5. DIVIDE AND CONQUER

Transcription

1 Divide-ad-coquer paradigm Lecture slides by Kevi Waye Copyright 005 Pearso-Addiso Wesley 5. DIVIDE AND CONQUER I mergesort coutig iversios radomized quicksort media ad selectio closest pair of poits Divide-ad-coquer. Divide up problem ito several subproblems (of the same kid). Solve (coquer) each subproblem recursively. Combie solutios to subproblems ito overall solutio. Most commo usage. Divide problem of size ito two subproblems of size /. Solve (coquer) two subproblems recursively. Combie two solutios ito overall solutio. O() time Cosequece. Brute force: Θ( ). Divide-ad-coquer: O( log ). O() time Last updated o 3/1/18 9:1 AM attributed to Julius Caesar Sortig problem 5. DIVIDE AND CONQUER Problem. Give a list L of elemets from a totally ordered uiverse, rearrage them i ascedig order. mergesort coutig iversios radomized quicksort media ad selectio closest pair of poits SECTIONS

2 Sortig applicatios Mergesort Obvious applicatios. Orgaize a MP3 library. Display Google PageRak results. List RSS ews items i reverse chroological order. Some problems become easier oce elemets are sorted. Idetify statistical outliers. Biary search i a database. Remove duplicates i a mailig list. No-obvious applicatios. Covex hull. Closest pair of poits. Iterval schedulig / iterval partitioig. Schedulig to miimize maximum lateess. Miimum spaig trees (Kruskal s algorithm) Recursively sort left half. Recursively sort right half. Merge two halves to make sorted whole. iput A L G O R I T H M S sort left half A G L O R I T H M S sort right half A G L O R H I M S T merge results A G H I L M O R S T 6 Mergig Mergesort implemetatio Goal. Combie two sorted lists A ad B ito a sorted whole C. Sca A ad B from left to right. Compare a i ad b j. If a i b j, apped a i to C (o larger tha ay remaiig elemet i B). If a i > b j, apped b j to C (smaller tha every remaiig elemet i A). Iput. List L of elemets from a totally ordered uiverse. Output. The elemets i ascedig order. MERGE-SORT(L) IF (list L has oe elemet) RETURN L. sorted list A ai 18 merge to form sorted list C sorted list B 11 bj 0 3 Divide the list ito two halves A ad B. A MERGE-SORT(A). T( / ) B MERGE-SORT(B). T( / ) L MERGE(A, B). Θ() RETURN L. 7 8

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sha1_base64="oidllbbcsvzr990sj0ljhkzac4e=">aaacwxicbvdlsgmxfe3hv6vvpdugkvwucqmccoicg5cvrc0kldjrtozlkso6izehv+dvu9r8ep8b0sbdvc0lozrk3n/feqrqwff+r4ksrq1vfddlw9s7u3vlyv6j1zh0oraatoomqupfdrroiraoalsyrwplyd6k0xmfzo9ycjflojgyjrf5yho6kyh17ra+q0gzjla+vg/gtometrnfqchdshxoqdihdc3khpwrftfbv0lgjmokk0okphp+xpiwgketmbsfwu+zmzkdgesalmloqmj5ka+g4qfgctptprxvty8f0af8btxtskfu7imejtamkdpkjwe7re3i/7rohvlbi5umieopmvuzyrftsc+0z4wwfgohgdccpdxyp+zyrydmwtdpm+wbcmyv8zjbjuwrir8rungzsxgxp/olmafy7t+fvw8u5ywyse5iickiofkhtyrbmkstt7io/kg4vvz/okxmmw6hxmnqdkibydh4oetjc=</latexit> A useful recurrece relatio Divide-ad-coquer recurrece: recursio tree Def. T () = max umber of compares to mergesort a list of legth. Recurrece. T () Solutio. T () is O( log ). 0 =1 T ( / )+T( / ) + >1 betwee / ad 1 compares Assorted proofs. We describe several ways to solve this recurrece. Iitially we assume is a power of ad replace with = i the recurrece. log Propositio. If T () satisfies the followig recurrece, the T () = log. T () = 0 =1 T (/) + >1 T () T ( / ) T ( / ) T ( / 4) T ( / 4) T ( / 4) T ( / 4) assumig is a power of = (/) = 4 (/4) = T ( / 8) T ( / 8) T ( / 8) T ( / 8) T ( / 8) T ( / 8) T ( / 8) T ( / 8) 8 (/8) = 9 T() = log 10 Proof by iductio Divide-ad-coquer: quiz 1 Propositio. If T () satisfies the followig recurrece, the T () = log. T () = Pf. [ by iductio o ] Base case: whe = 1, T(1) = 0 = log. Iductive hypothesis: assume T() = log. Goal: show that T() = log (). iductive hypothesis 0 =1 T (/) + >1 recurrece T() = T() + = log + assumig is a power of Which is the exact solutio of the followig recurrece? T () = A. T() = log B. T() = log C. T() = log + log 1 D. T() = log log + 1 E. Not eve Kuth kows. 0 =1 T ( / ) + T ( / )+ 1 >1 = k=1 log k o loger assumig is a power of = (log () 1) + = log (). 11 1

4 <latexit sha1_base64="plaap4pyckuiob8guiy/wl4ixim=">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</latexit> <latexit sha1_base64="plaap4pyckuiob8guiy/wl4ixim=">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</latexit> <latexit sha1_base64="plaap4pyckuiob8guiy/wl4ixim=">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</latexit> <latexit sha1_base64="plaap4pyckuiob8guiy/wl4ixim=">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</latexit> Aalysis of mergesort recurrece Digressio: sortig lower boud Propositio. If T() satisfies the followig recurrece, the T() log. T () Pf. [ by strog iductio o ] Base case: = 1. Defie 1 = / ad = / ad ote that = 1 +. Iductio step: assume true for 1,,..., 1. iductive hypothesis T() 0 =1 T ( / )+T ( / ) + >1 T( 1) + T( ) + 1 log 1 + log + 1 log + log + = log + ( log 1) + = log. o loger assumig is a power of = d/e d e l m apple d dlog e e l / m = dlog e / d eappled e log appledlog e 1 d eappled e a iteger e Challege. How to prove a lower boud for all coceivable algorithms? Model of computatio. Compariso trees. Ca access the elemets oly through pairwise comparisos. All other operatios (cotrol, data movemet, etc.) are free. Cost model. Number of compares. Q. Realistic model? A1. Yes. Java, Pytho, C++, A. Yes. Mergesort, isertio sort, quicksort, heapsort, A3. No. Bucket sort, radix sorts, Comparable[] a =...;... Arrays.sort(a); ca access elemets oly via calls to compareto() 14 Compariso tree (for 3 distict keys a, b, ad c) Sortig lower boud Theorem. Ay determiistic compare-based sortig algorithm must make yes b < c yes o a < b code betwee compares (e.g., sequece of exchages) yes o a < c height of prued tree = worst-case umber of compares o Ω( log ) compares i the worst-case. Pf. [ iformatio theoretic ] Assume array cosists of distict values a 1 through a. Worst-case umber of compares = height h of prued compariso tree. Biary tree of height h has h leaves.! differet orderigs! reachable leaves. a b c a < c b a c b < c yes o yes o h a c b c a b b c a c b a each reachable leaf correspods to oe (ad oly oe) orderig; exactly oe reachable leaf for each possible orderig 15! leaves h leaves 16

5 Sortig lower boud SHUFFLING A LINKED LIST Theorem. Ay determiistic compare-based sortig algorithm must make Ω( log ) compares i the worst-case. Pf. [ iformatio theoretic ] Assume array cosists of distict values a 1 through a. Worst-case umber of compares = height h of prued compariso tree. Biary tree of height h has h leaves.! differet orderigs! reachable leaves. h # leaves! h log (!) log / l Stirlig s formula Problem. Give a sigly liked list, rearrage its odes uiformly at radom. Assumptio. Access to a perfect radom-umber geerator. Performace. O( log ) time, O(log ) extra space. iput shuffled L ull L ull all! permutatios equally likely Note. Lower boud ca be exteded to iclude radomized algorithms Coutig iversios SECTION DIVIDE AND CONQUER mergesort coutig iversios radomized quicksort media ad selectio closest pair of poits Music site tries to match your sog prefereces with others. You rak sogs. Music site cosults database to fid people with similar tastes. Similarity metric: umber of iversios betwee two rakigs. My rak: 1,,,. Your rak: a 1, a,, a. Sogs i ad j are iverted if i < j, but a i > a j. Brute force: check all Θ( ) pairs. A B C D E me you iversios: 3-, 4-0

6 Coutig iversios: applicatios Coutig iversios: divide-ad-coquer Votig theory. Divide: separate list ito two halves A ad B. Collaborative filterig. Coquer: recursively cout iversios i each list. Measurig the sortedess of a array. Combie: cout iversios (a, b) with a A ad b B. Sesitivity aalysis of Google s rakig fuctio. Retur sum of three couts. Rak aggregatio for meta-searchig o the Web. Noparametric statistics (e.g., Kedall s tau distace). iput 1 Rak Aggregatio Methods for the Web Cythia Dwork Ravi Kumar Moi Naor Rak Aggregatio Methods for the Web Cythia Dwork Ravi Kumar Moi Naor D. Sivakumar cout iversios i left half A D. Sivakumar ABSTRACT cout iversios i right half B ABSTRACT cout iversios (a, b) with a A ad b B 1 1. INTRODUCTION Motivatio Motivatio 1. 8 output = 17 INTRODUCTION Coutig iversios: how to combie two subproblems? Coutig iversios: how to combie two subproblems? Q. How to cout iversios (a, b) with a A ad b B? Cout iversios (a, b) with a A ad b B, assumig A ad B are sorted. A. Easy if A ad B are sorted! Sca A ad B from left to right. Compare ai ad bj. Warmup algorithm. If ai < bj, the ai is ot iverted with ay elemet left i B. Sort A ad B. If ai > bj, the bj is iverted with every elemet left i A. Copyright is held by the author/ower. WWW10, May 1-5, 001, Hog Kog. ACM /01/0005. For each elemet b B, biary search i A to fid how elemets i A are greater tha b. Apped smaller elemet to sorted list C. Copyright is held by the author/ower. WWW10, May 1-5, 001, Hog Kog. ACM /01/ list A 7 list B sort A cout iversios (a, b) with a A ad b B 3 sort B ai 18 3 biary search to cout iversios (a, b) with a A ad b B bj 0 3 merge to form sorted list C

7 <latexit sha1_base64="3pxfe56q7vvcgldq8zgmwt6i7u=">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</latexit> <latexit sha1_base64="3pxfe56q7vvcgldq8zgmwt6i7u=">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</latexit> <latexit sha1_base64="3pxfe56q7vvcgldq8zgmwt6i7u=">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</latexit> <latexit sha1_base64="3pxfe56q7vvcgldq8zgmwt6i7u=">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</latexit> Coutig iversios: divide-ad-coquer algorithm implemetatio Coutig iversios: divide-ad-coquer algorithm aalysis Iput. List L. Output. Number of iversios i L ad L i sorted order. Propositio. The sort-ad-cout algorithm couts the umber of iversios i a permutatio of size i O( log ) time. Pf. The worst-case ruig time T() satisfies the recurrece: SORT-AND-COUNT(L) IF (list L has oe elemet) RETURN (0, L). T () = (1) =1 T ( / ) + T ( / ) + () >1 Divide the list ito two halves A ad B. (r A, A) SORT-AND-COUNT(A). (r B, B) SORT-AND-COUNT(B). (r AB, L) MERGE-AND-COUNT(A, B). T( / ) T( / ) Θ() RETURN (r A + r B + r AB, L) WAY PARTITIONING 5. DIVIDE AND CONQUER mergesort coutig iversios radomized quicksort media ad selectio closest pair of poits Goal. Give a array A ad pivot elemet p, partitio array so that: Smaller elemets i left subarray L. Equal elemets i middle subarray M. Larger elemets i right subarray R. Challege. O() time ad O(1) space. the array A SECTION p the partitioed array A L M R 8

8 Radomized quicksort Radomized quicksort Pick a radom pivot elemet p A. 3-way partitio the array ito L, M, ad R. Recursively sort both L ad R. Pick a radom pivot elemet p A. 3-way partitio the array ito L, M, ad R. Recursively sort both L ad R. the array A partitio A sort L p RANDOMIZED-QUICKSORT(A) IF (array A has zero or oe elemet) RETURN. Pick pivot p A uiformly at radom. (L, M, R) PARTITION-3-WAY(A, p). Θ() sort R RANDOMIZED-QUICKSORT(L). RANDOMIZED-QUICKSORT(R). T(i) T( i 1) ew aalysis required (i is a radom variable depeds o p) the sorted array A Aalysis of radomized quicksort Aalysis of radomized quicksort Propositio. The expected umber of compares to quicksort a array of distict elemets a 1 < a < < a is O( log ). Pf. Cosider BST represetatio of pivot elemets. Propositio. The expected umber of compares to quicksort a array of distict elemets a 1 < a < < a is O( log ). Pf. Cosider BST represetatio of pivot elemets. a i ad a j are compared oce iff oe is a acestor of the other. the origial array of elemets A a 7 a 6 a 1 a 3 a 11 a 8 a 9 a 1 a 4 a 10 a a 13 a 5 first pivot i left subarray first pivot (chose uiformly at radom) first pivot i left subarray first pivot (chose uiformly at radom) a3 ad a6 are compared (whe a3 is pivot) a9 a9 a3 a11 a3 a11 a1 a8 a10 a13 a1 a8 a10 a13 a a 4 a 1 a a 4 a 1 a 6 a 6 a 5 a 7 31 a 5 a 7 3

9 Aalysis of radomized quicksort Divide-ad-coquer: quiz Propositio. The expected umber of compares to quicksort a array of distict elemets a1 < a < < a is O( log ). Give a array of 8 distict elemets a1 < a < < a, what is the probability that a7 ad a8 are compared durig radomized quicksort? Pf. Cosider BST represetatio of pivot elemets. a i ad a j are compared oce iff oe is a acestor of the other. A. 0 B. 1 / C. / first pivot i left subarray first pivot (chose uiformly at radom) a ad a8 are ot compared (because a3 partitios them) D. 1 if two elemets are adjacet, oe must be a acestor of the other a9 a9 a3 a11 a3 a11 a1 a8 a10 a13 a1 a8 a10 a13 a a4 a1 a a4 a1 a6 a6 a5 a7 33 a5 a7 34 Divide-ad-coquer: quiz 3 Aalysis of radomized quicksort Give a array of distict elemets a 1 < a < < a, what is the probability that a 1 ad a are compared durig radomized quicksort? Propositio. The expected umber of compares to quicksort a array of distict elemets a 1 < a < < a is O( log ). A. 0 B. 1 / Pf. Cosider BST represetatio of pivot elemets. a i ad a j are compared oce iff oe is a acestor of the other. Pr [ a i ad a j are compared ] = / ( j - i + 1), where i < j. C. / a1 ad a are compared iff first pivot is either a1 or a D. 1 Pr[a ad a8 compared] = /7 compared iff either a or a8 is chose as pivot before ay of { a3, a4, a5, a6, a7} a9 a9 a3 a11 a3 a11 a1 a8 a10 a13 a1 a8 a10 a13 a a 4 a 1 a a 4 a 1 a 6 a 6 a 5 a 7 35 a 5 a 7 36

10 Aalysis of radomized quicksort Toy Hoare u m b e r ). 9.9 X is u s e d to r e p r e s e t i f i i t y. I m a g i a r y v a l u e s of x m a y o t be e g a t i v e a d rem v a l u e s of x m a y o t be s m a l l e r t h a 1. V a l u e s of Qd~'(x) m a y be c a l c u l a t e d e a s i l y by h y p e r g e o m e t r i c series if x is o t t o o s m a l l o r ( - m ) t o o large. Q~m(x) c a be c o m p u t e d f r o m a a p p r o p r i a t e set of v a l u e s of Pm(X) if X is e a r 1.0 or ix is e a r 0. L o s s of s i g i f i c a t d i g i t s o c c u r s for x as s m a l l as 1.1 if is l a r g e r t h a 10. L o s s of s i g i f i c a t d i g i t s is a m a j o r diffic u l t y i u s i g fiite p o l y o m i M r e p r e s e t a t i o s also if is l a r g e r t h a m. H o w e v e r, Q L E G h a s b e e t e s t e d i r e g i o s of x a d b o t h large a d small; procedure Q L E G ( m, m a x, x, ri, R, Q); v a l u e I, m a x, x, ri; r e a l I, m a x, x, ri; r e a l a r r a y R, Q; r e a l t, i,, q0, s; : = 0; i f m a x > 13 t h e := max +7; i f ri = 0 t h e ifm = 0the Q[0] : = 0.5 X 10g((x + 1 ) / ( x - 1)) else t : = / s q r t ( x X x - - 1); q0 : = 0; Q[O] : = t ; for i : = 1 step 1 util m do s := (x+x)x(i-1)xt Q[0]+ (3i-i i-) q0; q0 : = Q[0]; Q[0] : = s e d e d ; if x = 1 the Q[0] : = 9.9 I" 45; R [ + 1] : = x - s q r t ( x X x - 1); for i : = s t e p --1 u t i l 1 d o R[i] : = (i + m ) / ( ( i + i + 1) X x +(m-i1) X R [ i + l ] ) ; g o to t h e e d ; if m = 0 the b e g i i f x < 0.5 t b e Q[0] : = a r c t a ( x ) e l s e Q[0] : = - a r e t a ( 1 / x ) e d e l s e t : = 1 / s q r t ( x X x + 1); q0 : = 0; Propositio. The expected umber of compares to quicksort a array of distict elemets a1 < a < couet I a d J are o u t p u t v a r i a b l e s, a d A is t h e a r r a y ( w i t h s u b s c r i p t b o u d s M : N ) w h i c h is o p e r a t e d u p o b y t h i s p r o c e d u r e. P a r t i t i o t a k e s t h e v a l u e X of a r a d o m e l e m e t of t h e a r r a y A, a d r e a r r a g e s t h e v a l u e s of t h e e l e m e t s of t h e a r r a y i s u c h a w a y t h a t t h e r e e x i s t i t e g e r s I a d J w i t h t h e followig p r o p e r t i e s : M _-< J < I =< N p r o v i d e d M < N A[R] =< X f o r M =< R _-< J A[R] = X f o r J < R < I A[R] ~ X f o r I =< R ~ N The procedure uses a iteger procedure radom (M,N) which chooses equiprobably a radom iteger F betwee M ad N, ad also a p r o c e d u r e e x c h a g e, w h i c h e x c h a g e s t h e v a l u e s of its t w o parameters ; real X; iteger F; F : = r a d o m ( M, N ) ; X : = A[F]; I:=M; J:=N; for I : = I s t e p 1 u t i l N d o up: i f X < A [I] t h e g o to d o w ; I:=N; dow: f o r J : = J s t e p --1 u t i l M d o if A[J]<X the go to chage; J:=M; c h a g e : i f I < J t h e b e g i e x c h a g e (A[IL A[J]); I := I+ 1;J:= J - 1; g o to u p ed 4 else i f [ < F t h e b e g i e x c h a g e (A[IL A[F]) i I:=I+l ed else i f F < J t l l e b e g i e x c h a g e (A[F], A[J]) ; J:=J-1 ed ; ed partitio Iveted quicksort to traslate Russia ito Eglish. < a is O( log ). [ but could t explai his algorithm or implemet it! ] Leared Algol 60 (ad recursio). Pf. Cosider BST represetatio of pivot elemets. Implemeted quicksort. ai ad aj are compared oce iff oe is a acestor of the other. Pr [ ai ad aj are compared ] = / ( j - i + 1), where i < j. i=1 i=1 j=i+1 j=i+1 i=1 j=i+1 i=1 j=i+1 Expected umber of compares = all pairs i ad j <latexit sha1_base64="tsqczvbieugfpzvbrgcekd3sgfa=">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</latexit> <latexit sha1_base64="gjzqomht99byyg9epbf0urts8xc=">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</latexit> j j j i+ 1 i 1 i 1 = = = i+1 i+1 i+1 1 i+1 1 j1 i=1 j= j= j i=1 i=1 j= j i=1 j= 1 1 j1 j=1 j j=1 j=1 j j=1 1 1) (l + q[0] := t; 1 dx (l x dx + 1) x=1 x=1 x = l = l for R[ + for harmoic sum Remark. Number of compares oly decreases if equal elemets. ALGORITHM for the: for i : = ed QLEG; 61 P R O C E D U R E S 64F O R R A N G E A R I T H M E T I C ALGORITHM ALLAN GIBB* QUICKSORT U ia. v e r sr. i t y HOARE of A l b e r t a, C a l g a r y, A l b e r t a, C a a d a C. Elliott b e g i Brothers Ltd., Borehamwood, Hertfordshire, Eg. i : = step 1 util m do b e g i s : = (x + x) X (i -- 1) X t X Q[0I +(3i+iX i -- ) q0; qo : = Q[0]; Q[0] := s e d e d ; 1] : = x - s q r t ( x x + 1); i := step - 1 util 1 do R[i] : = (i + m ) / ( ( i -- m + 1) R[i + 1] --(i+i+ 1) X x); i : = 1 step util max do Rill : = -- Rill; 1 step 1 util mx do Q[i] : = Q[i - 1] X R[i] * T h i s p r o c e d u r e w a s d e v e l o p e d i p a r t u d e r t h e s p o s o r s h i p of t h e A i r F o r c e C a m b r i d g e R e s e a r c h C e t e r. p r o c e d u r e R A N G E S U M (a, b, c, d, e, f); procedure quicksort (A,M,N); value M,N; real a, b, c, d, e, f ; a r r a y A; i t e g e r M, N ; c o m m e t The term "rage u m b e r " was used by P. S. Dwyer, Q u i c k s o r t is a v e r y f a s t a d c o v e i e t m e t h o d of commet Liear Computatios (Wiley, 1951). Machie procedures for s o r t i g a a r r a y i t h e r a d o m - a c c e s s s t o r e of a c o m p u t e r. T h e rage arithmetic were developed about 1958 by Ramo Moore, e t i r e c o t e t s of t h e s t o r e m a y be s o r t e d, sice o e x t r a s p a c e is "Automatic Error Aalysis i Digital C o m p u t a t i o, " LMSD r e q u i r e d. T h e a v e r a g e u m b e r of c o m p a r i s o s m a d e is ( M - - N ) I Report 4841, 8 Ja. 1959, Lockheed Missiles ad Space Divi( N - - M ), a d t h e a v e r a g e m b e r of e x c h a g e s is oe s i x t h t h i s sio, Palo Alto, Califoria, 59 pp. If a _< x -< b ad c ~ y ~ d, a m o u t. S u i t a b l e r e f i e m e t s of t h i s m e t h o d will be d e s i r a b l e for the R A N G E S U M yields a iterval [e, f] such t h a t e =< (x + y) its i m p l e m e t a t i o o a y a c t u a l c o m p u t e r ; f. Because of machie operatio (trucatio or roudig) the i t e g e r 1,J ; machie sums a -4- c ad b -4- d may ot provide safe ed-poits if M < N the partitio (A,M,N,I,J); of the output iterval. Thus R A N G E S U M requires a o-local quicksort (A,M,J) ; real procedure A D J U S T S U M which will compesate for the q u i c k s o r t (A, I, N ) machie arithmetic. The body of A D J U S T S U M will be deed pedet upo the type of machie for which it is writte ad so ed quieksort is ot give here. (A example, however, appears below.) I t is assumed t h a t A D J U S T S U M has as parameters real v ad w, ad iteger i, ad is accompaied by a o-local real procedure C O R R E C T I O N 65 which gives a upper boud to the magitude ALGORITHM of the error ivolved i the machie represetatio of a umber. FIND The output A D J U S T S U M provides the left ed-poit of the C.output A. R.iterval HOARE of R A N G E S U M whe A D J U S T S U M is called Elliott Borehamwood, Hertfordshire, with i Brothers = --1, adltd., the right ed-poit whe called with i Eg. = 1 The procedures RANGESUB, R A N G E M P Y, ad R A N G E D V D procedure fid ( A, M, N, K ) ; v a l u e M, N, K ; provide for the remaiig fudametal operatios i rage a r r a y A; i t e g e r M, N, K ; arithmetic. R A N G E S Q R gives a iterval withi which the commet F i d will a s s i g to A [K] t h e v a l u e w h i c h it w o u l d square of a rage mber must lie. R N G S U M C, RNGSUBC, h a v e if t h e a r r a y A [ M : N ] h a d b e e s o r t e d. T h e a r r a y A will be R N G M P Y C ad R N G D V D C provide for rage arithmetic with p a r t l y s o r t e d, a d s u b s e q u e t e t r i e s will be f a s t e r t h a t h e first; complex rage argumets, i.e. the real ad imagiary parts are rage umbers~ Commuicatios of the ACM 31 Commuicatios of the ACM (July 1961) 37 ALGORITHM 63 PARTITION C. A. R. HOARE Elliott Brothers Ltd., Borehamwood, Hertfordshire, Eg. procedure NUTS AND BOLTS partitio (A,M,N,I,J); value M,N; a r r a y A; i t e g e r M, N, 1, J ; e : = A D J U S T S U M (a, c, - 1 ) ; f : = A D J U S T S U M (b, d, 1) ed RANGESUM; p r o c e d u r e R A N G E S U B (a, b, c, d, e, f) ; real a, b, c, d, e, f; c o m m e t R A N G E S U M is a o-local procedure; R A N G E S U M (a, b, - d, --c, e, f) ed RANGESUB ; p r o c e d u r e R A N G E M P Y (a, b, c, d, e, f); real a, b, c, d, e, f; c o m m e t A D J U S T P R O D, which appears at the ed of this procedure, is aalogous to A D J U S T S U M above ad is a olocal real procedure. M A X ad M I N fid the maximum ad miimum of a set of real umbers ad are o-local; real v, w; i f a < 0 A c => 0 t h e 1: b e g i v:=c; c:=a; a:=v; w:=d; d:=b; b:=w e d 1; i f a => O t h e 5. D IVIDE Problem. A disorgaized carpeter has a mixed pile of uts ad bolts. The goal is to fid the correspodig pairs of uts ad bolts. mergesort Each ut fits exactly oe bolt ad each bolt fits exactly oe ut. AND Toy Hoare 1980 Turig Award : b e g i i f c >= 0 t h e 3: e:= axe;f := bxd;goto8 ed 3 ; e:=bxc; ifd ~ 0the 4: f:=bxd; goto8 e d 4; f:=axd; goto8 5: e d ; ifb > 0 the 6: b e g i if d > 0 the e : = M I N ( a X d, b X c); f : = M A X ( a X c, b X d); go t o 8 e d 6; e : = b X c; f : = a X c; go t o 8 e d 5; f:=axc; i f d _-<O t h e 7: b e g i e:=bxd; goto8 ed 7 ; e:=axd; 8: e : = A D J U S T P R O D (e, - 1 ) ; f := A D J U S T P R O D (f, 1) ed RANGEMPY; p r o c e d u r e R A N G E D V D (a, b, c, d, e, f) ; real a, b, c, d, e, f; c o m m e t If the rage divisor icludes zero the program exists to a o-local label " z e r o d v s r ". R A N G E D V D assumes a o-local real procedure A D J U S T Q U O T which is aalogous (possibly idetical) to A D J U S T P R O D ; 38 i f c =< 0 A d ~ 0 t h e go to zer0dvsr; if c < 0 the 1: b e g i ifb > 0 the : e : = b / d ; go t o 3 e d ; e := b/c; 3: i f a -->_0 t h e 4: f : = a / c ; go t o 8 e d 4; f : = a / d ; go t o 8 ed 1; if a < 0 the 5: b e g i e : = a/c; go t o 6 ed 5 ; e := a/d; 6: i f b > 0 t h e 7: b e g i f : = b / c ; go t o 8 ed 7 ; f := b/d; 8: e : = ADJUSTQUOT (e, - 1 ) ; f : = ADJUSTQUOT (f,1) ed RANGEDVD ; p r o c e d u r e R A N G E S Q R (a, b, e, f); r e a l a, b, e, f; c o m m e t A D J U S T P R O D is a o-10cal procedure; if a < 0 the C ONQUER coutig iversios By fittig a ut ad a bolt together, the carpeter ca see which oe is bigger (but caot directly compare either two uts or two bolts). radomized quicksort media ad selectio closest pair of poits SECTION 9.3 Brute-force solutio. Compare each bolt to each ut Θ ( ) compares. Challege. Desig a algorithm that makes O( log ) compares. 39 C o m m u i c a t i o s o f t h e &CM 319

11 Media ad selectio problems Radomized quickselect Selectio. Give elemets from a totally ordered uiverse, fid k th smallest. Miimum: k = 1; maximum: k =. Media: k = ( + 1) /. O() compares for mi or max. O( log ) compares by sortig. O( log k) compares with a biary heap. max heap with k smallest Pick a radom pivot elemet p A. 3-way partitio the array ito L, M, ad R. Recur i oe subarray the oe cotaiig the k th smallest elemet. Applicatios. Order statistics; fid the top k ; bottleeck paths, Q. Ca we do it with O() compares? A. Yes! Selectio is easier tha sortig. QUICK-SELECT(A, k) Pick pivot p A uiformly at radom. (L, M, R) PARTITION-3-WAY(A, p). Θ() IF (k L ) RETURN QUICK-SELECT(L, k). T(i) ELSE IF (k > L + M ) RETURN QUICK-SELECT(R, k L M ) ELSE IF (k = L ) RETURN p. T( i 1) Radomized quickselect aalysis Selectio i worst-case liear time Ituitio. Split cady bar uiformly expected size of larger piece is ¾. Def. T(, k) = expected # compares to select k th smallest i array of legth. Def. T() = max k T(, k). T() T(3 / 4) + T() 4 ot rigorous: ca t assume E[T(i)] T(E[i]) Goal. Fid pivot elemet p that divides list of elemets ito two pieces so that each piece is guarateed to have 7/10 elemets. Q. How to fid approximate media i liear time? A. Recursively compute media of sample of /10 elemets. Propositio. T() 4. Pf. [ by strog iductio o ] Assume true for 1,,, 1. T() satisfies the followig recurrece: ca assume we always recur of larger of two subarrays sice T() is mootoe o-decreasig T() = Θ(1) if = 1 T (7/10 ) + T (/10 ) + Θ() two subproblems of differet sizes! otherwise T() + 1 / [ T( / ) + + T( 3) + T( ) + T( 1) ] + 1 / [ 8( / ) + + 8( 3) + 8( ) + 8( 1) ] + 1 / (3 ) = 4. tiy cheat: sum should start at T( / ) iductive hypothesis T() = Θ() we ll eed to show this 45 46

12 Choosig the pivot elemet Choosig the pivot elemet Divide elemets ito / 5 groups of 5 elemets each (plus extra). Divide elemets ito / 5 groups of 5 elemets each (plus extra). Fid media of each group (except extra). medias = = Choosig the pivot elemet Media-of-medias selectio algorithm Divide elemets ito / 5 groups of 5 elemets each (plus extra). Fid media of each group (except extra). Fid media of / 5 medias recursively. Use media-of-medias as pivot elemet. medias MOM-SELECT(A, k) A. IF ( < 50) RETURN k th smallest of elemet of A via mergesort. media of medias Group A ito / 5 groups of 5 elemets each (igore leftovers) B media of each group of 5. p MOM-SELECT(B, / 10 ) media of medias (L, M, R) PARTITION-3-WAY(A, p) IF (k L ) RETURN MOM-SELECT(L, k). ELSE IF (k > L + M ) RETURN MOM-SELECT(R, k L M ) ELSE RETURN p. =

13 Aalysis of media-of-medias selectio algorithm Aalysis of media-of-medias selectio algorithm At least half of 5-elemet medias p. At least half of 5-elemet medias p. At least / 5 / = / 10 medias p. medias medias p media of medias p media of medias p = = 54 5 Aalysis of media-of-medias selectio algorithm Aalysis of media-of-medias selectio algorithm At least half of 5-elemet medias p. At least / 5 / = / 10 medias p. At least 3 / 10 elemets p. At least half of 5-elemet medias p. medias p medias media of medias p media of medias p = = 54 54

14 Aalysis of media-of-medias selectio algorithm Aalysis of media-of-medias selectio algorithm At least half of 5-elemet medias p. At least / 5 / = / 10 medias p. At least half of 5-elemet medias p. At least / 5 / = / 10 medias p. At least 3 / 10 elemets p. medias p medias p media of medias p media of medias p = = Media-of-medias selectio algorithm recurrece Media-of-medias selectio algorithm recurrece Media-of-medias selectio algorithm recurrece. Media-of-medias selectio algorithm recurrece. Select called recursively with / 5 elemets to compute MOM p. Select called recursively with / 5 elemets to compute MOM p. At least 3 / 10 elemets p. At least 3 / 10 elemets p. At least 3 / 10 elemets p. At least 3 / 10 elemets p. Select called recursively with at most 3 / 10 elemets. Select called recursively with at most 3 / 10 elemets. Def. C() = max # compares o ay array of elemets. Def. C() = max # compares o ay array of elemets. C() C ( /5 ) + C ( 3 /10 ) C() C ( /5 ) + C ( 3 /10 ) media of medias recursive select computig media of 5 ( 6 compares per group) media of medias recursive select computig media of 5 ( 6 compares per group) partitioig ( compares) partitioig ( compares) Ituitio. Now, let s solve give recurrece. C() is goig to be at least liear i C() is super-additive. Assume is both a power of 5 ad a power of 10? Igorig floors, this implies that C() C( / / 10) + 11/5 Prove that C() is mootoe o-decreasig. = C(9 / 10) + 11/5 C()

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sha1_base64="uctptmzk5kuvff1msqyvcmsiy=">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</latexit> <latexit sha1_base64="uctptmzk5kuvff1msqyvcmsiy=">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</latexit> Divide-ad-coquer: quiz 4 Media-of-medias selectio algorithm recurrece Cosider the followig recurrece 0 1 C() = C( /5 )+C( 3 /10 ) >1 Is C() mootoe o-decreasig? A. Yes, obviously. B. Yes, but proof is tedious. C. Yes, but proof is hard. D. No. Aalysis of selectio algorithm recurrece. T() = max # compares o ay array of elemets. T() is mootoe o-decreasig, but C() is ot! T () Claim. T() 44. Pf. [ by strog iductio ] 6 <50 max{ T ( 1), T( /5 )+T( 3 /10 ) ) } 50 Base case: T() 6 for < 50 (mergesort). Iductive hypothesis: assume true for 1,,, 1. Iductio step: for 50, we have either T() T( 1) 44 or T() T( / 5 ) + T( 3 / 10 ) + 11/5 C(19) = 165 C(0) = 134. iductive hypothesis 44 ( / 5 ) + 44 ( 3 / 10 ) + 11/5 44 ( / 5) ( / 4) + 11/5 for 50, 3 / 10 / 4 59 = Divide-ad-coquer: quiz 5 Liear-time selectio retrospective Suppose that we divide elemets ito / r groups of r elemets each, ad use the media-of-medias of these / r groups as the pivot. For which r is the worst-case ruig time of select O()? A. r = 3 T() = T( / 3) + T( / 6) + Θ() T() = Θ( log ) B. r = 7 T() = T( / 7) + T( 4 / 14) + Θ() T() = Θ() C. Both A ad B. D. Neither A or B. 61 Propositio. [Blum Floyd Pratt Rivest Tarja 1973] There exists a compare-based selectio algorithm whose worst-case ruig time is O(). Time Bouds for Selectio* MANUEL BLUM, ROBERT W. FLOYD, VAUGHAN PRATT, RONALD L. RIVEST, AND ROBERT E. TARJAN Departmet of Computer Sciece, Staford Uiversity, Staford, Califoria Received November 14, 197 The umber of comparisos required to select the i-th smallest of umbers is show to be at most a liear fuctio of by aalysis of a ew selectio algorithm--pick. Specifically, o more tha comparisos are ever required. This boud is improved for extreme values of i, ad a ew lower boud o the requisite umber of comparisos is also proved. Theory. Optimized versio of BFPRT: compares. Upper boud: [Dor Zwick 1995].95 compares. Lower boud: [Dor Zwick 1999] ( + 80 ) compares. Practice. Costats too large to be useful. 6

16 Closest pair of poits 5. DIVIDE AND CONQUER Closest pair problem. Give poits i the plae, fid a pair of poits with the smallest Euclidea distace betwee them. mergesort coutig iversios radomized quicksort media ad selectio closest pair of poits Fudametal geometric primitive. Graphics, computer visio, geographic iformatio systems, molecular modelig, air traffic cotrol. Special case of earest eighbor, Euclidea MST, Vorooi. fast closest pair ispired fast algorithms for these problems SECTION Closest pair of poits Closest pair of poits: first attempt Closest pair problem. Give poits i the plae, fid a pair of poits with the smallest Euclidea distace betwee them. Brute force. Check all pairs with Θ( ) distace calculatios. Sortig solutio. Sort by x-coordiate ad cosider earby poits. Sort by y-coordiate ad cosider earby poits. 1D versio. Easy O( log ) algorithm if poits are o a lie. No-degeeracy assumptio. No two poits have the same x-coordiate

17 Closest pair of poits: first attempt Closest pair of poits: secod attempt Sortig solutio. Sort by x-coordiate ad cosider earby poits. Sort by y-coordiate ad cosider earby poits. Divide. Subdivide regio ito 4 quadrats Closest pair of poits: secod attempt Closest pair of poits: divide-ad-coquer algorithm Divide. Subdivide regio ito 4 quadrats. Obstacle. Impossible to esure / 4 poits i each piece. Divide: draw vertical lie L so that / poits o each side. Coquer: fid closest pair i each side recursively. Combie: fid closest pair with oe poit i each side. Retur best of 3 solutios. seems like Θ( ) L

18 How to fid closest pair with oe poit i each side? How to fid closest pair with oe poit i each side? Fid closest pair with oe poit i each side, assumig that distace < δ. Observatio: suffices to cosider oly those poits withi δ of lie L. Fid closest pair with oe poit i each side, assumig that distace < δ. Observatio: suffices to cosider oly those poits withi δ of lie L. Sort poits i δ-strip by their y-coordiate. Check distaces of oly those poits withi 7 positios i sorted list! why? L L δ = mi(1, 1) 1 3 δ = mi(1, 1) 1 δ 71 δ 7 How to fid closest pair with oe poit i each side? Closest pair of poits: divide-ad-coquer algorithm Def. Let s i be the poit i the δ-strip, with the i th smallest y-coordiate. Claim. If j i > 7, the the distace betwee s i ad s j is at least δ. Pf. Cosider the δ-by-δ rectagle R i strip whose mi y-coordiate is y-coordiate of s i. Distace betwee s i ad ay poit s j above R is δ. Subdivide R ito 8 squares. At most 1 poit per square. diameter is / < At most 7 other poits ca be i R. δ R si L s j ½δ ½δ CLOSEST-PAIR(p 1, p,, p ) Compute vertical lie L such that half the poits are o each side of the lie. δ 1 CLOSEST-PAIR(poits i left half). δ CLOSEST-PAIR(poits i right half). δ mi { δ 1, δ }. Delete all poits further tha δ from lie L. Sort remaiig poits by y-coordiate. Sca poits i y-order ad compare distace betwee each poit ad ext 7 eighbors. If ay of these distaces is less tha δ, update δ. O() T( / ) T( / ) O() O( log ) O() costat ca be improved with more refied geometric packig argumet RETURN δ. δ 73 74

19 Divide-ad-coquer: quiz 6 Refied versio of closest-pair algorithm What is the solutio to the followig recurrece? Q. How to improve to O( log )? A. Do t sort poits i strip from scratch each time. Each recursive call returs two lists: all poits sorted by x-coordiate, (1) =1 ad all poits sorted by y-coordiate. >1 Sort by mergig two pre-sorted lists. T () = T ( / ) + T ( / ) + ( log ) <latexit sha1_base64="dpqsxs9emydq5iq+l5by9kuo1i=">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</latexit> A. T() = Θ(). B. T() = Θ( log ). C. T() = Θ( log ). D. Theorem. [Shamos 1975] The divide-ad-coquer algorithm for fidig a closest pair of poits i the plae ca be implemeted i O( log ) time. Pf. T() = Θ(). (1) =1 T () = T ( / ) + T ( / ) + () >1 <latexit sha1_base64="3pqz3or04schqniwe5a09tyf+wa=">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</latexit> Divide-ad-coquer: quiz 7 Computatioal complexity of closest-pair problem What is the complexity of the D closest pair problem? Theorem. [Be-Or 1983, Yao 1989] I quadratic decisio tree model, ay algorithm for closest pair (eve i 1D) requires Ω( log ) quadratic tests. A. Θ(). B. Θ( log* ). C. Θ( log log ). D. Θ( log ). E. Not eve Tarja kows. (x1 - x) + (y1 - y) Theorem. [Rabi 1976] There exists a algorithm to fid the closest pair of poits i the plae whose expected ruig time is O(). Volume 8, umber 1 IhFCRMATION PROCESSING LE,TTERS A NOTE ON RABIN S NEAREST-NEIGHBOR ALGORITHM * Steve FORTUNE ad Joh HOPCROFT Jauary 1979 ot subject to Ω( log ) lower boud because it uses the floor fuctio Departmet of Computer Sciece, Corell Uiversity,Ithaca, NY, U.S.A. Received 0 July 1978, revised versio received 1 August 1978 Probabiktic algorithms, earest eighbor, hashig itroductio Oe otio that has received some attetio recetly is that of a probabilistic algorithm. A algo- tio to the distace betwee earest eighbors i the whole set. Usig this approximatio the distace betwee the earest eighbor i the whole set ca be foud i expected liear time. The algorithm s use of

20 Digressio: computatioal geometry Covex hull Igeious divide-ad-coquer algorithms for core geometric problems. problem brute clever closest pair O( ) O( log ) farthest pair O( ) O( log ) covex hull O( ) O( log ) Delauay/Vorooi O( 4 ) O( log ) Euclidea MST O( ) O( log ) ruig time to solve a D problem with poits Note. 3D ad higher dimesios test limits of our igeuity. The covex hull of a set of poits is the smallest perimeter fece eclosig the poits. Equivalet defiitios. Smallest area covex polygo eclosig the poits. Itersectio of all covex set cotaiig all the poits Farthest pair Delauay triagulatio Give poits i the plae, fid a pair of poits with the largest Euclidea distace betwee them. Fact. Poits i farthest pair are extreme poits o covex hull. 81 The Delauay triagulatio is a triagulatio of poits i the plae such that o poit is iside the circumcircle of ay triagle. poit iside circumcircle of 3 poits Some useful properties. No edges cross. Delauay triagulatio of 19 poits Amog all triagulatios, it maximizes the miimum agle. Cotais a edge betwee each poit ad its earest eighbor. o poit i the set is iside the circumcircle 8

21 Euclidea MST Computatioal geometry applicatios Give poits i the plae, fid MST coectig them. [distaces betwee poit pairs are Euclidea distaces] Fact. Euclidea MST is subgraph of Delauay triagulatio. Implicatio. Ca compute Euclidea MST i O( log ) time. Compute Delauay triagulatio. Compute MST of Delauay triagulatio. it s plaar ( 3 edges) 83 Applicatios. Robotics. VLSI desig. Data miig. Medical imagig. Computer visio. Scietific computig. Fiite-elemet meshig. Astroomical simulatio. Models of physical world. Geographic iformatio systems. Computer graphics (movies, games, virtual reality). airflow aroud a aircraft wig 84