Algorithms ROBERT SEDGEWICK | KEVIN WAYNE 2.3 QUICKSORT ‣ quicksort ‣ selection Algorithms ‣ duplicate keys FOURTH EDITION ‣ system sorts ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu Two classic sorting algorithms: mergesort and quicksort Critical components in the world’s computational infrastructure. ・Full scientific understanding of their properties has enabled us# to develop them into practical system sorts. ・Quicksort honored as one of top 10 algorithms of 20th century# in science and engineering. # Mergesort. [last lecture] # # ... # # Quicksort. [this lecture] ... "2 2.3 QUICKSORT ! quicksort ! selection Algorithms ! duplicate keys ! system sorts ROBERT SEDGEWICK | KEVIN WAYNE http://algs4.cs.princeton.edu Quicksort Basic plan. ・Shuffle the array. ・Partition so that, for some j – entry a[j] is in place – no larger entry to the left of j – no smaller entry to the right of j ・Sort each subarray recursively. input Q U I C K S O R T E X A M P L E shufe K R A T E L E P U I M Q C X O S partitioning item partition E C A I E K L P U T M Q R X O S not greater not less sort left A C E E I K L P U T M Q R X O S sort right A C E E I K L M O P Q R S T U X result A C E E I K L M O P Q R S T U X Quicksort overview "4 number). 9.9 X 10 45 is used to represent infinity. Imaginary conunent I and J are output variables, and A is the array (with values of x may not be negative and reM values of x may not be subscript bounds M:N) which is operated upon by this procedure. smaller than 1. Partition takes the value X of a random element of the array A, Values of Qd~'(x) may be calculated easily by hypergeometric and rearranges the values of the elements of the array in such a seriesTony if x is not tooHoare small nor (n - m) too large. Q~m(x) can be way that there exist integers I and J with the following properties : computed from an appropriate set of values of Pnm(X) if X is near M _-< J < I =< NprovidedM < N 1.0 or ix is near 0. Loss of significant digits occurs for x as small as A[R] =< XforM =< R _-< J 1.1 if n is larger than 10. Loss of significant digits is a major diffi- A[R] = XforJ < R < I culty in using finite polynomiM representations also if n is larger A[R] ~ Xfor I =< R ~ N than m. However, QLEG has been tested in regions of x and n The procedure uses an integer procedure random (M,N) which both・ large andInvented small; quicksort to translatechooses equiprobably Russian a random integer into F between English. M and N, and procedure QLEG(m, nmax, x, ri, R, Q); value In, nmax, x, ri; also a procedure exchange, which exchanges the values of its two real In, mnax, x, ri; real array R, Q; parameters ; begin・ real [t, buti, n, q0, s;couldn't explain his beginalgorithm real X; integer or F; implement it! ] n := 20; F := random (M,N); X := A[F]; if nmax > 13 then I:=M; J:=N; n := nmax +7; up: for I : = I step 1 until N do ・if riLearned = 0then Algol 60 (and recursion).if X < A [I] then go to down; begin ifm = 0then I:=N; Q[0] := 0.5 X 10g((x + 1)/(x - 1)) down: forJ := J step --1 until M do ・Implementedelse quicksort. if A[J]<X then go to change; begin t := --1.0/sqrt(x X x-- 1); J:=M; q0 := 0; change: if I < J then begin exchange (A[IL A[J]); Q[O] : = t; I := I+ 1;J:= J - 1; for i : = 1 step 1 until m do go to up begin s := (x+x)X(i-1)Xt end 2: begin Tony Hoare 4 × Q[0]+ (3i-i× i-2)× q0; else if [ < F then begin exchange (A[IL A[F]) i if c >= 0 then q0 := Q[0]; I:=I+l 3:begin 1980 Turing Award Q[0] := s end end; end e:= aXe;f := bXd;goto8 if x = 1 then else if F < J tllen begin exchange (A[F], A[J]) ; end 3 ; Q[0] := 9.9 I" 45; J:=J-1 e:=bXc; R[n + 1] := x - sqrt(x X x - 1); end ; ifd ~ 0then for i := n step --1 until 1 do end partition 4: begin R[i] := (i + m)/((i + i + 1) X x f:=bXd; goto8 +(m-i- 1) X R[i+l]); end 4; ALGORITHM 61 go to the end; f:=aXd; goto8 if m = 0 then ALGORITHMPROCEDURES 64FOR RANGE ARITHMETIC 5: end 2; begin if x < 0.5 tben QUICKSORTALLAN GIBB* ifb > 0 then Q[0] := arctan(x) - 1.5707963 else UniversityC. A. R. HOAREof Alberta, Calgary, Alberta, Canada 6: begin Q[0] := - aretan(1/x)end else Elliottbegin Brothers Ltd., Borehamwood, Hertfordshire, Eng. if d > 0 then begin t := 1/sqrt(x X x + 1); begin procedure RANGESUM (a, b, c, d, e, f); q0 := 0; procedure quicksort (A,M,N); value M,N; e := MIN(a X d, b X c); real a,b,c,d,e,f; q[0] := t; array A; integer M,N; f:= MAX(a X c,b X d); go to8 comment The term "range number" was used by P. S. Dwyer, for i : = 2 step 1 until m do comment Quicksort is a very fast and convenient method of end 6; Linear Computations (Wiley, 1951). Machine procedures for begin s := (x + x) X (i -- 1) X t X Q[0I sorting an array in the random-access store of a computer. The e:= bX c; f := aX c; go to8 range arithmetic were developed about 1958 by Ramon Moore, +(3i+iX i -- 2) × q0; entire contents of the store may be sorted, since no extra space is end 5; "Automatic Error Analysis in Digital Computation," LMSD qO := Q[0]; required. The average number of comparisons made is 2(M--N) In f:=aXc; Report 48421, 28 Jan. 1959, Lockheed Missiles and Space Divi- Q[0] := s end end; (N--M), and the average nmnber of exchanges is one sixth this if d _-< O then sion, Palo Alto, California, 59 pp. If a _< x -< b and c ~ y ~ d, R[n + 1] := x - sqrt(x × x + 1); amount. Suitable refinements of this method will be desirable for 7: begin then RANGESUM yields an interval [e, f] such that e =< (x + y) for i := n step - 1 until 1 do its implementation on any actual computer; e:=bXd; goto8 f. Because of machine operation (truncation or rounding) the R[i] := (i + m)/((i -- m + 1) × R[i + 1] begin integer 1,J ; end 7 ; machine sums a -4- c and b -4- d may not provide safe end-points --(i+i+ 1) X x); if M < N then begin partition (A,M,N,I,J); e:=aXd; of the output interval. Thus RANGESUM requires a non-local for i : = 1 step 2 until nmax do quicksort (A,M,J) ; 8: e := ADJUSTPROD (e, -1); real procedure ADJUSTSUM which will compensate for the Rill := -- Rill; quicksort (A, I, N) f := ADJUSTPROD (f, 1) machine arithmetic. The body of ADJUSTSUM will be de- the: for i : = 1 step 1 until nnmx do end end RANGEMPY; pendent upon the type of machine for which it is written and so Q[i] := Q[i - 1] X R[i] end quieksort procedure RANGEDVD (a, b, c, d, e, f) ; is not given here. (An example, however, appears below.) It end QLEG; real a, b, c, d, e, f; is assumed that ADJUSTSUM has as parameters real v and w, comment If the range divisor includes zero the program * This procedure was developed in part under the sponsorship and integer i, and is accompanied by a non-local real procedure exists to a non-local label "zerodvsr". RANGEDVD assumes a of the Air Force Cambridge Research Center. CORRECTION which gives an upper bound to the magnitude CommunicationsALGORITHM 65 of the ACM (July 1961) non-local real procedure ADJUSTQUOT which is analogous of the error involved in the machine representation of a number. FIND (possibly identical) to ADJUSTPROD; The output ADJUSTSUM provides the left end-point of the begin C.output A. R. interval HOARE of RANGESUM when ADJUSTSUM is called if c =< 0 A d ~ 0 then go to zer0dvsr; ALGORITHM 63 Elliottwith i Brothers= --1, and Ltd., the right Borehamwood, end-point when Hertfordshire, called with i Eng.= 1 if c < 0 then PARTITION The procedures RANGESUB, RANGEMPY, and RANGEDVD "5 procedure find (A,M,N,K); value M,N,K; C. A. R. HOARE provide for the remaining fundamental operations in range 1: begin array A; integer M,N,K; ifb > 0 then Elliott Brothers Ltd., Borehamwood, Hertfordshire, Eng. arithmetic. RANGESQR gives an interval within which the comment Find will assign to A [K] the value which it would 2: begin square of a range nmnber must lie. RNGSUMC, RNGSUBC, procedure partition (A,M,N,I,J); value M,N; have if the array A [M:N] had been sorted. The array A will be e := b/d; go to3 RNGMPYC and RNGDVDC provide for range arithmetic with array A; integer M,N,1,J; partly sorted, and subsequent entries will be faster than the first; end 2; complex range arguments, i.e.