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Quicksort Partitioning Overview 12 I J Lo J Hi 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 Quicksort t-shirt 3 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 5 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 valuesTony of x may Hoarenot 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 series if x is not too 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 largerInvented than 10. Loss of significant quicksort digits is a major diffi-to translateA[R] = RussianXforJ < R < I into English. 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 and[ butsmall; couldn't explain hischooses algorithmequiprobably a random integer or F implementbetween M and N, and it! ] 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, i, n, q0, s; begin real X; integer F; ・n :=Learned 20; Algol 60 (and recursion).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 Implementedri = 0then quicksort. 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 else 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]); Tony Hoare Q[O] : = t; I := I+ 1;J:= J - 1; for i : = 1 step 1 until m do go to up 1980 Turing Award begin s := (x+x)X(i-1)Xt end 2: begin 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 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 C.University A. R. HOARE of 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 6 begin C.output A. R. interval HOARE of RANGESUM when ADJUSTSUM is called ALGORITHM 63 if c =< 0 A d ~ 0 then go to zer0dvsr; 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 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.
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