Lowdegree minimum spanning trees
Discrete Comput Geom
LowDegree Minimum Spanning Trees 1
G. Robins 1
J. S. Salowe 0 1
I. Introduction
0 QuesTech Inc. , 7600A Leesburg Pike, Falls Church, VA 22043 , USA
1 1Department of Computer Science , Thornton Hall , University of Virginia , Charlottesville, VA 229032442 , USA
Motivated by practical VLSI routing applications, we study the maximum vertex degree of a minimum spanning tree (MST). We prove that, under the L j, norm, the maximum vertex degree over all MSTs is equal to the Hadwiger number of the corresponding unit ball; we show an even tighter bound for MSTs where the maximum degree is minimized. We give the bestknown bounds for the maximum MST degree for arbitrary L e metrics in all dimensions, with a focus on the rectilinear metric in two and three dimensions. We show that for any finite set of points in the rectilinear plane an MST exists with maximum degree of at most 4, and for threedimensional rectilinear space the maximum possible degree of a minimumdegree MST is either 13 or 14.

* Gabriel Robins was partially supported by NSF Young Investigator Award MIP9457412.
Jeffrey Salowe was partially supported by NSF Grants MIP9107717 and CCR9224789.
The Bounded Degree Minimum Spanning Tree (BDMST) Problem. Given a com
plete weighted graph and an integer D > 2, find a minimumcost spanning tree with
maximum vertex degree < D.
Finding a BDMST of maximum degree D = 2 is equivalent to solving the
traveling salesman problem, which is known to be NPhard [
7
]. Papadimitriou and
Vazirani have shown that, given a planar pointset, the problem of finding a
Euclidean BDMST with D = 3 is also NPhard [
17
]. On the other hand, they also
note that a BDMST with D = 5 in the Euclidean plane is actually a Euclidean MST
and can therefore be found in polynomial time. The complexity of the BDMST
problem when D = 4 remains open. Ho et al. [
12
], proved that an MST in the
rectilinear plane must have maximum degree of D < 8, and state (without proof)
that the maximum degree bound may be improved to D < 6. The results of Guibas
and Stolfi [
10
] also imply the D < 8 bound.
In this paper we settle the boundeddegree MST problem in the rectilinear plane:
we show that, given a planar pointset, the rectilinear BDMST problem with D < 3 is
NPhard, but that the rectilinear D > 4 case is solvable in polynomial time. In
particular, we prove that in the rectilinear plane an MST always exists with
maximum degree of D < 4, which is tight. We also analyze the maximum MST
degree in three dimensions under the rectilinear metric, which arises in
three. dimensional VLSI applications [
11
], where we show a lower bound of 13 and an
upper bound of 14 on the maximum MST degree (using the previously known
techniques, the best obtainable lower and upper bounds for three dimensions under
the rectilinear metric were 6 and 26, respectively).
More generally, for arbitrary dimension and Lp metrics we investigate:
(i) The maximum possible vertex degree of an MST.
(ii) The maximum degree of MSTs in which the maximum degree is minimized.
We prove that the maximum MST degree under the Lp metric is equal to the
socalled Hadwiger number of the corresponding Lp unit ball. The relation between
MST degree and the packing of convex sets has not been elucidated before, though
Day and Edelsbrunner [
6
] studied the related "attractive power" of a point. For
general dimension we give exponential lower bounds on the Hadwiger number and
on the maximum MST degree.
Our results have several practical applications. For example, our efficient
algorithm to compute an MST with low maximum degree enables an efficient
implementations of the Iterated 1Steiner algorithm of Kahng and Robins for VLSI routing
[
14
], which affords a particularly effective approximation to a rectilinear Steiner
minimal tree (within 0.5% of optimal for typical input pointsets [
21
]), and where the
central timeconsuming loop depends on the maximum MST degree [
1
], [
2
]. Our
results also have implications to newly emerging threedimensional VLSI
technologies [
11
], as well as for any other algorithms that use an MST as the basis for some
other construction.
The remainder of the paper is as follows. Section 2 establishes the terminology
and relates the maximum MST degree to the Hadwiger numbers (the central result
is Theorem 4). Section 3 studies the L 1 Hadwiger numbers and the maximum MST
degree for the L 1 metric. Section 4 considers the Hadwiger numbers and the
maximum MST degree for arbitrary Lp metrics. We conclude in Section 5 with open
problems. A preliminary version of this work has appeared in [
20
].
2. Hadwiger and MST Numbers
A collection of open convex sets forms a packing if no two sets intersect; two sets
that share a boundary point in the packing are said to be neighbors. The Hadwiger
number H(B) of an open convex set B is the maximum number of neighbors of B
considered over all packings of translates of B (a translate of B is a congruent copy
of B moved to another location in space while keeping B's original orientation).
There is a vast literature on Hadwiger numbers (e.g., [
5
] and [
20
]). Most results
address the plane, but there are several results for higher dimensions. In particular,
if S is a convex set in ~ k (i.e., kdimensional space), the Hadwiger number H(S)
satisfies
k 2 + k < H ( S )
< 3 k  1.
It is known that the regular ksimplex realizes the lower bound and the
khypercube realizes the upper bound [
22
]. Tighter bounds are known for khypersphei'es;
Wyner [
23
] showed that the Hadwiger number for spheres is at least 2~176 +oc1)),
and Kabatjansky and Leven~tein [
13
] showed that it is at most 20"401k(l+~ Only
four Hadwiger numbers for spheres are known exactly; these are the numbers in
dimensions 2, 3, 8, and 24, and they are 6, 12, 240, and 196560, respectively. The
threedimensional Hadwiger number has a history dating back to Newton and was
only determined much later.
For two points x = (Xl, x 2. . . . . x k) and y = (Yl, Y2. . . . , Yk) in kdimensional
P k
space 9~k, the Lp distance between x and y is Ilxyllp = V~.i=llxiyil p. For
convenience, if the subscript p is omitted, the rectilinear metric is assumed (i.e.,
p = 1). Let B(k, p, x) denote an open Lp unit ball centered at a point x in 9~k (we
use ~Rkp to denote kdimensional space where distances are computed under the Lp
metric). When x is the origin, B(k, p, x) is denoted as B(k, p), and we use H(k, p)
to denote the Hadwiger number of B(k, p).
Let I(k,p) be the maximum number of points that can be placed on the
boundary of an Lp kdimensional unit ball so that each pair of points is at least a
unit apart. With respect to a finite set of points P c ~Rkp and an MST T for P, let
v(k, p, P, T) be the maximum vertex degree of T (also referred to as simply the
degree of T). Let u(k, p, P ) = m a x r ~ ~ v(k, p, P, T), where T is the set of all MSTs
for P, and let v(k, p) = maxec ~, u(k, p, P). In other words, v(k, p, P) denotes the
maximum degree of any MST over P, and v(k, p) denotes the highest possible
degree in any MST over any finite pointset in kdimensional space under the Lp
metric.
We will need the following result, which is easily established.
Lemma 1. Ilxyllp > 2 if and only if B(k, p, x) A B ( k , p , y ) = 0 . Further, Ilxyllp = 2
if and only if B(k, p, x) and B(k, p, y) are tangent.
Lemma 2. The Hadwiger number H(k, p) is equal to I(k, p), the maximum number of
points that can be placed on the boundary of the unit ball B(k, p) so that all interpoint
distances are at least one unit long.
Proof. We first show that I(k, p) < H(k, p). Suppose that I(k, p) points are on the
boundary of B(k, p) and are each at least a unit distance apart. Consider placing an
Lp ball of radius ~1 around each point, including one at the origin. By L e m m a 1,
these balls form a packing, and all the balls touch the ball containing the origin.
Therefore, I(k, p) < H(k, p).
Next we show that H(k, p) < I(k, p). Consider a packing of Lp unit balls, and
choose one to be centered at the origin. Consider the edges connecting the origin to
each center of the neighboring balls. The intersections of these edges with the
boundary of B(k, p) yield a pointset where each pair of points is separated by at
least a unit distance, otherwise we would not have a packing of Lp unit balls. []
L e m m a 3. The Hadwiger number H(k, p) is equal to z,(k, p), the maximum MST
degree over any finite pointset in ~k.
PrOOf. We show that v(k, p) < I(k, p), the maximum number of points that can be
placed on the boundary of an Lp kdimensional unit ball so that each pair of points
is at least a unit apart. Let x be a point, and let Y l , . . . , Y~ be points adjacent to x in
an MST, indexed in order of increasing distance from x. Note that (y;, yj) must be a
longest edge in the triangle ( x , y i, yj), and that the MST restricted to x and
Yl. . . . . y~ is a star centered at x. Draw a small Lp ball around x, without loss of
generality a unit ball, and consider the intersection of the segments (x, Yi), 1 < i < v,
with B(k, p, x). Let these intersection points be called )~i, 1 < i < v, and suppose
there is a pair .vi and ~/, i < j, with II~iYjl[p < 1. Note that (Yi, Yj) is the shortest
edge on the triangle (x, .vi, )~j), and (x, ~j) is a longest edge. Now consider similar
triangle (x, Yi, z), where z is a point on the edge (x, yj). The path from yj to z to Yi
is shorter than the length of (x, yj), so (Yi, Yj) is not a longest edge in triangle
(x, y~, yj), a contradiction. We note this bound is tight for pointsets that realize
I(k, p). []
We next consider a slightly different number ~(k, p), which is closely related to
v(k, p). Recall that v(k, p, P, T) denotes the maximum vertex degree of the tree T.
Let ~(k, p, P ) = m i n r ~ , v(k, p, P, T), where ~" is the set of all MSTs for P, and let
~,(k,p) = m a x e c ~ k v(k,p, P). In other words, ~,(k,p, P) denotes the degree of an
MST over P that has the smallest possible degree, and ~,(k,p) denotes the
maximum of the degrees of all minimumdegree MSTs over all finite pointsets in ~R~
(recall that we use the phrase "the degree of T " to refer to the maximum vertex
degree of the tree T).
Although it is clear that ~,(k, p) < v(k, p), it is not clear when this inequality is
strict. In order to count ~(k, p), we define the MST number M(k, p) similarly to the
Hadwiger number H(k, p), except that the translates of the Lp unit ball B(k, p) are
slightly magnified. The underlying packing consists of B(k, p) as well as multiple
translated copies of (1 + e ) . B(k, p), and M(k, p) is the supremum over all 8 > 0
of the maximum number of neighbors of B(k, p) over all such packings. Clearly,
M(k, p) < H(k, p). We also define I(k, p ) as the number of points that can be
placed on the boundary of B(k, p) so that each pair is strictly greater than one unit
apart:
Consider a set S = {x1. . . . . x n} of n points in 9~kr. For convenience, let N = 2 ,
and let $ 1 , . . . , SN be the set of sums of the interdistances, one sum for each distinct
subset. Let
0 < 8 =
min
l<i<jgN
{IS i  Sgl: lS i  Sj] > O}.
A perturbation of a pointset S is a bijection from S to a second set S ' = {x~. . . . . x~,}
(for convenience, assume that the indices indicate the bijection); we say that a
perturbation of S is small if
8
i=1 [[xix~[[p < 2"
In discussing spanning trees of S and the perturbed set S', we assume that the
vertex set [n] consists of the integers 1 to n, where vertex i corresponds to point x i
or point x'i. The topology of a tree over vertex set In] is the set of edges in the tree.
Theorem 4. Let S be a set of points in fflk, and let S' be a set of points corresponding
to a small perturbation of S. Then the topology of an MST for S' is also a topologyfor an
M S T for S.
Proof. Let T be an MST for S, and let T ' be an MST for S'. Let I(T) and I(T') be
the lengths of T and T', respectively. Then l ( T )  8 / 2 < l ( T ' ) < l ( T ) + 8/2.
Consider the tree 7~ with the same topology as T ' but with respect to pointset S.
Now, I(T')  8 / 2 < I(T) < I(T') + 8/2, so I(T)  8 < I(T) < I(T) + 8. Since 8 is
the minimum positive difference between the sums of any two distinct subsets of
interdistances, I(T) = l(7~), and 7~ is also an MST for S. []
Lemma 5. The maximum of the degrees of all minimumdegree MSTs over all finite
pointsets in ~R~, is equal to the maximum number of slightly magnified unit balls that
can be packed around a given unit ball; that is, D(k,p) = M(k, p).
Proof. Let S be a set of points, and let 8 be defined as above. Place a small Lp
ball about each point x ~ S (without loss of generality a unit ball, though the intent
is that x is the only point inside B(k, p, x)), and connect each distinct pair (x, y),
x, y ~ S, with a line segment. Consider the intersections of these edges with
B ( k , p , x). Perform a small perturbation on S so that no two intersection points
have length 1. Repeat the argument used in the proof of Lemmas 2 and 3, this time
with balls of the form (1 + 8 ) . B(k, p), for small e > 0. The first part shows that
[(k, p) = M ( k , p), and the second that ~(k, p ) < f(k, p). This bound is tight for
pointsets that realize [(k, p). []
3. The Maximum Ll MST Degree
Hadwiger numbers are notoriously difficult to compute. In this section we detdrmine
the two and threedimensional Hadwiger numbers for the diamond and octahedron,
respectively. The first of these numbers is well known, but we could not find any
reference for the octahedron. We also study the MST numbers, obtaining a value of
4 in two dimensions, and bounds in higher dimensions. For notational convenience,
we define:
The Uniqueness Property. Given a point p, a region R has the uniqueness property
with respect to p if for every pair of points u, w ~ R, IIwull < max([lwpl[, Ilupll)
A partition of space into a finite set of disjoint regions is said to have the
uniqueness property if each of its regions has the uniqueness property.
Define the diagonalpartition of the plane as the partition induced by the two lines
oriented at 45~ and  4 5 ~ through a point p (i.e., partitioning ~ 2 _ {p} into eight
disjoint regions, four "wedges" of dimension two ( R 1  R 4 in Fig. l(a)), and four
"halflines" o f dimension one ( R 5  R 8 in Fig. l(a)). It is easy to show that the
diagonal partition has the uniqueness property, which in turn implies an upper
bound of 8 on the maximum MST degree in the rectilinear plane.
U
(b)
w
R2
P,6/
'R4
(a)
R3 p ~
R1
P
O u
(e)
L e m m a 6. Given a point p in the rectilinearplane, each region of the diagonalpartition
with respect to p has the uniqueness property.
Proof. We need to show that, for any two distinct points u and w that lie in the
same region, Ilwull < max(llwpll, Ilupll). This is obvious for the onedimensional
regions. Consider u, w in one of the twodimensional regions. Assume without loss
of generality that Ilupll < Ilwpll (otherwise swap the roles of u and v in this proof).
Consider the diamond D with left corner at p and center at c, such that u is on the
boundary of D (see Fig. l(c)). Let a ray starting at p and passing through w
intersect D at b. By the triangle inequality, Ilwull < Ilwbll + Ilbull < Ilwbll + Ilbcll +
Ilcull = Ilwbll + tlbcll + Ilcpll = Ilwpll. Thus every one of the eight regions of the
diagonal partition has the uniqueness property. []
Corollary 7. The maximum possible degree of an M S T over any finite pointset in the
rectilinearplane is equal to 8; that is, v(2, 1) = 8.
Proof. By L e m m a 6 the diagonal partition has the uniqueness property, which
implies that /(2,1) < 8. The pointset {(0, 0), ( + 1, 0), (0, + 1), ( + 89 + 1)} shows that
this bound is tight. Lemmas 2 and 3 imply that the maximum MST degree in the
rectilinear plane is equal to 8. []
We now show an analogous result for threedimensional rectilinear space.
Consider a cuboctahedral partition of ~ into 14 disjoint regions corresponding to the
faces of a truncated cube (Fig. 2(a), (b)), i.e., six congruent pyramids with square
cross section (Fig. 2(c)) and eight congruent pyramids with triangular cross section
(Fig. 2(d)). Most of the region boundaries are included into the triangular pyramid
regions as shown in Fig. 3(c), with the remaining boundaries forming four new
regions (Fig. 3(d)), to a total of 18 regions. Following the same strategy as in the
twodimensional case, we first show that the uniqueness property holds.
L e m m a 8. Given a point p in threedimensional rectilinear space, each region of the
cuboctahedral partition with respect to p has the uniqueness property.
Proof. We need to show that, for any two points u and w that lie in the same
region of the cuboctahedral partition, Ilwull < max(llwpll, Ilupll). This is obvious for
the twodimensional regions that are the boundaries between the pyramids (by an
argument analogous to that of L e m m a 6).
Consider one of the square pyramids R with respect to p (Fig. 2(c)), and let
u, w ~ R. Assume without loss of generality that Ilupll  Ilwpll (otherwise swap the
roles of u and w). Consider the locus of points D c R that are distance Ilupll from p
(Fig. 2(e)); D is the upper half of the boundary of an octahedron. Let c be the
center of the octahedron determined by D, so that c is equidistant from all points of
D. Let b be the intersection of the surface of D with a ray starting from p and
, D
........ L:::. d:""
"'..
Fig. 2. A t r u n c a t e d c u b e (a), (b) i n d u c e s a t h r e e  d i m e n s i o n a l c u b o c t a h e d r a l p a r t i t i o n o f space into
14 regions: six s q u a r e p y r a m i d s (c), a n d e i g h t t r i a n g u l a r p y r a m i d s (d). U s i n g t h e triangle inequality,
e a c h r e g i o n m a y b e s h o w n to c o n t a i n at m o s t o n e c a n d i d a t e p o i n t f o r c o n n e c t i o n w i t h t h e origin in
a n M S T (e), (19.
?
9"*'"
" R 2
 /
9. . . . . . . . . . . . ......~.,,."'.. 5"""
...'7
:: R 1
i/
" '  . R 4 f
. . . . . . . . . . . . . . . . . .
"i
':
QO~BQe~ele.
R3 . . . . . . . . . . .
,
"'~ ............
~ "
(c)
9
"'...o
.....o'"
..
i /"" . . . . . . . . . . . . . . . . :"",, / i
(b)
(d)
passing through w. By the triangle inequality, Ilwull ~ Ilwbll + Ilbull < Ilwbll + Ilbcll +
Ilcull = Ilwbll + Ilbcll + Ilcpll = Ilwpll (recall that the square pyramid regions do not
contain their boundary points). Thus, w is closer to u than it is to p , which implies
that the region R has the uniqueness property.
T o show the uniqueness property for the triangular pyramids, consider one of the
triangular pyramids R with respect to p (Fig. 2(d)), and let u , w ~ R . A s s u m e
without loss of generality that Ilupll < Ilwpil (otherwise swap the roles of u and w).
Consider the locus o f points D in R that are at distance Itupll f r o m p (Fig. 2(f)). Let
b be the intersection o f D with a ray starting f r o m p and passing through w. By the
triangle inequality, Ilwull < llwbll + Ilbu[I < Ilwbll + Ilbpll = Ilwpll (reeall that each
triangular region is missing one of its b o u n d a r y faces, as shown in Fig. 3(a), (b)). Thus,
w is closer to u than it is to p , which implies that region R has the uniqueness
property. Thus every one o f the 18 regions of the cuboctahedral partition has the
uniqueness property. [ ]
G. Robins and J. S. Satowe
Corollary 9. The maximum possible degree of an MST over any finite pointset in
threedimensional rectilinear space is equal to 18; that is, v(3,1) = 18.
Proof. By L e m m a 8 the cuboctahedral partition has the uniqueness property, which
implies that I(3, 1) < 18. The pointset {(0, 0, 0), ( + 1, 0, 0), (0, + 1, 0), (0, 0, +
1), ( + 89 + 89 0), (0, + 89 _+ 89 ( + ~,0,~ + 89 shows that this bound is tight. Lemmas
2 and 3 imply that the maximum MST degree in threedimensional rectilinear space
is 18. []
We can further refine the maximum MST degree bound of Corollary 7 by
applying the perturbative argument of T h e o r e m 4.
Theorem 10. The maximum degree of a rninimumdegree MST over any finite pointset
in the rectilinear plane is 4; that is, ~(2, 1) = 4.
Proof. The pointset {(0, 0), ( + 1, 0), (0, + 1)} establishes a lower bound of 4. To get
the upper bound of 4, consider/~(2, 1), the n u m b e r of points that can be placed on
the boundary of a unit ball (i.e., a diamond) in 9t2 such that each pair of points is
strictly greater than one unit apart. Consider the diagonal partition, as in the proof
of L e m m a 6; at most one point can be in the closure of each of the four
twodimensional regions, proving the result. []
T h e o r e m 10 has an interesting consequence on the complexity of the B D M S T
problem restricted to the rectilinear plane:
Instance. A planar pointset P = {x1. . . . . xn}, and integers D and C.
Question. Is there a rectilinear spanning tree with maximum degree < D and cost
< C?
If D = 4, the question can be decided in polynomial time. In fact, our methods
establish that such a boundeddegree MST can be computed as efficiently as an
ordinary MST. On the other hand, if D = 2, the problem is essentially a rectilinear
traveling salesman problem (a wandering salesman problem, since the tour is a path
rather than a circuit), and it is therefore NPcomplete. It turns out that the D = 3
question is also NPcomplete, using a proof identical to the one appearing in [
17
]
(this result is for the corresponding Euclidean problem, but since the construction is
restricted to a special type of grid graph, the proof holds in the rectilinear metric as
well). We summarize the rectilinear and Euclidean results in Table 1.
Next, we refine the maximum MST degree bound of Corollary 9.
Theorem 11. The maximum degree of a minimumdegree M S T over any finite pointset
in threedimensional rectilinear space is either 13 or 14; that is, 13 < b(3, 1) < 14.
Proof. For the lower bound, the following pointset shows that the maximum degree
of an MST is I ( k , p ) > 13: { ( 0 , 0 , 0 ) , ( + 1 0 0 , 0 , 0 ) , ( 0 , + 100,0),(0,0, + 100),
(47,  4, 49), (  6,  49, 45), (  49, 8, 43), (  4, 47,  49), (  49,  6,  45), (8,  49,
 4 3 ) , (49, 49, 2)}, since, for this pointset, all nonorigin points are strictly closer to
the origin than they are to each other, forcing the MST to be unique with a star
topology.
To obtain the upper bound of 14 on the maximum MST degree, consider the
cuboctahedral partition. Any two points lying in the closure of one of the 14 main
regions of the cuboctahedral partition must be within distance 1 of each other. []
We note that there is an elementary means to settle the 13 versus 14 question
raised in Theorem 11. Suppose we are trying to decide whether 14 points can be
placed on the surface of a unit octahedron so that each pair is greater than a unit
distance apart. The relationship between point (xi, Yi, zi) and point (x j, yj, zj) can be
phrased by the inequality Ixi  xj[ + lYi  Yj[ + Izi  Zfl > 1 subject to the
constraints Ixil + lYil + Izil = 1 and [xjl + lyyl + Izjl = 1. The absolute values can be
removed if the relative order between x 1 and x2, etc., is known. We can therefore
consider all permutations of the coordinates of the 14 points and produce the
corresponding inequalities. If the inequalities corresponding to a particular
permutation are simultaneously satisfied, ~(3,1) = 14, otherwise ~(3, 1) = 13. Feasibility can
be settled by determining whether a particular polytope contains a nonempty
relative interior (this approach is easily extended to any dimension k). We have not
settled the 13 versus 14 question, and the above procedure seems impractical due to
the large number of resulting inequalities.
M o n m a and Suri [
16
] used a similar perturbation argument to prove that for any
pointset in the Euclidean plane, there is an MST with maximum degree of 5. Cieslik
[
3
] has bounded the vertex degrees of Steiner minimal trees in Minkowski planes
(note, however, that we bound the minimum spanning tree degree, not the Steiner
minimal tree degree as Cieslik does, so the results of the two works are
complementary). We now address the Hadwiger and MST numbers for the kcrosspolytope.
Theorem 12. The maximum degree of a minimumdegree M S T over any finite pointset
in fft~ is at least ~'~(20"0312k); that is, ~(k, 1) = ~~(20"0312k).
Proof. Consider the family F ( j ) of points ( + l / j . . . . . + 1/j,O . . . . . 0), where j is an
integer between 1 and k. (Here, the j nonzero terms can be arbitrarily interspersed
in the vector.)
Each member of F ( j ) is distance 1 from the origin; the distance between
x ~ F ( j ) and y E F ( j ) depends on the positions and signs of the nonzero terms.
Given x = (Xl, x 2. . . . . x k) ~ F ( j ) , let ~ be the binary vector containing a one in bit
i if x / § 0 and a zero in bit i if x i = 0. If the Hamming distance between ~ and y is
at least j, then Ilxyll > 1. (The Hamming distance between two bit vectors is the
number of bit positions in which they differ.) We want to find a large set of ~ that
are mutually H a m m i n g distance greater than j apart.
Consider the set V(j) of bit vectors containing exactly j ones;
Form a graph G(t) = (V(t), E(t)) for which (~, y) E E(t) if and only if the H a m
ming distance between ~ and y is at most j. Note that G(t) is regular with degree
lj/2]
i=1 i
( k
To see this, we determine the number of edges adjacent to ~ = (1 . . . . . 1, 0. . . . . 0),
where there are j ones. The set of vectors in V(j) adjacent to ~ can be partitioned
into vectors that contain i zeros in the first j positions, 1 < i < [ j / 2 ] + 1. For a
giveni, t h e r e a r e ( J i ) w a y s t o c h o o s e t h e z e r o p o s i t i o n s a n d ( k ~ t J ) p o s i t i o n s t o
place the displaced ones in the last k  j positions.
Here is our strategy to find a subset of V(j) of large cardinality that are mutually
far apart: choose a vertex, delete its neighbors, and continue. The number of vertices
chosen must exceed IV(j)l/d(j). Suppose that cj = 16v~j = k. Then
,)
d(j)
( j / 2 + 1) j / 2
j / 2
( j / 2 + 1 ) ( 4 J / f  ~ ' ) ( k . ; 2 j )
>C t
Cj / 2
4 ' f f
[ C ~k/2c 1
Ctr
[ ~ )
"~ "
( ' )
Here, c' and c" are constants; the approximation to j / 2
for c gives the result.
is from [
9
]. Substituting
4. The Maximum Lp MST Degree
In this section we provide bounds on M(k, p) for general Lp metrics.
Theorem 13. The maximum degree of a minimumdegree M S T over any finite pointset
in ~Rk is at least b(k, p ) = l~(v~2 n~le~))), where ot = 1/2 p and E(x) = x lg(1/x) +
(1  x ) l g ( 1 / ( 1  x)).
Proof. Consider the vertices of the khypercube ( + 1. . . . . + 1). Each of these
points is k 1/p from the origin. On the other hand, if points x and y differ from each
i < a k
2 k
d U )
other in j positions, they are distance 2fl/p from each other. If 2fl/p > k 1/1', then x
and y are further from each other than they are from the origin.
We need to find the largest cardinality set of points on the khypercube that
differ in at least J = k / 2 p positions. To do this, construct a graph G whose vertex
set is the set of binary strings of length k, and for which there is an edge between
string a and string b if and only if the Hamming distance between a and b is at
most J. Proceed in the same manner as in the proof of T h e o r e m 12, except that
d ( J ) = ~=~=1
"
The number of vertices chosen must exceed 2k/d(J). Now,
f o r 0 < a < ~
1 (see Chapter 9, Problem 42, of [
9
]). Note that t~ = 1 / 2 p, so
= V~2k(1E(2P))
[]
T h e o r e m 13 shows that, for any fixed p > 1, ~(k, p ) grows exponentially in the
dimension. Note that this bound is less than the bound obtained by Wyner [
23
] (for
H ( k , 2), it is [~(2 0"189k) since E(88 = z3 lg3  1 = 0.189), but it is sufficient for our
purposes.
It is well known that H ( k , p ) _< 3 k  1 (e.g., [
22
]). In twodimensional space the
Hadwiger number is largest for L 1 and L~, the only planar Lp m e t r i c s with
Hadwiger number 8. For all other Lp metrics, the Hadwiger number is 6. On the
other hand, the planar MST number is smallest for L 1 and L~, having a value of 4,
and it is easily seen to be 5 for all other Lp metrics.
These observations raise an interesting question: how does the MST number
behave as a function of p ? Note that the maximum Hadwiger number is achieved by
parallelotopes. Next we derive the MST number for the L~ unit ball (i.e., the
khypercube), and show that the MST number is not maximized in the L~ metric in
any dimension.
Theorem 14. The maximum degree of a minimumdegree M S T over any finite pointset
in ~ is 2k; that is, ~,(k, oo) = 2 k.
Proof. We first show the result for p = oo; note that the Lp unit ball is a
khypercube. The upper bound is established by considering [(k, p), the number of
points that can be placed on the boundary of a unit ball in fitk such that each pair of
points is strictly greater than one unit apart. Note that at most one point can be
placed in each kant (the kdimensional analogue of "quadrant"). The lower bound
is established by considering the set of 2 k vertices of a khypercube. []
Theorem 15. For each k, there is a p such that the maximum degree of a
minimumdegree M S T over any finite pointset in ~ k space exceeds 2k; that is, for all k there exists
a p such that f~(k, p) > 2 k.
Proof. Consider the pointset (  1 , + 1. . . . . + 1), (e, + /~. . . . . + 8), and
(k 1/p, 0. . . . . 0), where (6 p + (k  1)8 p) = k. It is possible to choose E, /~, and p so
that each pair of points is on the surface of an Lp ball of radius k l/p, and all
interdistances are greater than k 1/p. []
5. Conclusion
Motivated by practical VLSI applications, we showed that the maximum possible
vertex degree in an Lp MST equals the Hadwiger number of the corresponding unit
ball, and we determined the maximum vertex degree in a minimumdegree Lp MST.
We gave an exponential lower bound on the MST number of a kcrosspolytope, and
showed that the MST number for an Lp unit ball, p > 1, is exponential in the
dimension. We concentrated on the L 1 metric in two and three dimensions due to
its significance for VLSI: for example, we showed that for any finite pointset in the
rectilinear plane there exists an MST with maximum degree of at most 4, and that
for threedimensional rectilinear space the maximum possible degree of a
minimumdegree MST is either 13 or 14.
We solved an open problem regarding the complexity of computing
boundeddegree MSTs by providing the first known polynomialtime algorithm for
constructing an MST with maximum degree 4 for an arbitrary pointset in the rectilinear
plane. Moreover, our techniques can be used to compute a boundeddegree MST as
efficiently as an ordinary MST. Finally, our results also enable a significant execution
speedup of a number o f c o m m o n VLSI routing algorithms. Remaining open
problems include:
1. Whether the MST number for L 1 in three dimensions is 13 or 14.
2. The complexity of computing a planar Euclidean MST with maximum
degree 4.
3. Tighter bounds on the Hadwiger and MST numbers for arbitrary k and p.
Acknowledgments References
We would like to thank the referees for their helpful comments.
1. T. Barrera , J. Griffith , S. A. McKee , G. Robins , and T. Zhang , Toward a Steiner Engine: Enhanced Serial and Parallel Implementations of the Iterated 1  Steiner Algorithm , Proc. Great Lakes Syrup. VLSI , Kalamazoo,MI, March 1993 ,pp. 90  94 .
2. T. Barrera , J. Griffith , G. Robins, and T. Zhang , Narrowing the Gap: NearOptimal Steiner Trees in Polynomial Time , Proc. IEEE Internat. ASIC Conf ., Rochester, NY , September 1993 , pp. 87  90 .
3. D. Cieslik , The Vertex Degree of Steiner Minimal Trees in Minkowski Planes, in Topics in Combinatorics and Graph Theory , R. Bodendieck and R. Henn (eds.), PhysicsVerlag, Heidelberg, 1990 , pp. 201  206 .
4. T. H. Cormen , C. E. Leiserson , and R. Rivest , Introduction to Algorithms, MIT Press, Cambridge, MA, 1990 .
5. J. T. Croft , K. J. Falconer , and R. K. Guy , Unsolved Problems in Geometry, SpringerVerlag, New York, 1991 .
6. W. H. E. Day and H. Edelsbrunner , Efficient Algorithms for Agglomerative Hierarchical Clustering Methods , J. Classification, 1 ( 1984 ), 1  24 .
7. M. R. Garey and D. S. Johnson , Computers and Intractability: a Guide to the Theory of NP Completeness , Freeman, San Francisco, CA, 1979 .
8. G. Georgakopoulos and C. H. Papadimitriou , The 1 Steiner Tree Problem , J. Algorithms , 8 ( 1987 ), 122  130 .
9. R. L. Graham and P. Hell , On the History of the Minimum Spanning Tree Problem , Ann. of History Comput., 7 ( 1985 ), 43  57 .
10. L. J. Guibas and J. Stolfi , On Computing all NorthEast Nearest Neighbors in the L1 Metric, Inform . Process. Leg., 17 ( 1983 ), 219  223 .
11. A. C. Harter , ThreeDimensional Integrated Circuit Layout , Cambridge University Press, New York, 1991 .
12. J.M. Ho , G. Vijayan, and C. K. Wong , New Algorithms for the Rectilinear Steiner Tree Problem , IEEE Trans. ComputerAided Design , 9 ( 1990 ), 185  193 .
13. G.A. Kabatjansky and V. Levengtein , Bounds for Packings of the Sphere and in Space, Problems Inform . Transmission, 14 ( 1978 ), 1  17 .
14. A. B. Kahng and G. Robins , A New Class of Iterative Steiner Tree Heuristics with Good Performance, IEEE Trans . ComputerAided Design, 11 ( 1992 ), 893  902 .
15. M. Kruskal , On the Shortest Spanning Subtree of a Graph, and the Traveling Salesman Problem , Proc. Amer. Math. Soc. , 7 ( 1956 ), 48  50 .
16. C. Monma and S. Suri , Transitions in Geometric Minimum Spanning Trees, Discrete Comput. Geom. , 8 ( 1992 ), 265  293 . 9
17. C. H. Papadimitriou and U. V. Vazirani , On Two Geometric Problems Relating to the Traveling Salesman Problem , J. Algorithms , 5 ( 1984 ), 231  246 .
18. B. T. Preas and M. J. Lorenzetti , Physical Design Automation of VLSI Systems , Benjamin/Cummings, Menlo Park, CA, 1988 .
19. A. Prim , Shortest Connecting Networks and Some Generalizations, Bell Systems Tech. J. , 36 ( 1957 ), 1389  1401 .
20. G. Robins and J. S. Salowe , On the Maximum Degree of Minimum Spanning Trees , Proc. A C M Symp. Computational Geometry , Stony Brook, NY , June 1994 , pp. 250  258 .
21. J. S. Salowe and D. M. Warme , An Exact Rectilinear Steiner Tree Algorithm , Proc. IEEE Internat. Conf. Computer Design , Cambridge, MA, October 1993 , pp. 472  475 .
22. G. F. T6th , New Results in the Theory of Packing and Covering , Convexity and its Applications 1983 .
23. A. D. Wyner , Capabilities of Bounded Discrepancy Decoding, A T & T Tech . J., 44 ( 1965 ), 1061  1122 .