Quicksort for Equal Keys

Abstract

When sorting a multiset of N elements with n < N distinct values, considerable savings can be obtained from an algorithm which sorts in time O(N log n) rather than O(N log N). In a previous paper, two Quicksort derivatives operating on linked lists were introduced which are stable, i.e., maintain the relative order of equal keys, and which achieve the previously unattainable lower bound in partition exchange sorting. Here, six more algorithms are presented which are unstable but operate on arrays. One of the algorithms is also designed for a speedup on presorted input. The algorithms are analyzed and compared to potential contenders in multiset sorting.

Full text

IEEE |RANS,\C-TIONS ON COUPUTIRS, VOL. C-3.1. NO. 4. ,\rRn_ l{)8-5
Correspondence
Quicksort fo. Equal Keys
LUTZ üT. WEGNER
Ärrlrddt -lvhen sorling a multiset of rV elementl with n < rV distinct
values, considerable savings can be obtained from an algorithm which
sorls in time O(N log n ) rather than O(N log N). In a previous paper, rwo
Quicksort derivalives operating on linked lists $ere introdüced lehich are
stable, i.e., mrinlain the relative order of equal kels, and which achie!e
the previously un:rttäinable lower bound in p:rrtition exchange sorting.
Here, six more l]lgorithms are presented which are unstable bu! operrte
on arrays. One of the algorithms is also designed for a speedup on pre-
sorted input. The algorithms are analyzed änd compared to potential
contenders in multiset sorting.
Inder Terns -A,nalysis of algorithms, arrays, multisets, presorted
files, Quicksort, sorting.
I. SNtoorH, STABLE, /v sfiu SoRTt\c
Consider an accounts rcceivable fiie which is sorted nccording to
customers to show which invoices, respectivelv. how many, are
outstanding from each customer. If a sort can take advantage from
the muLtiset structure of the input. considerable savings wiil result.
Naturally, any such algorithm is even more useful if it combines the
speedup for true multisets with no sisni[icxnt penalty for the case of
distinct keys since no d priorl knowledge of the nature ol the data
is assumed.
ln the tbllorving, algorirhms will be called -r/noor,4 iithey can sort
N distinct keys in time O(rV log N). on the average, and sort N
equal keys in time O(N) with a smooth transirion in between. This
term was introduced in I2l with respect to presoned lists.
The second desirable property in sorting multisets is stability. A
sorting algorithm is cailed srnble [5] if .ecords with equal keys
retain their original relative order. For the example above, assume
that the file is originally in ascending o.der according to invoice
number. Stable sorting according to customer ID will then leave rhe
invoices per customer in ascending order, an often needed property_
The third property to be mentioned concerns the space require-
ments. We say a sort is ili ri4r (in piace) if only ,((log rV)r) bits of
memory space are Lrsed for its variables besides space to store rhe N
records. Thus, Quicksort, whose recursion stack can be limited to
logarithmic size by sorting the shorter sublile lirst. is i/i .rit, but the
mergesoft, in general, is not. We extend the definition to inciude N
link fields if the input is given as a linked lisr.
The concept of a stable iinked iist Quicksort was discovered
independently by Motzkin l7l and by the aurhor Il I ], It2], who atso
extended the concept to include smoothness, demonstraled that
there are two algorithms as principal alternatives, and analyzed
them. The first algorithm, cailed TRISORT, features the elimination
of all keys equal to the pivot element during the parririoning phase.
It is therefore lermed a three-way splr algorithm. Using bina.y
Pascal compadsons as underlying cost measure, TRIsoRt is showü to
Manuscript r€ceived Aprit 16, 198,1; revised Oc(ober li, 198,1. An earlier
version ol this correspondence was presented at the Sevenreenrh Annuai Con-
ference on Informarion Sciences and Syslems, Johns Hopkins Universily.
Baltimore, MD, March 1983.
The author was with the Universität Karlsrühe, KäJlsruhe, Wesr Cermany.
He is now with the Fachhochschule Fulda, FB Angewandte Informatik und
Mathematik, D-6400 Fulda, Wesr Cemany.
require, on the average, 3(N + M)H\4! - ,lN key compdrisons to
sort a random multiset with N elements where each elemcnt occurs
M times (,ry. denores the Jirh harmonic number, ryt = ln(Ä) + I ti)r
all [ ? l). The generai pertbrmance, where each of thc n valucs
occurs ri times for I 5 -rr 5 n,-rr *.rr + .. +.r^ = N. is eivcn
rs well and is shorvn ro epproach rhe lower bound in prrirrir-,n
exchange sorting within a constant ftctor.
The second algorithm. callcd LtNKsoRT. fe:rtüres lhe eliminltion
of tll unrry subfiles in a sort. This can be achieved by applying lhc
following strategy. In the beginning of rhe partitioning phrsc, rhc
sublile is scanned from lcft to right üntil a key is discovercd which
is unequal to the pivot elemenr. [f no such key is found, thc subfilc
is unary and need not be treated any furthe.. Otherwise, tll pre
viously scanned keys are considered ls a prefix of the Icit subiile
which contains all keys S the pivot element, and the scan is con-
tinued for a nvo-*'a:- split. The cmcial point here is that for Iinked
list representations, the cost from simple scanning to partitioning is
one extra key comparison per treated subfile. Using again binary
Pascal comparisons as underlying cost measure, LTNKSORT is then
shown to require, on the average, less than 2.tvHf,.!l + rV key com-
parisons to sort a random multiset with rV elements wherc each
element occurs M times.
Here, we continue the discussion of Quickso.t for equil keys by
investigating several possibilities for array representations. A sum-
mary of the properties of the algorithms mentioned in this paper is
given in Fig. l.
ll. SrroorH QutcKsoFrT DERtvAnvEs FoR ARRAys
The question here is whethe. rhere is any way ofchanging euick-
sort to fealure a three-way splir or the elimination of unarv subfiles
*hen:pplied ro lntls. Con5t(lering Sedsewick \ wornini rem.rrL.
with respect to the elimination of all equal keys, namely thüt..no
efficient method for doing so has yet been devised" [9,p. 2,14], ir
seemed ar first that there was little hope.
The obvious problem for a three-way split is where to store keys
which have been detecred as being equal to the pivot element. Since
we insist on an in slrr soiutiol and since we do not know the split
position beforehand, we must consider intermediate storare ofequal
keys in some designated part of the array. Here, four different ways
of keeping track of equal keys are explored, namely.
. keeping them in the head and tail part of the anay and swirch,
ing them afterwards into the middle:
. keeping them only in the head part of rhe anay;
. moving them as a growing block through the ai.ay until they
reach their proper posirion;
. keeping them scatrered in the upper'.half" of the array and
collecting them in a second sweep.
- In the.followins, we depict the general siruation during the split
from which the invariant for the algorithm may be drawn. Note that
fl is the pivor element, e. g., first or last key, swap(.t, r) exchanges
the values of a and b, and circ(a,ü, c) is a trianguiar exchangJol
values which must have the following semantics to work prope;1y in
the extreme cases below: aux := c; c := b: b .= o; n := arx.
HEAD & TArL-soRT: Form a prefix and postfix of equal keys by
scanning the file f.om rhe leti and from the right sropping on keys
= 11, respectively, 5 H. Keys equai to,y are switched inro middle
after the split position is known (invoives no key comparisons).
ulirsoRT: Sca[ rhe ille from left to right stopping on keys - H;
collect keys equal to Il at front end and switch them lnto middle
afte rrv ards -
0018 -93,10/85/0400-03 6250 l 00 0 1985 IEEE

IE€€ TRj\NS^CIIONS ON COMPI]TERS, VOL, C_J.I. NO. .1, .\tRtL lgg5 l6,l
stable in situ
al qori thft
straight merge
quicksort
partitlon merge
Cook and Ki ni
spli t-sort-merge
no pdrtia I ly
yes yes
I iiked list Quick
sort (rRIs0RT/
LIN(SORT yes pdrtiälly
Yes pa.tial ly
Fig. l. Prope(ies ofdiscussed soding algorirhms
condition
atil > H. alil < H
aLil > H. a[j] = H
a[i) = H, a[jl < H
alil = H. aljl - H
swap(diil,rltjl): i :: j + t:j :=j - t
circ(dtji.z?[.r + l].alll):/ .- i + t..j :=j - lr.r
ci(r.rlrl.rit - l'.ollir, i .. t t:j .J - 1...l
swap(d[-r + l],d[i]);.r :=.r + li/ = i + I
lwap(a[) - ll,dtj]).] j= v - lij :=j _ r
condition
atil = H
a[i] > H
Fig. 3. Partitioning for uNlsoRT.
Fig. 2. Panirioning for HE,\D&TAtL-soR-r.
condition
ali) < H
a[i] > H. aljl < H
atil > H. alj) - H,t,üt.rt,!lr l/,.r :- _r r li I :- I
til,nUl);i := i + l.j :=j I
sLrDEsoRT: Scan the file from the left stoppins on keys + ,ry; then
scan the file from the right until a partner S H is found: keys equal
to Fl are afterwards in proper position.
DOUBLESoRT: Scan the file from the left stopping on keys = A
and from the righr stopping or keys <,y ; alier pointeri meer.
partition righr subfile again with the same method sropping rhis rime
on kevs + H and: H.
^ More sorts mav be creJted b1 combrnine .ome of the strJtesres
fromabore We nole thJt.rll iour algorirhm. u:e rhe.eme kei to
pcrtltron the llie- ."y rhe lir.r or median irom iirqt, mrddle rno ltsr
key, and all elimin;te all equal keys. Therefore. rhey achieve rhe
same spLit and may be compared on the basis of key comparisons
and/or exchanges per partitioning phase (see Section lll).
Here $ e close with .r rwo. ,Jy ,plir Jlgorithm. called L )l\RysORT.
whlch ellmrnales un:rr) \ubliles. The ba.ic ideJ is rhe,ame its lor
LINKSORT and the reader should have no trouble following through
the algorithm below (Fig. 6). As prelude to the following analysis,
we note that UN_ARYSORT eliminates exactly one key from"the recur,
sion Jnd need..V - 2 comp.rrr.ons to splii.V r I elemcnr. in.r lile
wh lch l\ not rnJl y. Note rl.o thar u e do * irhour tn ..eno ol_lile re.r.
in the iirsl r\ hile-roop whrch can cruse redundrnr le) comprri:un.
but.houid. in general. be more efficrenr rh.ln a re.t or rhe rn.errion
of a sentinel.
llL ANALysts oF SrrooTH euicKsoRT DERIVATIvES FoR ARRAys
We qt.rrr_$rth the cu.tomJr) cosl mcüsure, ncmely key cun_
pln5oni. lt ge d55ume three-$d) comptri:Ons r\llX j. lhen rhe
algofithm\ HE^D&TAtL-SORT. U),JISOR I, and SLIDESORT need N com_
parisons to partition N elements and eliminate all keys equal to the
pivo{ elemen(. Sorting ilgorirhm,. like eurch,ort. $nicn use thir
panluonrng to reduce .r glven lile ro t\\o or three subfil"s !Lhlch crn
Fig. .1. Panitioning lor slrDEsoRT

36+ IEEE rR^NSACTIONS ON COIIPUTERS. vol-. C-1.1. fto. .1. ^r'R lqtl5
L Scan abovc. 2- Scan below
condition
alil1 H. a[j] < H
aL(l * H, a[\) = Hsw.rp(.rttl,dtjl)l i:=l + I;j r=/ - I {ijrsr scan}
y3!qql!{-vl):.. :=.r + l:.r := r'- I {second scan}
p!öcedule UNARYSoRT (1, r: tntege!) I
{solt reculsivery an ärray dtr,.rl ellntnarr.s u.d.y subfttes
but exMlntnq €ach k€y exacrly once except !o! one key,
assue3 älN+1I > atlt fo. the ortglnal flre !^ att..NJl
w l, j: lncegelj
H, aux: keytype {el4ehls to be sorted conslst of keys only),
bEqh {INARYSORT }
H :- alllr 1 :- I + 1,
{a unä.y subfll€? }
!i!-l-l-g ä(11 'lr do ! :- 1+ r'
{.t.r'l ri H)
If1<!{notu.a!y}
!!-e!
beq!n {p.!tlrlon}
1:-1-1, l:-.+l;
!!1&i.j99
lepeat 1 :, ! + r unrit atll, Il
lepear j ,= j -rsg! a(il : H,
5'ap (ät11, atjl)
end {rhire },
{511tch bacr atiJ, at11, atjl)
circ tat i l, atU, atjt);
1f r < j - l !!9! u aRysoRr {1, j - 1),
!l I < ! !!S! uNÄRysom (1, !)
csl {tiien)
end IuNARlsoRr i;
Fig. 6. uNARysoRT-Quicksod wirh elimination of unary subfiles_
HEA.D&IArL-SORT v :[i] > H r. the lef! ro lighr scan
v aljl < s iD the liqht to left scan
UNISORT
SLIDESORT
V a{ll > ll in lhe left !o right scan
v alil I H in the left lo iighr scan
v a(il < H in the riqht to left scan
oouBLEsoRT v atil I H resuttiDg froF rhe seconai scan.
Fig. 7. Exra key comparisons obse!.r'ed fo. three way split.
be.orted recu15ively :rre Iermed prnitron-exchange so(s. Now the
following Iemma lollow s trrviaily.
Lemna J-l: If three-way comparisons are the underiying cost
measure, HEAD&TAIL-SoRT, uNIsoRT, and sLIoEsoRT are lower
bound pilrtition-exchange softs.
. .Proof.: The position ofeach key examined during the scan can
be determined by exactly one key comparison. The rJsult then fol_
lows fre6 5s6t"*1.k's lower bound theorem [9]. o
The lower bound result does not hold for the fourth alternative
(DouBLEsoRT) because the final pos ition of keys = H is determined
in a separote, second scan. If binary comparisons are the cost
measure, the analysis becomes more interesting. The central ques_
tlon nere is: Which keys Jre compared t*ice.,' fig. I giue, a
partral rnswet.
Assumingarandom inputof themultiset{,tr . 1,,..,_r" . nlwith
.rr + ' + -r" = 1y', we cao state the following result about the
average number of binan key compari:ons.
Theorem J.2: fo splir d random multiset wirh.r1 I ... i .1, =
N elements, on the average. uNrsoRT and DouBLEsoRT require
s -i- ,..., -,
r=i=,
key comparisons. HEAD&TAIL-SoRT requires Iess than
N + -l- t .l
N,+"
t_
-; ) Zt.'.r, + -.r.-rrr.r... - .-.("1/(,V - r.
/v =,="-:
key comparisons. tnd 5LIDESORT require\
l-
N ' 1,/ t ,.) " t 1.,, . - - .r.)rr.r, - ..- r")
+ 2(.r1 + ... i -r,))
key comparisons.
Proof: The first value is straighrforward. For N = n (N dis_
N +1
Fig. 5. Pilnitioning in two scans for DouBLEsoRr

o! co\rp! rERs. \nL. c-J{, N'). ,1, \tRtL l'lb5
tinct valucs) wc obtain N + (N + l)/2. For the sccond value,
which is only tn uppcr bound, we assumc that each key > H tbund
in thc lclt lo right scxn is exchanged against a key < H in the .ight
to lcti scirn :rnd thut.r; pivot keys account for.!i extra comparisons.
Tlris is not tme bccause keys > H, respcctivcly, kcys < H, can be
exchangcd rgirinst keys = H. Howevcr, the assumption gives a
relsonably close lpproximirtion and leads to the following lormuia:
kc,or,(.r',".rJ:N -l: (.;*, ) :,.p,,')
.v,:.'
where
p'(t)
(r,-. -.\.t t .r - ... - .'.., t /l N - r, \
-\ , /1.,, ...._.(,_, _,)/\r,_...-.,,/
is the probability that r keys > fl are encountered in the lef! to right
scan. Applying Vandermonde's convolution leads to the stated re-
sult. For N = n the bound is sharp and yields 4N/3.
In the case of sLtDEsoRT. we note that all keys + H in the left to
right scan and all keys 5 ,1 in the right to lelt scan require extra
comparisons. Since the number of keys > H on the left equiLls the
number of keys = .FI on the right, the total number of binary com-
parisons is N piüs the number of keys ( fI plus twice the number
ofkeys > Il in the lefr,!, + . . + -r, positions. From
",i ),"(".-.. ) . , r,,,, -"_,,,.?. ,.-:r'prrr)
with p(s), cespectivelv, p,(t), similar to above, the formula is de-
rived using again Vandermonde's convolution. For N = n we ob-
tain here 5N/3 - 2/3N which is ef,siiy checked using Sedservick's
well-known result that, on the average, (N - 2)/3 keys are out of
order which brrngs us to N + N/2 + lV/6: l0N/6 key com-
parisons ignoring the small additive constants. tr
This leaves ui with the following ranking for N : ri:
HEAD&TAIL-soRT uNrsoRT/DouBLEsoRT sLTDESoRT
Iess rhan 8N/b 9N/o lON b
From the general formüla above, the special ciLses for -r, = M
tl ! i l,nrrnda = I. i.e.. forJcon\tsntrepetrtron faclor'Lnd lor
unary files. can be denved whi.h yield no new insrght' our conlirm
the observation that uNrsoRT, DoUBLESoRT, and HEAD&TAIL,soRT
need 2N comparisons and 5LIDESORT only N comparisons ro sort"
N equal keys.
Next, we briefly glance at the number of exchanges needed in
each split, where we assign the triangular exchanges the same cost
as swaps even though they cost l/3 more.
Theorem 3.3: The average number of pairwise and triangular
exchanges inclüding the movemenr of pivot keys into the middle in
each partition phase for the multiset {r, 1,...,r" . n} is
5 l/2 the number of extra key comparisons + Zr,
for the HEAD&TArr--soRT,
fo. uNrsoRl,
,u,l ]=,tf' -, + -+ r^) ((.r, + + .!,) + (-r, + + .!,))
ior SLTDESoRT, and
''' + .rr-r) (,r, + ... + x" - l)
l-
.; ) r,(rr + +,r,)
165
Proof Similar to Theo.cm 3.2. o
For the intcrcsting casc N = ll we nolv hilve =(N/6) ibr HEr\D&-
TAIL-SoRT and DoUBLESoRI. but -(N/2) fo. UNIsoRT nnd sLTDESORTI
It therefore scems that the HE,\D&T^tL-soRT is thc most tttrüctivc
candidatc for its lowest number of comparisons and rcasonublc
number ol exchungcs. What the exlra cost of the control of thc
swapping of the head and tiil pivot keys into thc middle conccrns -
implementcd, e.g., by two for loops-we should be wilrned th it
can be substantial fo( the large number of smoll subfiles to!vxrds thc
"end" of the sort.
In all likelihood, DoUBLEsoRT is the most practical cirndiditc. As
a pleasant side effect we also know the exact numbcr of kcy cont-
pirisons needed to sort a random permutation with oouaissonr
Corollary 3.1: To sort a random permutttion of a multisct in an
array, DoUBLESoRT requires, on the average, the same numbcr ol kcv
compJnson\:Is TRISORT in a Iinked li:t.
For the two-way spLit algorirhm uN,\RysoRT we observcd,V + l
comparisons to split an arbit.ary mLlltiset file of1y' kcys. Thc number
of exchanges here is easily scen to be the same as for DouBLEsoRT
minus the exchanges in the second scan. Moreover, the rcsulting
split happens to be the same as for LINKSORT [12] and therelore a
result similar to the one ior DouBLEsoRT holds.
Corollary 3.5: To sort a random permutation of a multisct in tn
array, UNARYSORT requires, on the avcrage. the same numbcr ol kc1
comparisons as LINKSoRT in a linked list.
Thus, DOUBLESORT needs 3(N + M)H. - 41y' binary key com-
parisons and UNARysoRT needs less than 2NH, , + N binary kcy
comparisons to soat a random multisel of size ly' with const:lnt
repetition factorM : N/n.
In conclusion. the newly introduced smooth Quicksort three-way
split algorithms for arrays achieve the lower bound in partition
exchange sorring. Due to their binary comparisons implementation.
however, they are -just like their linked lis t relatives - inferior to
their two-way split competitor by a considerable marsin.
lV. QurcKSoRT FoR NEARLy SoRTED FrLEs
We close the presentation of new algorithms with a sort which
smoothl)/ handles presorted input, where the majority of keys is in
their proper place and the others are permuted at random. The
algorithm is called RUNSoRT for its ability to recognize runs. i.e..
sequences of keys in ascending (descending) order_ It was derived
from L:NARysoRT, i.e., it hopes for a run and in abandoning thar hope
performs a two-way split.
The finer point here is that we may not simpLy scan rhe subfile
from left to right until we find a pair ofkeys (a[i],dli - ll) for
which a[i] > ali + ll because ar rhat poinr neither d[i] nor
a[i + l] will do as a pivot element unless we backtrack.
As can be seen from the while,loop in Fig.8, the suggested
soiution is to scan the file from the leit and from the right with 3 key
comparisons and I index comparison tor each pair ofkeys. Thus, in
the prelude to an actual split, 3N'/2 key comparisons and N'/2
index comparisons are performed forN' keys which, taken withou!
the middle piece, rvould form a run. The prelude is follorved bv an
ordinary split for which we chose a so-called fictitious pivor element
computed from (d {i l + aljl div 2 (see [l2] for a discussion ol rhis
rder,. As Ihr trming re.ulrs of lhe nex.t \ecrion \how thrs is nur
a really fast Quicksort version and we do not explore this idea
any further.
V. How S\1ooTH IS SNIOOTH?
For all but the last algorithm we have given the result of their
multiset analysis or compared them analytically on the bnsis of
key comparisons and key exchanges per partitioning phase. For
Dijkstra's previously mentioned smoothsort, Hertel [3] has shown
that the sort is opiimal to within a constant factor both tbr totally
sorted sequences and for random sequences but not very smooth
+ .I. ! 'i for DOUBLESORT
N ,.-*"

IEEE tRi\NSr\CIIONS ON CO\lptTERS. VOt_. c_J{, No. 4, ^r,Rlt. lgll_5
procedure RUNSORT(I, r: thteger) ;
var l.J:lnleger, H: keyrype,
Frocdure rrap(var a, b! keytype),
lg*E (RuNsoRa )
4i rt11 > dlrj qs! sHap(atlt, atrl),
1 :- l, I :, r,
4ffs (1+r < J) 9gl (a(11 : ai1+lt) and
(alj-rt : atil) grg (at1+1 J : a{i_jl) 99
beqln
I :'l + r, j ': j - I
9!q;
1!1.J-l{nora!un}
then
&g1! (parrltloni
F :- (ai1t + atjt) 4E 2;
IS!93! !,= r - 1 unrrl atil : r;
repeat j := j - 1 glgl atit : s;
1-€ 1. r !-!s! s*aP(arU, atjJ)
unr1l 1 > i.
fl 1 . r - I ther RrjNsoRTlt, 1_ t);
1-E J + r . r rhs puNsoRr{j + r, r)
end il < j-l )
3!q (RuNsoRr );
Fig. 8. RUNsoRr - Quickso.t for multisets aFd presoded files.
in between when measured against the number of inversions in a
:"e-luence:nd is inferior ro. e.8.. Vehthorn., "tgorirt. 1bl 'ln
tnat resnect
ns lör rhe Cook and Kim 3lgonrhm JIl. no anjl)rical re\ulr
r5 I.noun excepr the obvious facr rhcr it is a OrlV ioe Nr rime
algorithm.
_ Jo bring the dozen algorithms mentioned in this correspondence
rnto some perspective. a modest competrtion on a number ol multi.
sers ano presorted tlles rras arranged. For rhdt purpose all \ons \r.ere
coded by rhe Jurhor in pascai and run on r ringle user pDp ll. To
k€ep rhe-effort reasonable, rhe file size *"". ""if".*fy ,"i i"
N = 1000 and the avelage time in seconds from 20 random oer-
mutatjons fo. each type of input was reported.
The tlpe: of inp.rt which were considered dre random oer
mutalion5 of a multi\et wilh consrtnt repeUtion iacror .V = .'V ,
where the n distinct values are 1.2.. . . ,'l (note thar M = 500
denotes x bin3ry lile Jnd .t1 - 1000 r un.rry iitc for.v = 100ü,
nearly \oled,files of N inregers I.1......V rvrrh p p.r."nr oirh"
Ke)c femoved fiom the Jscending run. permured. ,rnd in"ened rnro
tne vacated poSluon\ ff = 0 denOtes .r completely .orreO filc). rnd
r sequence ol r\!o nrns of length r. respectively.,V _.."tar"rt"
r fl: ,1" randoml) picked from the ordered sequence.
. labie IspeJks for i(seif Jnd !re refrain fromcoÄmen!sotherthirn
that Quicksort proved again its name.
al gori thm
quicksort
(tertbook )
Q!icksort
(sedgeNi ck)
TRISORT
llflKs0RT
HEAO&TIIL
uIrs0Rt
5L]DESORT
DOUELTSORT
UNARYSORT
gDooth5ort
(Di lkstra)
Kim sort i
Rufisofi |
F'l lH.2 H=t0 HrlO0Hr5C0v=1000 P.0X p= 5l p= loX p= 20! p - 50! ry"
0.91
2.52
5.43
0.54
r.31
1.08
0.99
1.03
0.90
I .01 t -00 1.03 I .08 I .r)5
0.95 0.90 0.89 0.95 LO2
0.86 O.57 0.26 0-12 0.lo
0,85 0.66 0.38 0.18 0.06
0.50 0.51 0.54 0.62 0.82
1.38
1.96
1.44
r.23
2.41
6.75
I .41
r.3t
t.2r 0.87 0.51 0.31 o.?6
2.05 1.50 0.69 0.38 0.25
1.68 1.39 0.45 0.15 0.05
1.28 0.85 0.40 o.t7 0.10
l.l4 0.82 0.42 0.18 0.05
2.4O 2.40 2.25 1.43 0.64
6.75 6.64 5.62 2.80 0.9t
1.39 1.31 t.32 1.15 0.64
I.2S 1 .09 O.t4 0.42 o.r?
2.52 2.50 2.5A 2.48 2.45
0.96 L42 1.80 2.62 4.65
0.25 0.44 0.58 0.79 !.21
0.ll 0.s4 0.56 O.7g 0.94

rlEe TRANS,\C ONS ON COTVIPUTERS. VOL, C-1.1, NO. 4, .\PRIL l9s5
V[. CoNcLUSIoN
ln this correspondencc it is proven thit Quicksort cxn eft'cctively
be lurned into a smooth multiset sort which achieves thc previously
unaftain:rble lowcr bound in prrtilion exchlngc sorting. illorcovcr.
a two-way split algorithm with climination of unir.y subfiles and r
Quicksort version for presorted files are introduced. Ho$ever, un-
like their linked list relatives. none of the six sorts is stable. The
efficiedcy of the new algorithms is demonstrated in a run time
comparison which also involves four contenders from the literature
rnrl r$o it:rble linkcd lrrr Qurcksort vrrirnts.
ACKNOWLEDGTIENT
The author would like to thank one of the .eferees tbr pointing oul
a mistake in thc analysis and for several helpful comments.
REFERENcES
ll C.R. Cook and D.J. Kim, Besl so.ting aigorithm for neirly soned
lists,' Cdnmr?r. ACM, vol. ?3, pp. 620-624, Nov. 1980.
[2] E.W Dijkstra, "Smoothsort. An altemrlive for soning ifl situ. .sci.
Comput. Progranning, vol. I, pp. 223-233, 1982i see also. 8n'rta.
Sci. Conput. Progrunning, vol- 2, p. 85, 1982.
[3] S. HedeL. "Smoothso( s behavior on p.esoned sequences. tfrrn. Pr.r-
cessing Leu., vol. 16, no. 4, pp. 165-170, lvlay 198J.
I.ll C.A.R. Hoare, 'Quickso(. Conput. J.. vol.5, pp. 10-15, 1961.
I5l D. E. Knuth, The An of Cornputer Programning, Vol. 3: So ing atkl
Searciirg. Reading, MA: Addison-wesley, 1973.
f6l K. Mehlhorn, "Sorting presorted files,' in Theoretical Conputer
Science. 4th GI Conference, Lecture Notes in Conputer Scietrce,
K. Weihrauch. Ed. Berlin: Springer, 1979, vol. 67. pp. 199-212.
[7] D. lvtolzkin, "A slable quicksod,'Softvare Pract. tver., vol- ll,
pp. 607-611, June l98l.
[8] L. Trabb Pardo, 'Stable sol1ing and mergihg with oplimal space and lime
bounds,'S1ÄM l. Comput., vol.6. no.2. pp.351-172. June 1977.
[9] R. Sedgewick. *Quickson with equal keys," SldV J. Conput., vol. 6.
no.2, pp.2,10-267, June 1977.
uol -. 'lnplementing quicksort programs," Cammln. ÄCu vol 2l
no. 10,pp.8,17 857, 1978: see also, Corrigendum." Camnun. ACM.
vol. 21, no. 6, p. 368, 1979.
I t] L. Wegner, "So(ing a linked list wilh equal keys," lrfo',l. Prccessilg
,e0., vol. 15, no. 5, pp. 205-208, Dec. 1982.
ll2l -,
-The L'nksod family-Design and analysis of fast, stable Quick-
sort deriva!ives, dissertation. Inst- Angewdndte inibrm., Univ.
KarLsruhe. Karlsruhe, West Cermany, Rep. 123, Feb. 198i.
An Adaptive Method for Unknown Distributions in Distributive
Partitioned Sorting
PHILIP J. JANUS AND EDMUND A. LAMACNA
Ärslracl -Distributive Partitioned Sort lDPS) is a fast internal sorting
algorithm which runs in O(ll) expected time on unilorinly dist but€d data.
Unfortunately, tde method i3 biased td;ard such inputs, and its per-
formance worsens as the data become incressingly nonuniform. such a5
\rith highly skerved distributions. An adaptation of DPS, which estimates
l\4anus.npI reier!ed \,1ry L lo8-l: rc!rsed \dvember tr. t'r8-t I1,,. oo,L
was supponed in pah by Air Force Sysrllng Coinmend. Rome Air Dcvel-
opment Center. under Contract F3060?-79-C-012,1-
P J. ldnus *rr wrlh rhe Dep.rnmenl oi Cbnpu(er s. ence rno S,.rrr\r,c,.
L nr\er 'rr) or RhJJe I.l-no. Kinesron. Rl0:881 He rrnu*q.th neSult*are
Engineenng Facility, Digilai Equipment Corporailon. Nashua. NH 01062.
E. A. Lanragna is with the Depanment of Computer Scieoce and Statistics,
University of Rhode Island. Kingston, Rl 02881.
367
the cumullrtive dislrihution function of the input datr from a randomly
s€lected sample, wns derehped rnd tested. The method runs onl!
2-.1 percent slower than DPS in the uniform case, but outpcrforms
DPS by t2-ll percent on erponentially distributed data for sufficiently
lÄrge files.
lnder lems -Anrlysis of algorithms, cumulative distribution func-
tion, distributive partitioning, Quicksort, sorting.
[. INTRoDUCTIoN
Distributive Partitioned So.t, or DPS, is an internal sorting tech-
nique described in a paper by Dobosiewicz I I l. Empirical evidence
presented there suggests that the algorithm runs considerably faster
rhan several cornmoniy used sorts including Quicksort [21. []1. Thc
expected running time of DPS on uniform data is O(n) rvhere l is the
number of items ro be sorted. This is possible because the proccdure
is not based upon key comparisons like most conventional so.ts
(e.g., Quicksort, Heapsort, Bubblesort). A comparison based sort
is not crpable of O(n) behavior. Furthermore, DPS s worst case
running time is O(n log z), which is no worse than lhat of any
comparison based method.
Untbrtunately, DPS is inherently biased toward uoiformly dis-
tributed datn. Its running time degrades on highly skerved inputs.
such as an exponentiaL distribution. The primary goal of this corre-
spondence is to develop an adaptation of DPS which will sort any
unknown distribution equally well, yet whose run time remains
competitive wi!h the.t of the o.iginal algorithm in the uniform case.
II. DISTRIBUTIVE PARTITIoNED SORTING (DPS)
DPS is a distributive soft resembling Knuth's lvlultiple List In-
sertion [4] in which the number of partitions, or buckets, created
equals the number of items to be sorted. DPS sorts 4 items as
foLlows.
SELECT: Determine the maximum. minimum. and median ele-
ments. all of which can be found in O(n) time [5]-[7].
PARTITIoN: Us'.ng these valües, divide the range of data between
the maximum and minimum ioto , buckets rvith r/2 equal length
intervals on one side of the median, and another n/2 equai intcrvals
on the other side.
DISTRIBUTET For each item, determine to which bucket it belongs.
RECURSE: For each bucket with more than one item, sort the items
in that pariition using DPS.
It is easy to see tha! in the best case this algorithm is O(n). For
equally spaced data, each bucket contains one ilem after the first
pass thereby producing the sorted set. In his original paper. Dob-
osiewicz shorved that the algorithm is O(n) on the average for
uniform distributions I ll.
The worse case occurs when for each n/2 buckets divided by the
median, all elements but one are distributed into the same bucke!.
The time I(n) is then governed by the recurrence
TG) : ,n + 2T(n/2), I(l) : co
T(n) : ,n log;n + O(n)
which is O(n log n).
There are many problems with DPS as originally pubLished in [1 ].
These range from lundamentaldifficulties with the algorithm itself,
to large time and storage overheads, to lheoretically bad worst case
running times. These problems are detailed in [8i, on which this
correrpondence i. based. fhe most imponanl improrements are
discussed below.
One aspect of DPS which contributes substantially to its running
time is the median location stage. Although the median of n items
0018-9340/85/0400-0367501 .00 0 1985 IEEE