Given a set $P$ of terminals in the plane and a partition of $P$ into $k$
subsets $P_1, ..., P_k$, a two-level rectilinear Steiner tree consists of a
rectilinear Steiner tree $T_i$ connecting the terminals in each set $P_i$
($i=1,...,k$) and a top-level tree $T_{top}$ connecting the trees $T_1, ...,
T_k$. The goal is to minimize the total length of all trees. This problem
arises naturally in the design of low-power physical implementations of parity
functions on a computer chip.
  For bounded $k$ we present a polynomial time approximation scheme (PTAS) that
is based on Arora's PTAS for rectilinear Steiner trees after lifting each
partition into an extra dimension. For the general case we propose an algorithm
that predetermines a connection point for each $T_i$ and $T_{top}$
($i=1,...,k$).
  Then, we apply any approximation algorithm for minimum rectilinear Steiner
trees in the plane to compute each $T_i$ and $T_{top}$ independently.
  This gives us a $2.37$-factor approximation with a running time of
$\mathcal{O}(|P|\log|P|)$ suitable for fast practical computations. The
approximation factor reduces to $1.63$ by applying Arora's approximation scheme
in the plane

Held, Stephan

Kämmerling, Nicolas

English

arXiv

Given a set P of terminals in the plane and a partition of P into k subsets P 1 , . . . , P k , a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree T i connecting the terminals in each set P i (i = 1, . . . , k) and a top-level tree T top connecting the trees T 1 , . . . , T k . The goal is to minimize the total length of all trees. This problem arises naturally in the design of lowpower physical implementations of parity functions on a computer chip. For bounded k we present a polynomial time approximation scheme (PTAS) that is based on Arora&apos;s PTAS for rectilinear Steiner trees after lifting each partition into an extra dimension. For the general case we propose an algorithm that predetermines a connection point for each T i and T top (i = 1, . . . , k). Then, we apply any approximation algorithm for minimum rectilinear Steiner trees in the plane to compute each T i and T top independently. This gives us a 2.37-factor approximation with a running time of O(|P | log |P |) suitable for fast practical computations. The approximation factor reduces to 1.63 by applying Arora&apos;s approximation scheme in the plane

Stephan Held

Nicolas Kämmerling

CiteSeerX

EuroCG 2015, Ljubljana, Slovenia, March 16–18, 2015
Two-Level Rectilinear Steiner Trees
Stephan Held∗ Nicolas Kämmerling∗
Abstract
Given a set P of terminals in the plane and a partition
of P into k subsets P1, . . . , Pk, a two-level rectilinear
Steiner tree consists of a rectilinear Steiner tree Ti con-
necting the terminals in each set Pi (i = 1, . . . , k) and
a top-level tree Ttop connecting the trees T1, . . . , Tk.
The goal is to minimize the total length of all trees.
This problem arises naturally in the design of low-
power physical implementations of parity functions on
a computer chip.
For bounded k we present a polynomial time ap-
proximation scheme (PTAS) that is based on Arora’s
PTAS for rectilinear Steiner trees after lifting each
partition into an extra dimension.
For the general case we propose an algorithm that
predetermines a connection point for each Ti and Ttop
(i = 1, . . . , k). Then, we apply any approximation
algorithm for minimum rectilinear Steiner trees in the
plane to compute each Ti and Ttop independently.
This gives us a 2.37-factor approximation with a
running time of O(|P | log |P |) suitable for fast practi-
cal computations. The approximation factor reduces
to 1.63 by applying Arora’s approximation scheme in
the plane.
1 Introduction
We consider the two-level rectilinear Steiner tree prob-
lem (R2STP) that arises from an application in VLSI
design. Consider the computation of a parity func-
tion of k input bits using 2-input XOR-gates. Due to
the symmetry, associativity, and commutativity of the
XOR function, this can be realized by an arbitrary
binary tree with k leaves, rooted at the output sim-
ply by inserting an XOR-gate at every internal vertex
[11, 12]. Throughout this paper we consider the parity
function as a placeholder for any fan-in function of
the type x1 ◦ x2 ◦ · · · ◦ xk, where ◦ is a symmetric,
associative, and commutative 2-input operator, i. e.
◦ ∈ {⊕,∨,∧}.
On a chip such a tree has to be embedded into
the plane and all connections must be realized by
rectilinear segments. If each input and the output are
single points on the chip, a realization of minimum
length and thus power consumption is given by a
∗Research Institute for Discrete Mathematics, University of
Bonn,
held@or.uni-bonn.de, kaemmerling@or.uni-bonn.de.
p1
p2
p3
p1
p2
p′1
p′2
p3
Figure 1: On the left, we have two inputs p1 and
p2 and a single output p3. The XOR-gate should be
placed at the median of the three terminals. If the
inputs have the side outputs p′1 and p
′
2, the XOR-gate
should be placed at p3, saving the horizontal length.
minimum length rectilinear Steiner tree. This is a tree
connecting the inputs and the output by horizontal
and vertical line segments using additional so-called
Steiner vertices to achieve a shorter length than a
minimum spanning tree. At each Steiner vertex of
degree three an XOR-gate is placed. Higher degree
vertices can be dissolved into degree three vertices
sharing their position. Figure 1 shows an example of
an embedded parity function on the left.
In practice input signals may be needed for other
computations on the chip and thus delivered to other
side outputs. Similarly, the result may have to be
delivered to multiple output terminals. Thus, each
input and its successors and the output terminals must
be connected by separate Steiner trees as well. These
trees are then connected by a top-level Steiner tree into
which the XOR-gates will be inserted. Considering the
additional terminals allows to construct a potentially
shorter Steiner tree as shown in Figure 1 on the right.
Algorithms ignoring the side outputs cannot guarantee
an approximation factor better than two, as we will
see in Section 2.
This motivates the definition of the minimum two-
level rectilinear Steiner tree problem, where we are
given a set P ⊂ R2 of n terminals and a partition of
P into k subsets P1, . . . , Pk.
A two-level rectilinear Steiner tree T =
(Ttop, T1, . . . , Tk) consists of a Steiner tree Ti
for each i ∈ {1, . . . , k} connecting the terminals
in Pi and a (group) Steiner tree Ttop connecting
the embedded trees {T1, . . . , Tk}. We call Ttop the
top-level tree. Note that all trees are allowed to cross.
The objective is to minimize the total length of all
trees
l(T ) :=
k∑
i=1
l(Ti) + l(Ttop),
where l(T ′) :=
∑
{x,y}∈E(T ′) ‖x− y‖1 is the `1-length
This is an extended abstract of a presentation given at EuroCG 2015. It has been made public for the benefit of the community and should be considered a
preprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.
31st European Workshop on Computational Geometry, 2015
of a Steiner tree T ′.
For each i ∈ {1, . . . , k} the top-level tree and Ti
intersect in at least one point. We can select one such
point qi ∈ Ttop ∩ Ti and call it connection point for Ti
and Ttop. Then Ttop is a Steiner tree for the terminals
{q1, . . . , qk} and each Ti is a Steiner tree for Pi ∪ {qi}.
Obviously, this problem is NP-hard as it contains
the minimum rectilinear Steiner tree problem in two
ways: if k = 1 or if |Pi| = 1 for i ∈ {1, . . . , k}.
Designing the top-level tree as a stand-alone problem
is hard. If all subtrees Ti (i ∈ {1, . . . , k}) are fixed,
Ttop cannot be approximated to arbitrary quality, as
the group Steiner tree problem for connected groups
in the Euclidean plane cannot be approximated within
a factor of (2− ε) [9]. However we are in a more lucky
situation as we can tradeoff the lengths of bottom-level
and top-level trees.
To the best of our knowledge the two-level rectilinear
Steiner tree problem has not been considered before
despite its practical importance [11, 12]. It is loosely
related to the hierarchical network design problems
[1, 5, 6] or multi-level facility location problems [3, 4].
However, those problems are structurally different,
typically considering problems in graphs, and do not
apply to our case.
In [11], ordinary rectilinear Steiner trees were used
to build power efficient fan-in trees, when each input
and the output consists of a single terminal. In prac-
tice designers are also interested in the depth of the
constructed circuit [12]. However, for finding good
power versus depth tradeoffs a better understanding
of short solutions is an essential prerequisite and the
aim of our work.
1.1 Our Contribution
In Section 2 we show that the näıve approach of picking
a random terminal from each partition as a connection
point to the top-level tree and building the bottom-
level trees and top-level as separate instances gives as a
2α-factor approximation, where α is the approximation
factor of the used minimum Steiner tree algorithm.
Then in Section 3 we show how to lift our in-
stance into an equivalent (2+k)-dimensional rectilinear
Steiner tree instance. If the number k of partitions is
bounded by a constant, we obtain a PTAS by applying
Arora’s PTAS for rectilinear Steiner trees [2].
As our main result we improve the approximation
guarantee for unbounded k from (2 + ε) to 1.63 in
Section 4. Using spanning tree heuristics this approach
turns also into a fast practical algorithm with running
time O(n log n) and approximation factor 2.37.
2 Simple Bottom-Up Construction
A simple bottom-up approach, which works for any
metric space, is to compute a Steiner tree Ti for Pi
T1T2
Ttop
T1T2
Ttop
q1q2
Figure 2: A tight example when choosing connection
points as arbitary points of Pi.
(i = 1, . . . , k). In each Ti we fix a connection point
qi ∈ Pi arbitrarly, compute a Steiner tree Ttop for
{q1, . . . , qk}, and return T = (Ttop, T1, . . . , Tk).
Theorem 1 The simple bottom-up approach is a 2α-
factor approximation algorithm for the minimum two-
level Steiner tree problem, if we use an α-factor ap-
proximation algorithm for the minimum Steiner tree
problem as a subroutine.
Proof. Let T be the two level Steiner tree computed
by the simple bottom-up approach and let T ? =
(T ?top, T
?
1 , . . . , T
?
k ) be a minimum two-level Steiner tree.
Let be q?i ∈ T ?top ∩ T ?i the connection point of the
optimum two-level Steiner tree. Since T ?i is a Steiner
tree on {q?i } ∪ Pi, we have dist(q?i , qi) ≤ l(T ?i ). Thus,
l(T ) ≤ α · l(T ?top) + α
k∑
i=1
dist(q?i , qi) +
k∑
i=1
α · l(T ?i )
≤ α · l(T ?top) + 2α
k∑
i=1
l(T ?i ) ≤ 2α · l(T ?).

Figure 2 shows that the factor (2 + ε) is sharp. For
the instance P1 = {(0, 0), (1, 0)}, P2 = {(0, 0), (−1, 0)}
a minimum two-level rectilinear Steiner tree of length
2 is shown on the left with l(Ttop) = 0. On the right, a
bad choice of connection points and minimum Steiner
trees Ttop, T1, and T3 yield a total length of 4.
3 PTAS for a bounded number of partitions
We can reduce the two-level rectilinear Steiner tree
problem in the plane to an ordinary rectilinear Steiner
tree problem in a higher dimensional space, where we
can apply Arora’s PTAS [2].
The idea of the PTAS is to lift every subset
P1, . . . , Pk to an additional dimension. We assume
k > 1. Otherwise the two-level Steiner tree problem is
an ordinary Steiner tree problem. Let P1, . . . , Pk ⊂ R2
be the subsets of a two-level Steiner tree instance, we
define a Steiner tree instance in R2+k. The set of
terminals P ′ is comprised as follows.
For each original terminal x ∈ Pi ⊂ R2 (i ∈
{1, . . . , k}), we add a terminal x′ := (x,K ·ei) ∈ R2+k,
where ei ∈ Rk is the unit vector with value one at the i-
th coordinate and K is a large constant, e. g. we could
choose K as l(B(P )). Now for x ∈ Ph and y ∈ Pi the
distance of their high dimensional copies x′, y′ ∈ P ′ is
‖x′−y′‖1 = ‖x−y‖1+2K‖eh−ei‖1 = ‖x−y‖1+2Kδh,i,
EuroCG 2015, Ljubljana, Slovenia, March 16–18, 2015
T2
T1
Ttop
Ttop-space
T1-space
T2-space
Figure 3: A flat Steiner Tree in a lifted instance.
where δh,i is one if h = i and zero otherwise. An ex-
ample of a lifted two-level Steiner tree is given in
Figure 3.
A (k + 2)-dimensional Steiner tree is called flat if
all Steiner points have either the form (x, 0) ∈ R2+k
or (x,K · ei) ∈ R2+k, where x ∈ R2 and ei ∈ Rk is a
unit vector. The following Lemma has essentially been
proven by Snyder [10], who shows that an optimum
Steiner tree can be found in the d-dimensional Hanan
grid [7].
Lemma 2 A (k + 2)-dimensional Steiner Tree T for
P ′ of length l(T ) can be transformed in strongly poly-
nomial time into a (k+ 2)-dimensional flat Steiner tree
T ′ of length at most l(T ).
The following lemma shows the equivalence between
the original two-level rectilinear Steiner tree problem
in the plane and the lifted regular rectilinear Steiner
tree problem.
Lemma 3 If k > 1, a two-level rectilinear Steiner tree
T for P1, . . . , Pk of length l(T ) can be transformed into
a (k + 2)-dimensional Steiner tree T ′ for P ′ of length
at most l(T ) + kK and vice versa.
Proof. We only sketch the proof, details can be found
in [8]. We get T ′ by connecting the lifted components
of T at its connection points by an edge of length
K. Conversely, we flat T by applying Lemma 2 and
replacing edges, s. t. for each j ∈ {3, . . . , k + 2},
T ′ contains at most one edge in direction j, without
increasing the length of the tree. Projecting the edges
onto the first two coordinates we obtain a feasible
two-level Steiner tree of length at most l(T ′)− kK.

Theorem 4 For bounded k there is a PTAS for the
two-level rectilinear Steiner tree problem.
Proof. Choose K = l(B(P )) and for ε > 0 set
ε′ := 1k+1ε. Then compute an (1 + ε
′)-approximate
(k+ 2)-dimensional Steiner tree T ′ for the lifted termi-
nal set P ′ with Aroras PTAS [2] that has a poly-
nomial running in bounded dimension. Then we
apply Lemma 3 to obtain a two-level Steiner tree
T = (Ttop, T1, . . . , Tk) for P1, . . . , Pk with length at
most l(T ′)− kK. Let T ′? and T ? be optimum Steiner
trees for P ′ and P . Since l(T ?) ≥ K the length of T
is
l(T ) ≤ l(T ′)− kK ≤ (1 + ε′)l(T ′?)− kK
≤ (1 + ε′)l(T ?) + (1 + ε′)kK − kK
≤ (1 + (k + 1)ε′)l(T ?) = (1 + ε)l(T ?).

4 Predetermined Connection Points
In all algorithms of this section we predetermine a
connection point qi for each set Pi (i = 1, . . . , k) and
then call a Steiner tree approximation algorithm for
{q1, . . . , qk} to get Ttop and Pi ∪ {qi} to get Ti (i =
1 . . . , k). We use the fact that we consider rectilinear
instances to obtain better approximation factors than
in Section 2.
4.1 Bounding Box Center
A natural approach is to choose each connection point
as the center of the bounding box B(Pi).
Theorem 5 Using bounding box centers as connec-
tion points, we get a 1.75α-factor approximation algo-
rithm for the two-level rectilinear Steiner tree problem,
when using an α-factor approximation algorithm for
rectilinear Steiner trees as a subroutine.
A detailed analysis can be found in [8], which also
provides a simple example attaining this factor.
4.2 Adjusted Bounding Box Center
We can improve the approximation factor by a more
careful choice of the connection point. For a set Pi
(i ∈ {1, . . . , k}), we call the coordinate system with
origin in the central point of its bounding box the
coordinate system of Pi.
If a set Pi of terminals contains an element in each
quadrant of its bounding box B(Pi), we call the bound-
ing box B(Pi) complete. For subtrees with a complete
bounding box we choose the connection point to the
top-level tree as the central point of the bounding
box as in Section 4.1 and compute a Steiner tree T ′i
for Pi ∪ {qi} as follows: We compute a Steiner tree
T ′i for Pi. Thereby, we embed maximal paths in T
′
i
containing only Steiner vertices with degree two so
that each such path has minimum distance to qi while
preserving its length. We then add an edge from qi to
ai, where ai is a point in T
′
i minimizing the distance
to qi.
Let T ? = {T ?top, T ?1 , . . . , T ?k } be an minimum two-
level Steiner tree. Again we choose T ?top under all
minimum two-level Steiner trees as large as possible so
that there is a connection point q?i ∈ T ?top∩T ?i ⊆ B(Pi).
We present the main ingredient for the analysis of
the approximation ratio.
31st European Workshop on Computational Geometry, 2015
B(Pi)
u v
w
p
p′
qi
Figure 4: An example of the situation in the proof
of Lemma 6. The green diamond is the l1-circle with
radius h around qi.
Lemma 6 Let i ∈ {1, . . . , k} and T ′i , ai be con-
structed as above. If B(Pi) is complete, then l(B(Pi))+
‖qi − ai‖1 ≤ l(T ?i ).
Proof. Define h := ‖qi − ai‖1. Since B(Pi) is com-
plete, T ′i intersects at least three of the four axes of
the coordinate system to Pi. We assume w.l.o.g. that
T ′i intersects the left, upper and right axis.
By the choice of T ′i and h there exist (see also Fig-
ure 4) for all z ∈ [0, h]
p ∈ {(x, y) ∈ Pi : x ≤ −h, y ≤ 0},
p′ ∈ {(x, y) ∈ Pi : x ≥ h, y ≤ 0},
u ∈ {(x, y) ∈ Pi : x < 0, y ≥ h},
vz ∈ Vz := {(x, y) ∈ Pi : x ≥ z, y ≥ h− z}.
Let vh ∈ Vh. If the unique T ?i -paths from p to
u and from p′ to vh intersect, then T
?
i connects the
lines {(x, y) : x = 0} and {(x, y) : x = h} twice, and
therefore l(B(Pi)) + h ≤ l(T ?i ).
Otherwise, we can choose a minimum z ≥ 0 such
that there is a v ∈ Vz and the unique T ?i -paths from
p to u and from p′ to v are disjoint. The lines {(x, y) :
y = 0} and {(x, y) : y = h− z} are connected twice in
T ?i . Therefore we get l(B(Pi)) + h− z ≤ l(T ?i ).
If z = 0 we are done. Otherwise, if our statement is
false there is an 0 < ε ≤ z such that ε = l(B(Pi))+h−
l(T ?i ) and a w ∈ Vz− ε2 . Since the unique T
?
i -paths from
p to u and from p′ to w are not disjoint, T ?i connects
the lines {(x, y) : x = 0} and {(x, y) : x = z− ε2} twice.
Therefore we get the contradiction
ε = l(B(Pi)) + h− l(T ?i )
≤ l(B(Pi)) + h− l(B(Pi))− h+ z − z + ε2 =
ε
2 .

With Lemma 6 we tradeoff between the cost ‖q?i −
qi‖1 ≤ l(B(P )) to connect qi to T ?top and the cost
‖qi − ai‖1 to connect qi to T ′i . From this we could
derive an approximation factor of 13/8 if all partitions
are complete. In general this is not the case and for
incomplete bounding boxes we shift the connection
points (and the coordinate system) towards the actual
terminals as in Figure 5. A careful analysis using
a similar version of Lemma 6 on the shifted coordi-
nate system (details can be found in [8]) gives us the
following result:
B(Pi)
qi
Figure 5: An example for an incomplete bounding
box. The connection point qi is shifted diagonally to
the upper right until the red box hits a terminal.
Theorem 7 There is a 2.37-factor approximation al-
gorithm with runtime O(n log n) and an 1.63-factor
approximation algorithm in poly-time for the two-level
rectilinear Steiner tree problem.
References
[1] E. Alvarez-Miranda, I. Ljubic, S. Raghavan, and P.
Toth: Recoverable Robust Two-Level Network De-
sign Problem. INFORMS Journal on Computing, to
appear, 2014.
[2] S. Arora: Polynomial time approximation schemes
for Euclidian Traveling Salesman and other geometric
problems. Journal of the ACM, 45(5): 753–782, 1998.
[3] M. Baou and F. Barahona. A polyhedral study of a
two level facility location model . RAIRO - Operations
Research, 48: 153–165, 2014.
[4] J. Byrka and B. Rybicki. Improved LP-Rounding
Approximation Algorithm for k-level Uncapacitated
Facility Location. ICALP ’12, 157–169, 2012.
[5] A. Balakrishnan, T. L. Magnanti, and P. Mirchandani:
A dual-based algorithm for multi-level network design.
Management Science, 40(5): 567–581, 1994.
[6] J. R. Current, C. S. ReVelle, and J. L. Cohon: The hi-
erarchical network design problem. European Journal
of Operational Research, 27(1), 57–66, 1986.
[7] M. Hanan: On Steiner’s problem with rectilinear
distance. SIAM Journal on Applied Mathematics,
14(2): 255–265, 1966.
[8] S. Held and N. Kämmerling: Two-Level Rectilinear
Steiner Trees. http://arxiv.org/abs/1501.00933,
2015.
[9] S. Safra and O. Schwartz: On the complexity of
approximating TSP with neighborhoods and related
problems. Computational Complexity, 14(4): 281–307,
2006.
[10] T.L. Snyder: On the exact location of Steiner points
in general dimension. SIAM Journal on Computing,
21(1): 163–180, 1991.
[11] H. Xiang, H. Ren, L. Trevillyan, L. Reddy, R. Puri,
and M. Cho: Logical and physical restructuring of
fan-in trees. ISPD ’10, 67–74, 2010.
[12] H. Xiang, L. Reddy, L. Trevillyan, and R. Puri: Depth
Controlled Symmetric Function Fanin Tree Restruc-
ture. ICCAD ’13, 585–591, 2013.


Two-Level Rectilinear Steiner Trees

Abstract

Similar works

Full text

Available Versions

CiteSeerX