Volume *, nurrtir * INFORMATION

PRCXXSSINGLWi ERS February1978

DIVIDE AND CONQUER FOR LLYEAREMECFED TIME *

Jon Louis BENTLEY and Michael Ian SHAMOS

LWmento of clomputer sclsnce undMht#wm@ics,C aaqk-Rfellon University, Pittsbcqgh, PA 1.5213, U&A.

Received 29 Match 1977; revisedversion received 17 Octuber 1977

Avcragscasc analysis, computational geometry, convex hull, divide-and-conqLer,

expected time, line= programming,

rand6mJets

1. Introduction and thus takes advantage of the fact that h may be

smah. Unfortunately, if k is not known in advance,

Divide-and-conqueris one of the most frequently the algorithm may take quadratic time. Eddy [2] has

used methods for the design or fast algorithms. The developed a hull algorithm analogous to QUKX ORT

most common application of the technique involves that has good empirical performance but atso has a

breaking a problem of size N into two subproblems of quadratic worst case. * In this paper we use informa-

size N/2, solving these subproblems, then doing work tion about the probability distribution of h to obtain

proportional to N to marry the partial answers into an algorithm with O(N) expected running time with-

a solution for the entire problem; this scheme leads to out sacrificing O(N log N) worst-case behavior.

This new convex hull algorithm leads to tinear

algorithms of O(N log N) worstsasc time complexity.

In this paper we investigate a similar divide-and-con- expected-time solutions to a host of other geometry

quer technique which can be used to construct algo- problems that are related to hull-finding. Among these

are determining the greatest distance between two

rithms with linear average-casetime complexity.

points of a set, the smallest circle enclosing a set, ild

The problem of determining the convex hull of a

set of points in two and three dimensions has pro- constructing linear pattern classifiers. Analogous

techniques yield a linear average-case algorithm for

duced a rash of recent papers [4,8,15,16], all con-

linear programming in two variables.

taining algorithms with o(N logA!) worst-case per-

The divide-and-conquer scheme we use to achieve

formance. That this is optimal follows from the fact

the above results seems to be a general method suitable,

that in the worst case all N points may be vertices of

for the construction of fast average-case algorithms,

the convex hull, and since the vertices of a convex

It achieves fast expected time at the cost of making

palygon occur in sorted angular order about each

only relatively weak assumptions about the under-

interior point, any convex hull dgorit.?m ust be

m

lying probabifity distribution of the inputs. Whereas

able to sort [ 14,8] e If the boundary of the convex

many fast average-case algorithms display poor worst-

hull contains very few points, however, this lower

case behavior (QUICK:SORT, for example; see [ 13]),

bound does not apply, and a faster algorithm ma:{ be

those that we give in this paper have optimal worst-

possible. The algorithm of Jarvis [S] runs in time

case performance. These algorithms sre not merely

Q&N), where it is the number of actual hull vertices,

of asymptoiic interest - they are faster than previous

methods even for very small problem sizes (IV> 40,

* This researchwas supportedin part by the Office of Naval

for example).

Re~~ch under Contract NOM3l4~76-C-0829.

1 R.W.Floyd is able to show that E,ddy sdg6rithm runs in in reading this paper, one must be very careful to

linear expected time for cert% symmetricdistributions keep in mind the distinrtion between worst-case and

(personalcommunication).

87

murapcm andyaw. For emmpb,whiIeany camx

huiIdgwithmmustrunintimeS&N~N)foraome

ldlgkm~thmwithlilKtuaxpecdsd

*A=

nutninatime(for some d&tribution ofinputs).Notim

tlut there ir no contmdkdon betwean 8 wontsue

lowetboundofSZQW~A )8nd8n8ve~upper

boundd0#.

B8sic realIt8from stochlrtic geometry am daabod

inSection2;thcsaresulbformthebrrbofourp+a.

bilistic wlysis of the 8lgorithmapre8ented. In Sw

tion 3 we give 8 fast expected-time 8@&thm for

findiryconwhulbInthepl8wndinvut&teb

d&8ilthW&Jln8U8lIdlf0h4A@hm.SOCdon4

ahow8howthismethodc8nbe8pplIedtootherprob=

lean8md wed IS 8 bullding block for developing 8ddk

tionrl fast expected-time algorithms.Sect&ma cuak

5

t8ins su*ticns for further work 8lq the38 line&

stoch8atic @wmary e& with the propartia of

d

mndcxnsets of p&ts, lines urd other mtric ob.

jectsurdirmeasentWoolforMJydnlthe8ve~

c8se of guomehic 8lgorithm&M8ny phenomenr in

g8ometlic8l prob8bility M counter-intuitivd 8nd dir*

&Uh to explrin Without ti took Ofpmb8buhtrc

mea theory. For tutam*, the strtement, Clmom

~pdnh8rr8ndomintheplura,bme8ninlkr,

without 8 predw sp8ciflc8tion ofdistribution from

which the points ~IUto be chosen. Furthermom, not

rU conceiv8ble distributions utisfy the axiomsof

probrbility. Points c8n be choa8n unifomdyin the

@n43 nly from 8 Mt OfbOUItd8d

O Ldb88@BC mduw6

[6 ], So the ilktUitiVC&

8ttnctiVe not&onOf 8 uniform

mndom selection from the whole plum must be dis

CudHI.

The problem of determ~&Q3), the expected

number of wrtica of the cawex hull of N pdntr,

hW@UJiWd8lpoddWiOf8ttenti~ [1$,9,11);8

8Umm8ryofthisworltm8ybefoundin[I2).Wenow

quoteaever8ire8&8th8twiubeumdl8terh8n8lyB

fnlour~thms:

Fubruary 1978

asthe originaL The nsult of each of the recursive

Notethr 8 wtcx of tke OQ~BW.uil cf a iJni&

h

calls is a convex polygon whose expected number o

&&nerIr anti set ir m&Am& f@l suw assipmnt of

OfI! points. wztkcs is O(No), w&h p

piur8llb dw8~ttorll~~&utiec

set is now just the hti of the union of the hulls found

TMa implica that fw diat? butio~ trt&wng the Me-

in the subproblems. Shames [ 1S] has given an a&

p@nW ~rs~ption of J Morem4, &hocxpcctcd

rfthm to find the hull Gf the union of two convex

m&et of vertices of tba convex hull ;ubounded by

polygons in tkt prop1 tionel to the total number of

.

IF@) ;

tite points is rcgif_sent ;_d a pair of in*>:gcrs

as which

f~exarn~ ! .iiN

true,wemayhave~Q)1=c);-kj

define the left an ;I right endpoints of a ~gmcna of

pointsare~tedunifomLh~o.~ trekut -/ofa

the array. Dlvisicv into further subs& :.A PCsccom-

circle, then rcgv) - fv. As we & k *.; irl the next

plishctdby taking the arithmetic mean ^the crtd-

sectioa, the onJyxsumption A. Jut ?hcdirtnbution

points as deftnmg two new segments, e 2.; note that

of p&u &at needr to be made i.u o. :Icr to obtain a

the division preserves randomless. In i;.~ple~ncr ting

linexr expected-t&neelgorithm is that I::J) = Q(P),

this 3lgorithm recursively, it,z;crucial 1.) pas 0; tly

forsomepn,The convex huIl of a random set of points,

independently and at random to meet Ke but not Kt,

Biometrika52 (1965) 331-344.

and we define Hi to be the closed half-plane bounded

141 R.L. Graham,An efficient algorithmfor determining

by Lt that contains Kr, consider E(u), the expected the convex huIIof a planarset, Information Processing

number of vertices of the intersection of all the Hi. Lett. I (1972) 132-133.

Preliminary results were obtained by Renyi and R.A.. Jarvis, On the identification of the convex hull of

a fbrite set of points in the plane, Information Pro-

Sulanke (lo] and Ziezold 1181 has shown by duality

cessing Lett. 2 (1973) 18-21.

that E(u) is of the same asymptotic order as the ex-

[62 M.G. Kendall and P.A.P. Moran, Geometrical Probability,

pected number of points on the hull of a set of N Griffin (1963) 125 pp.

points drawn uniformly within Kr. If K1 shrinks to a f71H.T. K&g, M. Schkomick and C.D. Thompson, On the

point, then E(u) approaches the constant n2/2. In any average number of maxima in a set of ve.ctors, submitted

for publication.

event, under fairly conservative assumptions we will

VI F.P. Prqarata and S.J. Hong, Convex hulls of tlnite

have E(u) = O(w), p

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