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WORKING PAPER
ALFRED P. SLOAN SCHOOL OF MANAGEMENT
On Central Limit Theorems
In Geometrical Probability
Florin Avram
Dimitris Bertsimas
WP# 3312-MS
July 14, 1991
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
On Central Limit Theorems
In Geometrical Probability
Florin Avram
Dimitris Bertsimas
WP#3312-MS
July 14, 1991
M.I
T.
AUG
LIBRARIES
5 1991
RECtlVfcu
1
On
central limit theorems in geometrical probability
Florin
Avram
'
^
Dimitris Bertsimas
July 14, 1991
Abstract
We
prove central limit theorems (CLT)
probability
when
points are generated
in
for the following
the
[0, 1]'^
problems
in
geometrical
cube according to a Poisson point
process with parameter n:
1.
The
length of the /bth-nearest graph Nk^„, in which each point
is
connected to
its
k-th nearest neighbor.
2.
The
length of the Delaunay triangulation Z)„ of the points.
3.
The
length of the Voronoi diagram Ki of the points.
We
show using the technique of dependency graphs of Baldi and Rinot that the
pendence
rajige in all these
problems converges quickly to
addition, our approach leads to
class of
more
efficient sequenticd
and
de-
with high probability. In
parallel zJgorithms for this
problems on randomly distributed points.
Introduction
1
In a pioneering
[11]
it
paper by Beardwood, Halton and Hammersley
[4]
and continued
in
Steele
was shown that the lengths L„ of several combinatorial optimization problems (the
'Florin Avriui, Department of Mathematics, Northeastern University.
The
research of the author was
partially supported from a grant from Northeastern University.
'Dimitris Bertsimas, Sloan School of Management,
MIT. The research
ported by the National Science Foundation under grants DDM-9014751,
Young
Investigator
Award with matching funds from Draper
1
of the author
was
partially sup-
DDM-9010332 and by a
Laboratories.
Presidential
minimum
traveling salesman,
etc.) satisfy laws of large
cube
1,... ,n in the
It
is
matching,
minimum spanning
minimum
tree,
numbers, when their input consists of n random
iid
Steiner tree,
points Xk, t
=
[0,1]''.
example Steele
also believed (see for
[12]),
but yet unknown, that they satisfy
central limit theorems (CLT), since, while the edges of the above optimal graphs are not
independent, the dependence "seems to be local". In trying to make this intuitive idea precise
we succeeded
constructions
1.
The
in
proving
CLTs
for three graphs,
computational geometry (see
in
length
A'^jt,„
for
which are among the most fundamental
example, Preparata and Shamos
of the fcth-nearest graph, in which each point
is
connected to
[9]):
its
k-th
nearest neighbor.
2.
The
length V„ of the Voronoi diagram of the points, which
polygons
point
O
point in
3.
The
for
the set of
is
X
all
a Voronoi polygon around a
points in the plane which are closer to
D„
O
of the Delaunay triangulation of the points, which
on the given points with an edge between them
1).
,
the collection of Voronoi
than to any other
(see figure 1).
length
figure
X
each point. Given a collection of points
is
This graph
is
indeed a triangulation,
is
the graph defined
if
they are Voronoi neighbors (see
i.e.,
a subdivision of the convex hull
of points in triangles.
Both the Voronoi diagram and the Delaunay triangulation are fundamental constructions in computational geometry, since
are based on their construction.
tant for the
in
the
MST
efficient
MST,
since
it
many
algorithms for solving geometrical problems
In particular, the
contains the
MST
Delaunay triangulation
as a subgraph,
have to be also Voronoi neighbors (see figure
algorithm for the
MST,
O(nlogn) time and then runs
algorithm for the
MST
as well.
its
quite impor-
i.e.,
points which are neighbors
1).
This property leads to an
since one first constructs the
the greedy algorithm on
is
Delaunay triangulation
in
graph, leading to an O(nlogn)
Figure
We
MST
1:
The Voronoi diagram and
believe that our results give
as well.
Ramey
[10]
some
the Delaunay triangulation
partial insight
has attempted to prove a
on why a
CLT
for the
CLT
might hold
MST,
for the
but his approach,
although very interesting, did not succeed since he needed some unproven, but plausible,
lemmas from continuous
Figure
The CLT
2:
percolation.
The
MST
for jVi,„ has already
complicated fourth
moment
is
a subgraph of the Delaunay triangulation
been obtained by Bickel and Breiman
estimates.
They wrote: "Our proof is
long.
[5],
We
who
used very
believe that this
is
due to the complexity of the problem. Nearest neighbor distances are not independent".
We
used instead a simple geometrical approach, which works very well
and
in all these
problems
has the potential to work for other problems also.
it
Our
results are established for the Poisson model,
which
is
asymptotically equivalent to
the usual Euclidean model, but more convenient to work with.
Poisson model
Let X,,
in
i
=
1,
.
.
,
.
Nn he
the points of a Poisson process with intensity n on
a d-dimensional cube, so that
The paper
for all three
a Poisson
^"-^/^"U
<I>(x) is
x}
=
mean
n.
.^(x).
(1)
the cumulative distribution of a standard
and the Voronoi diagram
E[U]
:?
n
The constants
=
A'^(0, 1)
normal. In Section
we
3,
nearest neighbor graph, the Delaunay triangulation
Jt-th
satisfy:
Pd.
(2)
J~
Pj appearing
in (2)
have been explicitly computed by Miles
sharp contrast with the problems studied
we
variable with
y/Var[Ln]
show that the lengths E[Ln] of the
that
random
lie
structured as follows. Section 2 contains the proof of our main result that
is
limPr{
lim
is
which
problems:
n-.oo
where
Nn
R'',
calculated recently in
[1].
We
in [4], [11]
with the exception of the
[7],
which
is
in
MST constant
include for convenience simple direct computations
for the constants.
Finally in the final section
efficient sequential
and
we remark that our approach
parallel algorithms for this class of
leads to
more
problems on randomly distributed
points.
2
Central limit theorems
In this section
ology and
we prove CLTs
comment on
its
for the 3 problenns
we
consider.
We
first
outline the method-
applicability to other combinatorial problems.
Intuitive idea
2.1
We
identify a certain geometrical configuration An, such that
Pr{An}
—
1,
whose oc-
currence implies independence of the configuration at points which are far enough.
precisely, there
a cutoff distance, such that
is
the event -4„ happened, then determinis-
if
the configuration around a given point
tically,
More
is
not influenced by that of points further
than the cutoff distance.
For
all
the three problems, the event
= j-^
into p„
points
Xk
only a
finite
An
the event that, in the subdivision of
is
equal subcubes, each subcube contains at least one and at most
If all
the neighboring subcubes around a point are nonempty,
number of them determines the
fc-th
e log
[0,
1]*^
n of the
we then show that
nearest neighbors or the Voronoi neighbors
of a point. Thus, conditional on the event An, the three problems exhibit "m-dependence"
(at the subdivision level) for
[2]
As mentioned
see
[10],
how
for the
and
lie
it
m. Applying the theory of dependency graphs from
a similar approach for the
we described
thesis of
for the three
minimum spanning
tree. In
order to
problems we considered might generalize
consider the event Bn,k that there exist 2 points which are Voronoi neighbors
some distance
all
Ph.D
the introduction, our approach was inspired by the
who attempted
MST,
which are
than
in
the approach
at
finite
CLTs.
yields the
Ramey
some
shorter than
of each other, but can also be connected by a chain of edges
/
/
edges for some large
decision not to connect
and the shortest such chain between the two points uses more
Such two points
k.
them
ts
will
not be neighbors in the
affected hy the position of far
away
MST
and the
points. Let B'^^. be the
complement of the event Bn,k
Conditioning
in this
case on >i„
and thus showing that Pr{fl„jt}
(modulo some
make
technicalities).
fl
—
*
5J, ^
0,
m
finite
for the
MST
makes the problem m-dependent with
as n
—
oo would establish the
CLT
Further analogies with continuous percolation, when d
=
2,
the above quite plausible, even though a rigorous proof has not been submitted.
To summarize,
lation,
in the nearest neighbor, the
Voronoi diagram and the Delaunay triangu-
we have established the geometrical configuration whose probability of occurrence
tends to
the
and which impHes
1
MST
finite
range dependence and thus the CLT. In contrast, for
we know the geometrical configuration, but we can not show
occurrence tends to
Finally, in the
1.
TSP
probabiHty of
and minimum matching problems we do not
even have a clear geometric argument of why dependence
2.2
its
is
local.
Dependency graphs
The fundamental concept
that captures the idea of local dependence
graphs. Introduced in Petrovskaya and Leontovitch
Baldi and Rinott
Let Xa, a G
V
[8]
follows:
G=
be a collection of random variables. The graph
dependency graph
for
has one endpoint
in
Xa,
if
for
any pair of
Ai and the other
in
disjoint sets A\,
A2, the sets of
Xa, a S A2 are independent. The following
is
that of dependency
and applied to several problems by
dependency graphs are defined as
[2], [3]
is
{V,
E)
is
said to be a
A2 £ V such that no edge
random
variables Xa, a
in
E
£ A\ and
the fundamental theorem that captures local
dependence.
Theorem
Gn =
1
([3]) Let {Xa,n,
{Vn,En), n
maximum
degree of
>
I
a
Let Sn
.
Cn
\Pr{^ILZll^ <
(^^d
x}
€ V„}
be
= HoeVn
^<»n,o-^
< Bn
suppose \Xan\
-
$(x)|
<
random
32(1
variables having a dependency graph
=
Var[Sn]
<
00.
Let
Dn
denote the
almost surely. Then
+ V^k '^"':^"^" )^.
(3)
n
Thus
^"
if
l
l
ff^"
—
as n
^ 00
^^^—^^-^(0,1).
(4)
Remark:
The above theorem
quantifies the fact that
enough independence (D„ small), enough bound-
ness (S„ small) and enough variability (cn large) imply the
CLT.
The CLT
2.3
Let
a set
»
=
We
C
(»i,
We
be the length of the graphs Nk,n,Dn and V„.
Lji
.
of m'' subcubes with m''
,
.
.
ij),
where each
decompose the
Ln =
ij
=
subdivide the cube
The subcubes
(i^t;^)-
takes values from
1
to
will
[0, 1]''
into
be denoted by indices
m.
L„ as
total length
E hn
(5)
Tec
where L'
is
the
sum
of
all
edges with both ends
in the
subcube
portion of the edges within the subcube with only one endpoint
be the number of points falling
non-empty and contain
>in
We
first
Lemma
at
in
most
e
the cube
log
i
in
»',
plus the
the subcube
n points,
lim PriAr,}
n—'OO
-
1.
Proof
Poisson with parameter A
Pr{\ < ^,^ < eA}
But Pr{Nj^
=
0}
=
where we used
=
1
- Pr{Af,„ =
= -^ =
0}
-
logn.
Pr{N^,^
>
Thus
eA}.
e-^ and
eStirling's
1
bounds
PriK
(eA)!
for the
N^
Let
A^-'
will
be
(6)
2
is
of the
i.e.,
show that
variable A^^^
i.
and An be the event that the subcubes
= n^ec{i<^r.n<«'°gM-
The random
sum
<
"
x_^_^<e-\
(e
- 1)^^
next to
s/X
last inequality.
-A
=
<eA} > l-2e-^
Therefore,
1--2
Since we are using the Poisson model, the
random
variables
are independent and
A'''
therefore
Pr{^„}>(l--r' = (l--)i^.
n
Taking
limits
we obtain
lim
and thus lemma
We
n
introduce
Pr{An} >
_
lim e
2
»«"
=
and
j:
1
2 follows.
now
a distance between subcubes
d(ij)
=
i
maii<r<d{|»r
-
>]}•
Let
SlR={UC:
subcubes of radius
S^j^ denotes the sphere of
i.e.,
sets of subcubes,
we
d{lj)<R},
R
around
i.
Moreover,
if
A,B
are two
let
d{A,B} = minj^^j^gd(ij).
The
following proposition
is
the heart of our development of the
CLT
and captures the
idea of "local dependence" in the three problems we consider.
Proposition 3 Conditional on
R
the event
A^ having
(depending only on the dimenston d) such that
with
{L-r
d{A,B) > R, then
,
i
6 B}
if
occurred, there exists a fixed
A,B
the corresponding collections of
is
any pair of
random
number
sets of subcubes
variables {L^„,
«
6 A] and
are independent.
Proof
Let X-' denote the set of
random
of the points in a subcube.
We
variables {Xy, v
note
first
£i},
i.e
the collection of the locations
that the point process on
[0, 1]''
obtained from
the Poisson point process by conditioning on An, will retain the property that
independent.
X^
will
be
prove the proposition for d
=
the proof holds with no changes in any
d.
We
will
neighbor graph we present the case k
=
where the geometry
2,
<^(».
R =
J)
£
4.
2,
Note
since
if
that
first
>
d{i,j)
if
3,
easier to visualize, but
Similarly, for ease of exposition in the k nearest
I.
=
For the nearest neighbor graph (with d
with
is
two points Xk €
then a point
and
2
it
=
1),
we prove the proposition
and X; € j are nearest neighbors, then
i
a subcube at distance one of
in
i
is
guaranteed
to be closer than A'/ (see also figure 3).
Nearest neighbor
X.
Figure
3:
\
.
X
Local dependence in the nearest neighbor graph
Moreover, the decision which of the points Xr belonging
is
the nearest neighbor of
Xk G
that the analog of this property
the event An, {L^^'
some function f{X^,
since
Xj
^ ^)
*
i
'^
i,
is in
fails
and Xi 6
some function
f{X-^,
i
€ Sa,2} and
But, given
similarly {Lj^,
i
€ B}
4 implies that Sa.2, Sb,2 are disjoint
turn our attention to the Voronoi diagram.
»
is
included
in
the sphere
j are Voronoi neighbors then d{i,j)
to the Voronoi polygon of both. In fact, with
lies
the sphere S^^
MST, matching and TSP).
to hold for the
the set of points which are nearest to the point
Xt
in
no way affected by any point Xt outside Sj^. (Note
6 Sb.j}- Since d{A,B) >
polygon of a point Xk €
d(i,J)
subcubes
is
and
are independent, the proposition follows.
We now
if
in
>
4,
<
Sj^j,
Arguing as before, the Voronoi
since the Voronoi polygon of
Xk than any
4,
more
other point Xi. Thus,
since the midpoint of Xjt
effort
we can check that
then any circle through Xk, Xi has to contain a
(since all
full
subcube
if
Xk
is
Xk €
»
and Xi belongs
d{i,j)
f,
<
3,
since
where a point
subcubes are non-empty) and thus Xk, Xi cannot be Voronoi neighbors.
The
decision which of the points
neighbors
is
the nearest neighbor
Thus, arguing as
Xr belonging
oiXk €
is in
i,
the nearest neighbor case,
in
subcubes
in
in
the sphere S^^ are Voronoi
no way affected by any point Xr outside S^
/?
=
8
is
sufficient to
..
prove the proposition.
D
Since we want to apply theorem
is
1
we want an upper bound on the
length of
L-'
,
which
provided by the following proposition.
Proposition 4 Given
the event
An
Proof
Since every distance
is
at
.,n
Given the event
>4„, A^;^^
<
e
Finally, in order to apply
is
provided
in
^
most the length
log n
-
.,n
and since
theorem
1
of the diagonal of the subcube,
m=
we have that
^
(j^^)^ we obtain
(7).
D
we need a lower bound on the variance of L„. This
the following proposition.
Proposition 5 There
exists a constant /j
depending only on the dimension such that
Var[Ln]>fdn^.
(8)
Proof
We
will
use a general technique from
subdivide the cube
[0, 1]^
in
Ramey
[10],
which we
illustrate for
n squares of dimension -^ with area ^ each.
d
=
We
2.
We
identify
a particular configuration, which has-a strictly positive probability of occurring and has
nonzero
e
=
g-T^.
1.
We
variability.
Consider
all
describe the construction for the nearest neighbor graph.
the subcubes that have the following properties:
They contain exactly
and radius
e,
Let
2 points inside a circle with center the center of the
and no other points
in
the subcube.
10
subcube
2.
A
ring of points exist near the outside of the
another.
In particular,
that surround the
initial
we demand that
all
boundary
1
most
2e of
the 36 smaller subcubes of dimension
subcube are non-empty
1
at distance at
(see figure 4).
i
For the t-nearest graph the construction
points inside the
that
circle, so
outside subcubes contains k
+
I
is
similar except
we require that there
points.
They contain exactly one point
subcube. The three points
in the
the circle in six equal circular regions.
second
The
subcubes with
all
e
and no other points
We
circle are located as follows:
subdivide
As
3 points are located in alternate regions.
neighbor graph a ring of points exist outside the boundary at distance at
in the nearest
most
of the
inside a circle with center the
center of the subcube and radius | and exactly 3 points inside a circle
in the
its
+1
.
For the Voronoi diagram and the Delaunay triangulation we consider
the following properties:
and
it-nearest neighbors are inside the circle
all
are i
2e of each other.
The
probability of such configuration
is
pi
>
Note that the
0.
point inside has Voronoi neighbors only the three points in the outside circle. Fixing the
position of the three points in the outside circle there
the Voronoi polygon and
easily
above we
Un be the
let
the positions of
all
A^i
be the number of subcubes
<t— algebra determined by the
the points lying outside the
N
^,
a certain variability
Delaunay triangulation of the point
its
computable. Letting
is
inside, all of
/ satisfying
random
variables
^
for
which are
the two properties
»i,
.
.
,
.
»Af,
subcubes and inside the larger
and by
circle.
Then,
Var[U] =
Var[E[Lr,\Un]]
Var[E[U\Ur,]]
+
E[E[Ll\Ur,]
n
d,-
We now
is
have
Theorem
sum
the
all
avpx >
^
=
d.|C/„]]
=
0,
of the Voronoi polygon (Delaunay triangulation) of the inside point.
the ingredients to prove the
CLT.
6 The length Ln of the k-nearest neighbor graph, the Voronoi diagram and the
Delaunay tnangulaiton
^lun^Pr{
n-.oo
where ^(x)
{E[Ln\Un]?]
+ E[Var[U\Un]] > E[Var[Yl d, +
—E[N] =
where
-
satisfies the
^"-^/^"U
CLT
x}.<^(x),
(9)
yVar[Z,„]
is
the cumulative distribution of a standard normal.
12
—
Proof
We
is
establish that the
first
random
')'
variable
2isymptotically normal. Indeed,
we
given that the event
-^
proposition
the
3.
An has occurred
Bn <
-
—
^
Lj-^
and Lj ^
if
the sense of section 2.2
in
<
d(i,j)
1
dependency graph
this
we have that
=
iKnl
at
is
and D„
m"^
R
most
<
72'^
,
i.e.,
finite
is
defined
Moreover,
independent of
from proposition
3.
from proposition 4 and Var[L„] > fdn~^ from proposition
°^i'
R
R, where
Proposition 3 establishes the correction of this construction.
maximal degree Dn of
Applying theorem
dependency graph
define a
between the L^^ by putting an edge between
in
"
••• '
y/Var[L„]
n.
In addition,
5.
Therefore, since
where uj
is
'y
^i^)[
\Vn\DlBl
<
;t < ^d
;
—
»•
oo.
Applying now theorem
has occurred
We now
"<i-
= j^^
1,
we
need to show that
(/„
=
0,
i
ns
establish that
asymptotically normal N(0,
is
we obtain
Pognrl
Var[Ln]'i
n
<
a constant depending on the dimension only. Since m!^
\Vn\DlBl
as
nl+i
]
dTTT
"~
'
"I
given that the event
An
1).
"
1
?~
1
is
asymptotically normal. But
Pr{Un <x} = Pr{Un < x\An}Pr{An} + Pr{Un < x\A'JPr{A'n}
Taking
we
linnits
and using lemma
2
and the asymptotic normality of
(7„
given the event
An
establish the theorem.
Remark:
We
can establish rates of convergence to the normal distribution by using
^/Var[Ln\
3
(3).
We
see that
n*
Computations of the expectations
In this section
in (2),
we
will find exactly £'[L„],
i.e.,
we
will
compute the constants
/Jj,
appearing
using the toroidal rather than the usual Euclidean distances, so that there are no
13
boundary
However, the boundary
effects.
can be shown to be negligible (Jaillet
effects
[6]),
and so the same results holds.
The
k-th nearest neighbor graph
Consider a given point under the toroidal model. Then the length of the nearest neighbor
=
graph L„
"/?„,
where Rn
the distance of a given point to
is
its
t-th nearest neighbor.
Also,
E[Rn]= I
Pr{Rr,
>
r}dr
=
Jo
2^
L
where
cj
'^''
7i
^•!
the volume of the unit sphere
is
Letting u
=
Letting n
—
n
cj
r''
dimension
in
d.
we obtain that
oo we obtain that
A ro + ;-i)
--~i;^-^S 0-1)!
E[u\_
1
<'">
The length of the Voronoi diagram
Let
Rn be
E[Ln\
=
the length of the perimeter of the Voronoi polygon around a given point O.
n
^^
^2"
contribution to
^'''
Rn from
a point
OC
tioning on the distance
E[hn] given that
perform the calculation
OC =
=
C
for d
=
2.
Then
of the Poisson process.
r (see figure 5), let £'[/i„(r)]
Let
£'[/»„]
£'[/2n]
=
Then
be the expected
nE[hn]- Condi-
be the expected contribution to
Then,
r.
E[hn]
=
E[hn{r)]2Krdr.
/
Jo
Moreover, E[/i„(r)]
=
2/o
Pr{the length
diagram}^/. But, Pr{the length
Pr{no perpendicular
belongs
in
dl at
dl at distance
distance
bisector cuts the segment
the circle with center
B
and radius
72
E[hn]
= 4^ /
Jo
/
from
A
is
/
from
e~'**?'
"*"'
re-'((?)'+'')M/dr
Jo
14
part of the Voronoi
=
point of the Poisson process
,x/2
/
is
part of the Voronoi diagram}
OB} = Pr{no
BO} =
A
^".
Thus
Computing the
integral
we obtain
lim Ef/inln?
=
4
n—>oo
which leads to
lim
2.
n-^oo
Figure
The Delaunay
Let
its
Rn be
the
5:
The expectation
,Jn
of the length of the Voronoi diagram
triangulation
sum
of the distances in the Delaunay triangulation from a given point
Voronoi neighbors. Then, the expected length of the triangulation
Let h{r)rdr dO
= Pr{A
is
a Voronoi neighbor of
r2T
E[Rn]
=
compute h{r) we consider the
the Poisson process B, which
is
in
rdr d6}. Then
rh{r)rdrde.
circle passing
to the right of the
contains no other point of the process.
a point
is
Jo
Jo
In order to
.4 is
to
/-x/a
/
/
O—
O
It is
well
15
through O,
segment
known
OA
(see for
A and
a third point of
with the property that
example
[9])
that O,
A
it
are
.
,
L(r,t)
R(r,t)
Figure
6:
Voronoi neighbors
circle to the left of
of the length of the Delaunay triangulation
The expectation
if
and only
O^
if
no points of the Poisson process
lie
in the
segment of the
(see figure 6).
Let Tr the radius of the circle and L{r,t), R{r,t) the areas shown
in figure 6.
Therefore,
rv/2
h{r}
=
2
<
e-^<"'''"dPr{T,
/
t}
Jo
where we multiplied by
2 because
we considered the point
C
could have considered the case in which
<0
Pr{Tr
is
the area of the right segment of the
simple geometry L{r,t)
integrals
we
=
<i,t^
-
on the
is
=
circle.
^^
U
left.
4>
=
E[Rn]s/n
=
64
—
which leads to
,.
hm
E[U]
—
= —
-=-
y'n
16
OA, where we
Moreover,
sin'^j^
=
and R[r,t)
n—oo
to the right of
e"^*''"".
find that
Hm
C
32
07r
(tt
-
is
the angle
<p)i^
+
^^.
ACB,
then from
Computing the
On
4
the relation of
The approach we used
CLTs
to efficient algorithms
leads also to a faster algorithm to
and the Delanay triangulation
for
compute the Voronoi diagram
randomly distributed points.
From
proposition
3,
in
order to find the portion of the Voronoi diagram inside a subcube, we need only to consider
only those subcubes that are at distance at most
An
event
occurs, there are
diagram inside a subcube.
R =
4
from
this
subcube. Thus,
if
the
O(logn) points that contribute to the portion of the Voronoi
Running an 0{k
log k) algorithm with
it
= O(logn)
gives an
0(log n loglog n) algorithm to find the Voronoi diagram inside a subcube. Since there are
j^^ subcubes, we obtain an 0(n log
log n) algorithm.
If
the event
An does not
simply run any O(nlogn) algorithm. Since the probability of An goes to
time O(nloglogn) with probability
runs
in
done
in parallel,
1.
In addition, since this
1,
occur,
we
this algorithm
computation can be
the above appraoch leads to an 0(log nlog log n) parallel algorithm with
i-^^ processors.
In general,
if
such an approach were to work
for other
combinatorial problems, we would
obtain not only a CLT, but faster sequential as well as parallel algorithms for their solution.
References
[1]
F.
Avram and
D. Bertsimas, (1990), "The
rical probability
minimum spanning
tree constant in geomet-
and under the independent model; a unified approach",
to appear in
Annals Appl. Prob.
[2]
P. Baldi,
Y. Rinott, (1989), "Asymptotic normality of some graph related
statistics",
Jour. Appl. Prob., 26, 171-175.
[3]
P. Baldi,
Y. Rinott, (1989), "On normal approximations of distributions
dependency graphs". Annals
[4]
J.
in
terms of
Prob., 17, 1646-1650.
Beardwood, J.H. Halton and J.M. Hammersley, (1959), "The shortest path through
many
points", Proc. Cambridge Phtlos. Soc, 55, 299-327.
17
[5]
P. Bickel
and
L.
moment bounds,
[6]
P. Jaillet, (1990),
[7]
R. Miles, (1970),
Breiman, (1983), "Sums of functions of nearest neighbor distances,
limit
theorems and goodness of
fit
tests",
Ann. Prob.,
11, 185-214.
personal communication.
"On
the
homogeneous planar Poisson point process". Math. Bio-
sciencts, 6, 85-127.
[8]
M. Petrovskaya and A. Leontovitch,
of
random
Appi,
[9]
F.
(1982),
variables with a slowly growing
"The central
limit
theorem
for a
sequence
number of dependencies", Theory
Prob.
27, 815-825.
Preparata and M. Shamos, (1985), Computational Geometry, Springer- Verlag,
New
York.
[10]
M. Ramey,
[11]
J.M. Steele, (1981) "Subadditive Euclidean Functionals and Nonlinear Growth
(1983), Ph.D. thesis,
Department of
ometric Probability", Ann. Prob.,
[12]
9,
Statistics,
Yale University.
in
Ge-
365-376.
J.M. Steele, (1988) "Growth Rates of Euclidean Minimal Spanning Trees with Power
Weighted Edges", Annals of Prob., 1767-1787.
18
55U
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071
Date Due
OCT
Lib-26-67
iI/i^AKlCs
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