Geometric Applications of A Matrix Se~ching Algorithm Alok Aggarwal 1 Maria M. Klawe 2 Shlomo Moran 1'4 Peter Shor 3 Robert WIlber 2 1. IBM T. J. Watson Center, Yorktown Heights 2. IBM Almaden Research Center, San Jose 3. Math. Sciences Research Institute, Berkeley 4. On leave from The Technion, Haifa, Israel. I. Introduction

points in the plane. In fact, using some geometric properties of a convex polygon and the fact that any point can be the nearest neighbor of at most six

The all-farthest neighbor problem for a set of n points in the plane, P, is to find for each point Pie P,

other points in the plane, Lee and Preparata (LP78)

another point pj~ P with j # i such that

obtained a O(n) algorithm for the all-nearest neighbor problem on a convex polygon. However, since

d(Pi, pj)

-

max

l 1 and j >

and . 4 I ' i + 1 . . . . .

the

n;j, .... m]

T h e o r e m 3.1: Let A be an n x m m a t r i x , w h e r e n is

submatrices

(when i < n

a p o w e r of 2.

1)

e= 1 orm.

and

L e t h be a positive integer and let

Setf=m+

1-e.

Suppose that up to

m a x ( m -- 2, 0) cells h a v e already been queried (i.e.,

j < m). T h e time required b y this algorithm is given

h a v e been assigned fixed values).

by the r e c u r r e n c e

Also, suppose

that no cells h a v e been queried in A f, that a n y cells f(n,m)

< m + max

that h a v e been queried in A e h a v e been set to h, and

If(rn/21 - l,j) + 1)},

l r 2 a n d j =

• o t h e r w i s e set A (i, j ) t o h + 1. 2)

If i < r 2 a n d

j#

N o t e t h a t L satisfies t h e c o n d i t i o n s of t h e t h e o -

1 o r if i > r 2 a n d j # m ,

set

r e m , w i t h p a r a m e t e r s h t a n d e t, w h e r e e ~ = 1 a n d hr=hife=

A (i, l ) t o a l o w v a l u e .

landh'-h+

life=m.

Similarly, R

satisfies t h e c o n d i t i o n s of t h e t h e o r e m , w i t h p a r a m W h e n t h e first s t a g e e n d s , e x a c t l y m - 1 cells of A have been queried and the values fixed are cons i s t e n t w i t h all m a x i m a b e i n g in c o l u m n

1 and

in r o w s 1 t h r o u g h r 2 - 1 all m a x i m a

in r o w s r 2

t h r o u g h n b e i n g in c o l u m n m. Q u e r i e d cells in c o l umns 2 through m -

e t e r s h " a n d e", w h e r e e " = m - c + 1 a n d h " = h if e=mandh'=h+ queries

life=

of cells w i t h i n

1. I n t h e s e c o n d s t a g e L or R are handled

by

recursively applying the adversary strategy to the t w o s u b m a t r i c e s , a n d q u e r i e s t o cells o u t s i d e of L a n d R a r e a n s w e r e d in s u c h a w a y t h a t t h e y i m p o s e

1 all h a v e v a l u e s less t h a n 1.

n o c o n s t r a i n t s u p o n t h e p o s i t i o n s of t h e m a x i m a w i t h i n L o r R. A f t e r t h e first s t a g e is c o m p l e t e d a c o l u m n c a n d t w o s u b m a t r i c e s L a n d R a r e s e l e c t e d as f o l l o w s . T h e r u l e s f o r a n s w e r i n g a q u e r y of cell A (id) a r e

F o r 0 _< j < m, let sj b e t h e n u m b e r of q u e r i e d cells in c o l u m n s 1 t h r o u g h j of A (s o = 0).

Let c be the

as f o l l o w s .

s m a l l e s t i n t e g e r in [ 1, m ] s u c h t h a t sc = c - 1 ( t h e r e is s u c h a n i n t e g e r b e c a u s e S,n = m - 1 ). U s i n g t h e

1) If i < k 1 a n d j = 1, o r if i > k 4 a n d j = m, t h e n s e t

A ( i , j ) t o h + 1.

a c t t h a t t h e sj's a r e a n o n d e c r e a s i n g s e q u e n c e of int e g e r s it is e a s y t o s h o w b y i n d u c t i o n t h a t f o r a l l j in

2) I f k 2 < i < k

3andj=cthensetA(i,j)

toh+2.

[0, c - 1 ], sj > j. I n p a r t i c u l a r , s c_ 1 > c - 1. S i n c e Sc-1