UC Berkeley
UC Berkeley Previously Published Works
Title
Asymptotics for Euclidean minimal spanning trees on random points
Permalink
https://escholarship.org/uc/item/68r8t9th
Journal
Probability Theory and Related Fields, 92(2)
ISSN
0178-8051
Authors
Aldous, D
Steele, JM
Publication Date
1992-06-01
DOI
10.1007/BF01194923
Peer reviewed
eScholarship.org Powered by the California Digital Library
University of California
Probab. Theory Relat. Fields 92, 247-258 (1992)
Probability
Theory a.d
Related Fields
9 Springer-Verlag 1992
Asymptotics for Euclidean minimal spanning trees
on random points
David Aldous 1'* and J. Michael Steele 2'**
1 Department of Statistics, University of California, Berkeley, CA 94720, USA
2 Department of Statistics, Wharton School, University of Pennsylvania, Philadelphia, PA 19104,
USA
Received March 1, 1991; in revised form September 3, 1991
Summary.
Asymptotic results for the Euclidean minimal spanning tree on n ran-
dom vertices in
R d can
be obtained from consideration of a limiting infinite forest
whose vertices form a Poisson process in all R d. In particular we prove a conjecture
of Robert Bland: the sum of the d'th powers of the edge-lengths of the minimal
spanning tree of a random sample of n points from the uniform distribution in the
unit cube of R d tends to a constant as n --, oe.
Whether the limit forest is in fact a single tree is a hard open problem, relating
to continuum percolation.
1 Introduction
Let x = {xl,..., x, } be a finite set of points in R d, d > 2. The minimal spanning
tree (MST) t of x is a connected graph with vertex-set x such that the sum of the
edge-lengths of t is minimal, i.e.
lel = min ~ le[
e~t G e~G
where [el = Ixi - xjl is the Euclidean length of the edge e = (x, xj) and the
minimum is over all connected graphs G. Minimal spanning trees are one of the
most studied objects in combinatorial optimization, and even the probability
theory of MST is rather well developed. See Steele [13] for an overview. In that
paper it was shown that, if (xi: 1 _< i __ n) are i.i.d, with compactly supported
density f and if 0 < ~ < d, then
n -(d-~)/a •
lel ~
~ c(~, d) ~
(f(x))td-~)/ddx
a.s.
e~t R d
(1)
* Research supported by N.S.F. Grants MCS87-01426 and MCS 90-01710
** Research supported in part by N.S.F. Grant DMS88-12868, A.F.O.S.R. Grant 89-0301, ARO
Grant DAAL03-89-G-0092 and NSA Grant MDA-904-H-2034
248 D. Aldous and J.M. Steele
where c(:~, d) depends only on e and d. The most concrete goal of this paper is to
extend partially this result to the extreme case e = d (see Proposition 3a). The proof
of (1) used subadditivity, a technique going back to the original probabilistic
analysis of the traveling salesman problem by Beardwood et al. [4], who stated
that the e = 1, d = 2 case of (1) could be proved by their method. This paper uses
a completely different approach. We study the analog of the MST for an infinite set
of points, the points of a Poisson process on all R d. Since empirical distributions
of i.i.d, sequences look locally like Poisson processes, we can relate their MST
to the MST of the unbounded Poisson process. This approach also gives another
proof of the existence of a limit distribution for degrees of vertices in the MST
(Proposition 3b).
To formalize this approach, let x = (x~) be a finite or countably infinite subset of
R e, d > 2. Call x nice if
(i) x is locally finite, i.e. has only finitely many elements in bounded subsets of
Re; and
(ii) the interpoint distances (Ixj - x~l, i <j) are all distinct.
Given a pair (x, x) with x nice and x s x we can define trees tm (x, x) with vertices
from x as follows. Let ~1 = x, and let tl be the single vertex ~1. Let tz be the tree
consisting of the vertex ~1 and the vertex ~2 ~ x\{~l } which is closest to ~l in
Euclidean distance, together with the edge (straight line segment) connecting ~i
and 42. Inductively define tm= tin(x, x) to be t,,_l together with a new edge
(~Jm, ~,,), where Jm < m- 1 and ~,, ~x\{~l ..... ~m-1} are chosen so that the
edge-length [~,,- ~jm[ is minimal (over all possible edges connecting tin-1 to
x\t,,_ 1.) In the finite case where n = Ix[ < 0% this procedure terminates with the
tree t, (x, x). Here we have described a variant of the well-known greedy algorithm
for constructing the Euclidean MST (see Sect. 4). That is, in the finite case t(x, x) is
the unique Euclidean MST on vertices x, and so in particular it does not depend on
the choice of the starting vertex x.
What happens when x is infinite is less simple. In the infinite case, write
too (x, x) = •. t. (x, x).
Lemma 1 Let g = g(x) be the graph on an infinite nice vertex-set x defined by taking
(xl, x2) as an edge in g fit is an edge in either too (xl, x) or too (Xz, x). Then the graph
g is a forest and each component ofg is an infinite tree.
This lemma is proved in Sect. 2. An equivalent definition ofg is given in Lemma 12.
The central character of the paper is the random tree 3- defined as follows.
Take a Poisson point process ~A# = {q~ } of rate 1 in R e, d __> 2. Let ~Ar ~ = X u {0}.
Throwing away a null set, ~A# ~ is a random nice subset ofR e. Let ~ = g(o/V ~ be the
forest constructed via Lemma 1.
It is natural to conjecture that N is in fact a.s. a tree, but this seems to be related
to deep issues in continuum percolation (Sect. 5). Here we finesse the issue by
writing ~- for the component of N containing the vertex 0. The next lemma records
some simple facts about r This lemma and Proposition 3 will be proved in Sects.
2 and 3.
Lemma 2 Let D be the degree of vertex 0 in J-. Let L~ ..... LD be the lengths of the
edges of J incident at O. Then
(a) D <= be, a constant.
(b) ED = 2.
1
~" EL ~
(c) le=~z,~ ~ <~176
Random minimal spanning trees 249
Now let Y, denote the point process consisting of n points (t/i: 1 < i < n) that are
independent and uniformly distributed on the unit cube [0, 1] e. Throwing away
a null set, ~Ar, is a nice subset of R d. Let 50, = t,(t/1, ~.) be the MST on these
n vertices. It is intuitive that 5P,, suitably rescaled, converges locally to fr This idea
is formalized in Proposition 9 and leads to a proof of our main result.
Proposition
3 (a)
Let
(levi)
be the lengths of the edges of S~.. Then
n-- 1 L 2
~ ]elf ,le as n---~o~.
1
(b)
Let A.,i be the proportion of vertices of ~. with degree i. Then for each i
EA.,i ~ P(D = i) as n ~ oo .
Discussion 1.
This idea of getting asymptotics by considering a kind of MST on the
Poisson limit process is rather natural to modern theoretical probabilists. The
purpose of this paper is to provide the details, which are slightly less obvious than
the first author thought initially. Additional motivation was provided by the
emergence of Conjecture 13 below, which seems both interesting and difficult. Our
results in themselves do not provide useful numerical bounds for the limiting
constants. In work complementary to the present paper, Avram and Bertsimas [-3]
give a unified treatment of the Euclidean model and the i.i.d, model (see below) and
discuss numerical bounds; they do not explicitly use the Poisson limit process. As
remarked in Steele [13], for d = 2 the existence of a limit in (a) was conjectured by
Robert Bland on the basis of simulations. Indeed, in this case there is an absolute
bound
on
Ee~t
[el 2 regardless of n or the positions of the n points in [0, 1] 2.
2. In the setting of (1), where the MST is built from i.i.d, points with some
density with compact support K, it seems intuitively clear that
n- 1 L 2
~ lell a ' lelKI as n--, oo
1
where [Kt is the d-dimensional Lebesgue measure of K. But we have not attempted
to write out the details.
3. A simpler "i.i.d. model" for MSTs discards the geometry of R e and supposes
that the inter-point distances are i.i.d, with some distribution 4. The analog of part
(a) of Proposition 3 has been well-studied by combinatorial methods in the case
where ~ has uniform distribution. See Timofeev 1-15], Avram and Bertsimas [3] and
Aldous [2] for recent results on more general distributions.
4. The existence of limits in (b) was proved by different methods in Steele et al.
[14], who also proved a.s. convergence. The same question in the i.i.d, model was
solved explicitly in Aldous [2], using "limit process" arguments in the spirit of this
paper.
5. Proving central limit theorems in this area seems technically difficult. Ramey
[-12] shows that the CLT for total length of the Euclidean MST can be reduced to
a technical "conditional independence" conjecture.
2 Technicalities
Let us first record one simple fact.
250 D. Aldous and J.M. Steele
Lemma 4
There is a bound be on the degree of any vertex in any Euclidean MST
in R d.
Proof.
An easy application of the triangle inequality shows that (in any dimension)
two edges of a MST meeting at a vertex cannot make an angle of less than 60 ~
We now work toward the proof of Lemma 1. Let c~ be the graph with vertex-set
x and such that
(xi, xj)
is an edge of cl iff Ixj - xil < l. Write
t~(x)
for t~(x, x).
Lemma 5
If(y1, Y2) is an edge of t~ (x) for some x ~ x, then it is also an edge of either
t~(yl)
or t~(y2).
Proof
In the construction of t~o (x) with vertices (x = ~1, ~2,- 9 -), we may suppose
that the edge (Yl, Yz) occurs as (~,i, ~j), say, with i < j and has length I. If the edge is
not in t~ (~i) then the component of cz containing ~ must be infinite; but then the
edge cannot appear in to (x).
Proof of Lemma 1.
By Lemma 5 and the definition of g, a component of g which
contains a vertex x must contain all edges of t~ (x). Thus all components of g are
infinite. Next, suppose g contains a circuit
(yl, Y2 ..... Yk, Y~)
with the maximal
edge-length attained by the edge e =
(Yk, Yl),
say. Consider
t~(yl).
This cannot
contain the edge e, because the algorithm would first have added the vertices
{Y2, - 9 9 ,
Yk } which can be connected with Yz using shorter edges. Similarly,
too (Yk)
cannot contain edge e, and so 9 does not contain e.
There is a natural notion of convergence for locally finite sets x,, x. We define
x, ~ x to mean: we can label x as x~, x2,.., and x, as x,, ~, x,,2, 9 9 9 such that as
n ---~ oo
x,.i --+ xl,
0 < i < co; (2)
Ix. ~ CLI-' [x c~ CLI (3)
for each L such that x has no point on the boundary of
CL - [- L, L]d.
Now suppose h, and h are graphs with respective vertex-sets x, and x. Define
convergence of graphs h, ~ h to mean: x, ~ x, and, for each L as above, for all
n > some no (L), we have the two properties
if
(x.,i, x.,j)
is an edge of h. with x.,i ~ Cr then (xi, xj) is an edge of h (4)
and
if (x~, xj) is an edge of h with
xi ~ CL
then
(x,,~, x,,j)
is an edge of
hn 9
(5)
Note from this definition that convergence h.---> h and x..i ~ xl implies that the
degree
d(x.,i)
of
x..i
in h. converges to the degree
d(xi)
of
xl
in h.
Recall the trees
tk(X,
X) defined in the introduction. Define a graph 9k(X) by:
(xi, xj)
is an edge in gk iffit is an edge in either
tk(X~,
X) or
tk(Xj,
X). It is clear that,
for nice sets x. and x such that x is infinite,
if x, ~ x, x, --* x, then
tk(x,,
X~) --+
tk(x,
X) for each k
and hence, for fixed k,
if x, ~ x then 9k(x,) and gk(X) satisfy (5). (6)