summary o Sortig Algorithms Algorithm ime Notes selectio-sort O( ) i-place slow (good or small iputs) ertio-sort O( ) quick-sort O( ) expected heap-sort O( ) merge-sort O( ) i-place slow (good or small iputs) i-place, radomized astest (good or large iputs) i-place ast (good or large iputs) sequetial data access ast (good or huge iputs) 55 By Pro. Daiji Kim, Fall 018

56 Comparo-Based Sortig May sortig algorithms are comparo ased. hey sort y makig comparos etwee pairs o ojects Examples: ule-sort, selectio-sort, ertio-sort, heap-sort, merge-sort, quick-sort,... Let us thereore derive a lower oud o the ruig ti me o ay algorithm that uses comparos to sort el emets, x 1, x,, x. Is x i < x j? o yes 56 By Pro. Daiji Kim, Fall 018

57 Coutig Comparos Let us just cout comparos the. Each possile ru o the algorithm correspods to a root-to-lea path i a decio tree x i < x j? x a < x? x c < x d? x e < x? x k < x l? x m < x o? x p < x q? 57 By Pro. Daiji Kim, Fall 018

58 Decio ree Height he height o the decio tree a lower oud o the ruig time Every iput permutatio must lead to a separate lea output I ot, some iput 4 5 would have same output orderig as 5 4, which would e wrog Sice there are!=1 leaves, the height at least (!) miimum height (time) x i < x j? x a < x? x c < x d? (!) x e < x? x k < x l? x m < x o? x p < x q?! 58 By Pro. Daiji Kim, Fall 018

59 he Lower Boud Ay comparo-ased sortig algorithms takes at least (!) time hereore, ay such algorithm takes time at least (!) hat, ay comparo-ased sortig algorithm must ru i ( ) time. ( ) ( ). 59 By Pro. Daiji Kim, Fall 018

60 Bucket-Sort Let e S e a sequece o (key, elemet) items with keys i the rage [0, N 1] Bucket-sort uses the keys as idices I to a auxiliary array B o sequeces (uckets) Phase 1: Empty sequece S y movig each etry (k, o) ito its ucket B[k] Phase : For i 0,, N 1, move the etries o ucket B[i] to the ed o sequece S Aalys: Phase 1 takes O() time Phase takes O( N) time Bucket-sort takes O( N) time Algorithm ucketsort(s): Iput: Sequece S o etries with i teger keys i the rage [0, N 1] Output: Sequece S sorted i ode creasig order o the keys let B e a array o N sequeces, eac h o which iitially empty or each etry e i S do k = the key o e remove e rom S ert e at the ed o ucket B[k] or i = 0 to N 1 do or each etry e i B[i] do remove e rom B[i] ert e at the ed o S 60 By Pro. Daiji Kim, Fall 018

61 Example Key rage [0, 9] 7, d 1, c 3, a 7, g 3, 7, e Phase 1 1, c 3, a 3, 7, d 7, g 7, e B Phase 1, c 3, a 3, 7, d 7, g 7, e 61 By Pro. Daiji Kim, Fall 018

62 Properties ad Extesios Key-type Property he keys are used as idices ito a array ad caot e aritrary ojects No exteral comparator Stale Sort Property he relative order o ay two items with the same key preserved ater the executio o the algorithm Extesios Iteger keys i the rage [a, ] Put etry (k, o) ito ucket B[k a] Strig keys rom a set D o possile strigs, where D has costat size (e.g., ames o the 50 U.S. states) Sort D ad compute the rak r(k) o each strig k o D i the sorted sequece Put etry (k, o) ito ucket B[r(k)] 6 By Pro. Daiji Kim, Fall 018

63 Lexicographic Order A d-tuple a sequece o d keys (k 1, k,, k d ), where key k i said to e the i-th dimesio o the tuple Example: he Cartesia coordiates o a poit i space are a 3-tuple he lexicographic order o two d-tuples recursively deied as ollows (x 1, x,, x d ) (y 1, y,, y d ) x 1 y 1 x 1 y 1 (x,, x d ) (y,, y d ) I.e., the tuples are compared y the irst dimesio, the y the secod dimesio, etc. 63 By Pro. Daiji Kim, Fall 018

64 Lexicographic-Sort Let C i e the comparator that compares two tuples y their i-th dimesio Let stalesort(s, C) e a stale sorti g algorithm that uses comparator C Lexicographic-sort sorts a sequece o d-tuples i lexicographic order y executig d times algorithm stalesort, oe per dimesio Lexicographic-sort rus i O(d()) time, where () the ruig time o stalesort Algorithm lexicographicsort(s) Iput sequece S o d-tuples Output sequece S sorted i lexicographic order or i d dowto 1 stalesort(s, C i ) Example: (7,4,6) (5,1,5) (,4,6) (, 1, 4) (3,, 4) (, 1, 4) (3,, 4) (5,1,5) (7,4,6) (,4,6) (, 1, 4) (5,1,5) (3,, 4) (7,4,6) (,4,6) (, 1, 4) (,4,6) (3,, 4) (5,1,5) (7,4,6) 64 By Pro. Daiji Kim, Fall 018

65 Radix-Sort Radix-sort a specializatio o lexicographic-sort that uses ucketsort as the stale sortig algorithm i each dimesio Radix-sort applicale to tuples where the keys i each dimesio i are itegers i the rage [0, N 1] Radix-sort rus i time O(d( N)) Algorithm radixsort(s, N) Iput sequece S o d-tuples such that (0,, 0) (x 1,, x d ) ad (x 1,, x d ) (N 1,, N 1) or each tuple (x 1,, x d ) i S Output sequece S sorted i lexicographic order or i d dowto 1 ucketsort(s, N) 65 By Pro. Daiji Kim, Fall 018

66 Radix-Sort or Biary Numers Cosider a sequece o -it itegers x x 1 x 1 x 0 We represet each elemet as a -tuple o itegers i the rage [0, 1] ad apply radix-so rt with N h applicatio o the radixsort algorithm rus i O() time For example, we ca sort a sequece o 3-it itegers i liear time Algorithm iaryradixsort(s) Iput sequece S o -it itegers Output sequece S sorted replace each elemet x o S with the item (0, x) or i 0 to 1 replace the key k o each item (k, x) o S with it x i o x ucketsort(s, ) 66 By Pro. Daiji Kim, Fall 018

67 Example Sortig a sequece o 4-it itegers By Pro. Daiji Kim, Fall 018

68 he Selectio Prolem Give a iteger k ad elemets x 1, x,, x, take rom a total order, id the k-th smallest elemet i th set. O course, we ca sort the set i O( ) time ad the idex the k-th elemet. k= Ca we solve the selectio prolem aster? 68 By Pro. Daiji Kim, Fall 018

69 Quick-Select Quick-select a radomized selectio algorithm ased o the prue-ad-search paradigm: Prue: pick a radom elemet x(called pivot) ad partitio S ito L: elemets less tha x E: elemets equal x G: elemets greater tha x Search: depedig o k, either aswer i E, or we eed to recur i either L or G L k < L E x x G L < k < L E (doe) k > L E k = k - L - E 69 By Pro. Daiji Kim, Fall 018

70 Partitio We partitio a iput sequece as i the quick-sort algorithm: We remove, i tur, each elemet y rom S ad We ert y ito L, E or G, depedig o the result o the comparo with the pivot x Each ertio ad removal at the egiig or at the ed o a sequece, ad hece takes O(1) time hus, the partitio step o quick-select takes O() time Algorithm partitio(s, p) Iput sequece S, positio p o pivot Output susequeces L, E, G o the elemets o S less tha, equal to, or greater tha the pivot, resp. L, E, G empty sequeces x S.remove(p) while S.Empty() y S.remove(S.irst()) y < x L.addLast(y) else y = x E.addLast(y) else { y > x } G.addLast(y) retur L, E, G 70 By Pro. Daiji Kim, Fall 018

71 Quick-Select Vualizatio A executio o quick-select ca e vualized y a recursio path Each ode represets a recursive call o quick-select, ad stores k ad the remaiig sequece k=5, S=( ) k=, S=( ) k=, S=( ) k=1, S=(7 6 5) 5 71 By Pro. Daiji Kim, Fall 018

72 Expected Ruig ime Cosider a recursive call o quick-select o a sequece o size s Good call: the sizes o L ad G are each less tha 3s4 Bad call: oe o L ad G has size greater tha 3s Good call Bad call A call good with proaility 1 1 o the possile pivots cause good calls: Bad pivots Good pivots Bad pivots 7 By Pro. Daiji Kim, Fall 018

73 Expected Ruig ime, Part Proailtic Fact #1: he expected umer o coi tosses required i order to get oe head two Proailtic Fact #: Expectatio a liear uctio: E(X Y ) = E(X ) E(Y ) E(cX ) = ce(x ) Let () deote the expected ruig time o quick-select. By Fact #, () < (34) *(expected # o calls eore a good call) By Fact #1, () < (34) hat, () a geometric series: () < (34) (34) (34) 3 So () O(). We ca solve the selectio prolem i O() expected time. 73 By Pro. Daiji Kim, Fall 018

74 Determitic Selectio We ca do selectio i O() worst-case time. Mai idea: recursively use the selectio algorithm itsel to id a good pivot or quick-select: Divide S ito 5 sets o 5 each Fid a media i each set Recursively id the media o the ay medias. Mi size or L Mi size or G 74 By Pro. Daiji Kim, Fall 018