) n. ALG 1.3 Deterministic Selection and Sorting: Problem P size n. Examples: 1st lecture's mult M(n) = 3 M ( È

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1 Algorithms Professor Joh Reif ALG 1.3 Determiistic Selectio ad Sortig: (a) Selectio Algorithms ad Lower Bouds (b) Sortig Algorithms ad Lower Bouds Problem P size fi divide ito subproblems size 1,..., k solve these ad "glue" together solutios k T() = Â T( i ) + g() i=1 time to combie solutios Examples: 1st lecture's mult M() = 3 M ( È ) + q() Mai Readig Selectios: CLR, Chapters 7, 9, 10 Auxillary Readig Selectios: AHU-Desig, Chapters ad 3 AHU-Data, Chapter 8 BB, Sectios 4.4, 4.6 ad F( ) È fast fourier trasform F() = biary search B() = B (È ) merge sortig S() = S ( È + q ) + q() + q()

2 Examples Selectio, ad Sortig Decisio Tree Model o Time =( # comparisos o logest path) iput a,b,c Biary tree a<b? root Y b<c? N N Y a<c? N facts: with L Leaves has = L-1 iteral odes È () max height log L Y a<c? b,a,c b<c? a,b,c Y a,c,b N c<a<b Y b,c,a N c,b,a 3 4

3 Mergig lists with total of keys iput X 1 < X <... < X k ad Y 1 < Y <... < Y -k output ordered merge of two key lists Algorithm Isert Case k=-1 iput (X 1 < X <... < X k ), (Y 1 ) Algorithm : Biary Search by Divide-ad-Coquer goal provably asymptotically optimal algorithm i Decisio Tree Model use this Model because it allows simple proofs of lower bouds time = # comparisos so easy to boud time costs [1] Compare Y 1 with X È k [] if Y 1 > X È isert Y 1 ito ( X È k k + 1 <... < X k) else Y 1 X È ad isert Y k 1 ito ( ) X 1 <... < X È k 5 6

4 Case: Mergig equal legth lists Iput (X 1 < X <... < X k ) Total Compariso Cost: È log (k+1) = È log () Sice a biary tree with =k+1 leaves has depth > È log(), this is optimal! where k = (Y 1 < Y <... < Y -k ) Algorithm [1] i 1, j 1 [] while i k ad j k do {if X i < Y j the output (X i ) ad set i i+1 else output (Y j ) ad set j j+1 [3] output remaiig elemets Algorithm clearly uses k-1 = -1 comparisos 7 8

5 Lower boud: cosider case X 1 < Y 1 < X < Y <...< X k < Y k ay merge algorithm must compare claim: X i with Y i for i=1,..., k () Y i with X i+1 for j=1,..., k-1 (otherwise we could fi so requires flip Y < i X i with o chage) k-1 = -1 comparisos! Sortig by Divide-ad-Coquer Algorithm Merge Sort iput set S of keys [1] partitio S ito set X of È ad set Y of keys Î [] Recursively compute Merge Sort (X) = (X 1, X,..., X È [3] merge above sequeces usig -1 comparisos keys Merge Sort (Y) = (Y 1, Y,..., Y ) ) Î 9 [4] output merged sequece 10

6 Time Aalysis T() = T ( È T(l) = 0 fi T() = È log = q ( log ) ) ( + T Î - È log ) Lower Bouds o Sortig (o decisio tree model).... depth È log (!)! distict leaves 11 1

7 Easy Approximatio (via Itegratio) log(!) = log() + log(-1) log() + log Ú Ú log xdx log xdx Ú log x dx 1 Better boud -1 1 (Sice log k k Ú log xdx) 1 k - log - log e + log e usig Sterlig Approximatio iput output Selectio Problems X 1, X,..., X ad idex k x (k) = the k'th best e {1,...,}! p ( e) ( ) fi log (!) log - log e + 1 log (p) 13 14

8 History: Rev C.L. Dodge (Lewis Carol) wrote article o law teis touramet i James Gazett, 1883 Selectio of the champio X - X is easily determied i -1 compariso felt prizes ujust because: - although wier X always gets lst prize - secod X () may ot get d prize X -1 X X X X () X() X 1 X - X requires -1 comparisos X ote: X () ot declared d best if it is left brach proof everyoe except the champio X must lose at least oce! Carol proposed his ow (ooptimal) touramet

9 Selectio of the secod best X () È usig - + log comparisos [] Let Y be the set of players kocked out by champio X Y È log Algorithm [1] form a balaced biary tree for touramet to fid X usig -1 comparisos [3] Play a touramet amog the Y's x [4] output X () = champio of the Y's usig È log -1 more comparisos x Y Y x log height Y x 17 18

10 Lower Bouds o fidig X () requires - + È log comparisos Claim m 1-1, sice at ed we must kow X as well as X () proof #compariso m 1 + m +... where m i = #players who lost i or more matches Claim m (#who lost to X )-1 sice everyoe (except X ) who () lost to X must also have lost oe more time. 19 0

11 lemma (#who lost to X i worst case ) È log declare X i > X j if (a) X i previously udefeated ad X j lost at least oce proof Use oracle who "fixes" results (b) both udefeated but X i played more matches (c) otherwise, decide cosistetly with previous decisios of games so that champio X plays È log matches fi forces path from X to have legth È log to root 1

12 Selectio by Divide-ad-Coquer Algorithm Select k (X) iput set X of keys ad idex k [1] if < c o the output X (k) by sortig X ad halt [] divide X ito sequeces Î d of d elemets each (with < d leftover), ad sort each sequece [3] let M be the medias of each of these sequeces [4] m Select M (M) [5] let X - = {x e X x < m} let X + = {x e X x > m} [6] if X - k the output Select k (X - ) else if - X + = k the output m else output Select k-( - X + ) (X + ) 3 4

13 Î smallest d Colums = sequeces ot X + d m M i sorted order for a sufficietly large costat c 1 ot X - (assumig d is costat) largest If say d=5, T() 0c = O() 1 d+1 Propositio X -, X + each - Î Î d 3 4 { T() c T ( 1 Î if < co ) + d 3 T ( 4 ) + c 1 5 6

14 proof Fix a path p from root to leaf The comparisos doe o p defie a relatio R p Let R p + = trasitive closure of R p Lower Bouds for Selectig X (k) iput X = {x 1,..., x }, idex k Theorem Every leaf of Decisio Tree has depth -1 Lemma If path p determies X m = X (k) the for all iπm either x i R p + xm or x m R p + xi proof Suppose x i is u related to x m by R p + The ca replace x i i liear order either before or after x m to violate x m = x (k) 7 Let the "key" compariso for x i be whe x i is compared with x j where either j=m + () x i R p x j ad x j R p xm, or + (3) x j R p x i ad x m R p xj Fact x i has uique "key" compariso determiig + + either x i R xm p or x m R xi p fi So there are -1 "key" comparisos, each distict! 8

15 A hard to aalyze sort: D Î () while D > 0 do for i = D +1 to do iput keys SHELLSORT X1,..., X icremet sort begi j i - D while j>0 do if x j > x j+ D the ed else j 0 ed D Î D / begi SWAP (x j, x j+ D ) j j - D AHU Data Structures & Alg., pp

16 procedure passes of SHELLSORT: 1 icremet sort ( X k, X + k) for k=1,..., icremet sort ( X, X k, X, X +k +k 4 4 for k=1,..., k) ( for icremet sort (Y 1,..., Y l ) i = z by 1 util i> or Xi-1 < Xi facts if X i, X do for j=1 by -1 util 1 do if X j-1 > X j the swap ( X j-1, X j) sorted i pass p + 1 p fi they remai sorted i later passes () distace betwee comparisos dimiish as,,..., 4,... p (3) The best kow time boud is 0 ( 1.5 ) 31 3

17 procedure RADIXSORT iput X 1,..., X e {1,..., } [1] for j=1,..., do iitialize B[j] to be the empty list [] for i=1,..., do add i to B[X i ] [3] let L = (i 1, i,..., i ) be the cocateatio of B[1],..., B[] -- Costs O() time o uit cost RAM -- avoids W(log) lower boud o SORT by avoidig comparisos istead uses idexig of RAM -- geeralizes (i c passes) to key domais {1,..., c } [4] output X i1 X i... X i 33 34

18 ope problems i sortig Complexity of SHELLSORT -- very good i practice claims Sedgewick -- Is it q( 1.5 )? () Complexity of variable legth -- sort o multitape TM or RAM -- Is it W( log )? 35