A DVANCES IN S WITCHING N ETWORKS
D.-Z. Du and H.Q. Ngo (Eds.) pp. 307 - 357
c 2000 Kluwer Academic Publishers
Notes on the Complexity of Switching Networks
Hung Quang Ngo
Department of Computer Science and Engineering
University of Minnesota, Minneapolis, MN 55455
E-mail: hngo@cs.umn.edu
Ding-Zhu Du
Department of Computer Science and Engineering
University of Minnesota, Minneapolis, MN 55455
E-mail: dzd@cs.umn.edu
Contents
1 Introduction
308
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
1.2 Basic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
1.3 A historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
2 The complexity of checking whether a graph is a superconcentrator or concentrator
313
3 Expanders
3.1 Algebraic graph theory . . . . . . . . . . . .
3.2 The eigenvalue characterization of expansion
panders . . . . . . . . . . . . . . . . . . . .
3.3 On the second eigenvalue of a graph . . . . .
3.4 Explicit Constructions of Expanders . . . . .
. . . . . . . . .
rate for regular
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . .
strong ex. . . . . .
. . . . . .
. . . . . .
316
316
318
326
328
4 Concentrators and Superconcentrators
329
4.1 Linear Concentrators and Superconcentrators . . . . . . . . . . . . . . . 329
4.2 Superconcentrators with a given depth . . . . . . . . . . . . . . . . . . . 333
4.3 Explicit Constructions and other results . . . . . . . . . . . . . . . . . . 336
307
5 Connectors
5.1 Rearrangeable connectors . . . . . . . . . . . . . . .
5.2 Non-blocking connectors . . . . . . . . . . . . . . .
5.3 Generalized connectors and generalized concentrators
5.4 Explicit Constructions . . . . . . . . . . . . . . . .
6 Conclusions
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References
1
Introduction
1.1 Overview
There are various complexity measures for switching networks and communication networks in general. These measures include, but not limited to, the number
of switching components, the delay time of signal propagating through the network, the complexity of path selection algorithms, and the complexity of physically designing the network. This chapter surveys the study of the first measure,
and partially the second measure. It is conceivable that the number of switching
components, or the “size” of a network, affects directly the third and fourth measures.
The most common and most intuitively obvious way to study a particular type
of switching networks is to model the network by a graph. It is customary to use
a directed acyclic graph with a degsinated set of vertices called inputs and another
disjoint set of vertices called outputs. Depending on the functionality of the network under consideration, this graph has to satisfy certain conditions. Our job is
then to determine the minimum number of edges of graphs satisfying these conditions, and construct an optimal one if possible. For example, a typical functionality
of a network is rearrangeability. The corresponding graph model is the connectors, also referred to as rearrangeable networks. For different types of networks
and their desired functionalities, the reader is referred to [16, 72, 40]. This chapter
assumes that the reader is familiar with concepts such as rearrangeability, strictly
nonblockingness or wide sense nonblockingness of multistage interconnection networks.
There have been several articles and books discussing aspects of switching
network complexity or their associated graphical models. For example, the articles
by Pippenger (1990, [72]), and Bien (1989, [17]); the classic book by Beneš (1965,
[16]), a book by Hui (1990, [39]), and a recent book by Hwang (1998, [40]). We
308
shall see that this chapter’s material, although has some small overlaps with the
articles and books above, have been collected in one place for the first time. Many
open questions shall also be presented along the way.
We shall attempt to make this chapter as self-contained as possible, requiring
mostly elementary background on probability and linear algebra. This objective
will certainly affect our choice of which results to be covered. Moreover, we shall
also try to select results that are more intuitive, which might not be the best results
known on the problems under discussion. Pointers to more advanced results shall
be given as needed.
1.2 Basic models
We now give definitions of several important graphs and their components arising
in studying the complexity of switching networks, settling down the main terminologies for the rest of the chapters.
, where and are the biparti
, !"#$ , and for every
%'& % )(*,+.64 587 4,5:9 % %
%
/0 132 <;=; %
%
where /> is the set of neighbors of all vertices , !? denotes the maximum
degree of vertices in , is called the density, and is called the expansion rate
of the expander . It is a strong -expander if the above inequality holds
% @ . A family of linear expanders of density and expansion is a5
for all
sequence ABDCFEHCJG I,K , where DC is an LCMN -expander and LC,OQP , LCJR,KS+HLC,O
as TUOVP .
Definition 1.2. An XWY -network is an directed acyclic graph (DAG) with distinguished vertices called inputs and a disjoint set of W distinguished vertices
called outputs. When Z[W , we call it an -network for short. The size of a
Definition 1.1. A bipartite graph
-expander if
tions, is called an
such that
, we have
network is its number of edges. The depth of a network is the maximum length of
a path from an input to an output.
W
XWY
An
-network is our main model for a switching network with inputs
and
outputs. With respect to physical switching networks, the vertices of our
DAG represent links between switching components of the underlying physical
networks, and there is an edge between two links if these two links are an input
and an output link of some switch. Moreover, the depth of the DAG implicitly
represent the delay of signal propagating from inputs to outputs.
309
A classical objective of studying switching networks of various types was to
design these networks with as few switching components as possible. The main
hope was that reducing this number also reduces several other complexity measures. Hence, one of our objectives is to find the minimum size
-network
(or -network for that matter) satisfying certain properties, and construct one with
as small a size as possible. We shall see that this objective is usually a trade-off
with another objective which is to have a network with small depth, which presumably has small signal propagation delay. Determining this trade-off in some precise
manner is thus another research topic.
XWY
XWY
] W
XWY
W
X,
_
LCM3
\2*W
Definition 1.3. An
-concentrator is an
-network where
, such
that for any subset of
inputs there exists a set of
vertex disjoint paths
-concentrator. An
connecting to the outputs. An -concentrator is an
-concentrator is an
-concentrator with at most
edges. An
concentrator is an
-concentrator. A family of linear concentrators of density is a sequence
such that each
is an
-concentrator, where
as
.
]
XW^)
XWY
)
X)
AB`CFE CJG I,K
DC
LCLOQP TUOVP
Definition 1.4. An -superconcentrator is an -network with inputs and outputs
such that for any ] K @ and ]La @ with ] K )<]a_3b , there exist a set of
vertex disjoint paths connecting ]cK to ] a . An Sd -superconcentrator is an superconcentrator with at most d) edges. A family of linear superconcentrators of
density d is a sequence ABDCFEHCJG I,K such that each DC is an 6CMSd -superconcentrator,
where LCLOQP as TUOVP .
Concentrators and superconcentrators are models for various communication
networks and parallel architectures [72, 39]. For example, we can think of concentrators as a model for switching networks in which a set of processors communicate
with a set of identical memories. While if a set of processors can request a particular set of memories, we need superconcentrators.
The graphs we just introduced are only part of the set of models we shall discuss. They were defined here to facilitate the history discussion in the next section.
Other models shall be defined in associated sections.
1.3 A historical perspective
In this section, we briefly summarize the study of superconcentrators, concentrators
and expanders. An objective is to give the reader a feel of why certain results
were mentioned or covered in this chapter. Another reason for discussing these
graphs’ history is that they have rich and deep connections to many other areas
of Computer Science and Mathematics, motivating many beautiful and difficult
310
problems. Other complexity models, although shall be discussed later on in the
chapter, will not be mentioned here. We shall briefly summarize their development
in the corresponding section.
Concentrators were first introduced by Pinsker (1973 [65]) in the context of
telephone switching networks. The notion of a superconcentrator, according to
Pippenger (1996, [73]), was first introduced by Aho, Hopcroft and Ullman (1975,
[1]), who attributed it to conversations with W. Floyd. They wanted to use superconcentrator as a tool to establish non-linear lower bounds for the complexity of
circuits computing Boolean functions. Valiant (1975, [85, 84]) constructed linear
size superconcentrators using Pinsker’s concentrators, thus negating the intention
of Aho, Hopcroft and Ullman. Paul, Tarjan and Celoni, continuing the theme, applied superconcentrators to demonstrate the optimality of certain algorithms (1977,
[63, 64]). In 1973, Pippenger [66] also independently obtained some initial results
on concentrators. From 1977 to 1979, Pippenger [69, 68, 67] and Fan Chung [24]
studied the size bounds of minimum sized superconcentrators and concentrators.
The concepts of connectors, generalizers, generalized connectors were also introduced by Pippenger in these papers as models for different switching networks.
Pinsker, Valiant, Pippenger and Chung works were roughly based on the same
line of reasoning: the probabilistic method. Their proofs of the existence of good
concentrators and superconcentrators were based on the existence of good bounded
concentrators. The proofs were not constructive and thus of little practical value,
although certainly very interesting. In 1973, Margulis [53] gave the first explicit
construction of a family of expanders. As we shall see, expanders can be used to
construct bounded concentrators; hence, Margulis construction yields an explicit
construction of superconcentrators. Margulis construction was fairly technical using deep results from the Representation Theory of finite groups. He showed that
such that for
, each graph
of his
there exists a constant
constructed family of graphs is an
-expander. Extending this work, Gabber
and Galil (1981, [35]) showed that
works. Their proof was a bit
less technical, involving relatively elementary analysis. Moreover, they specified
-expanders (larger density). They also
a way to construct a family of
specify how to use linear expanders to construct linear bounded concentrators and
then to use the later to construct linear superconcentrators.
dYegf
h iW a XWkj\l
Sp)Sd avuxw y
dqrdts? z
{tS-|d}sB
om n
As we can see, the explicit constructions were too specialized to the given
parameters, and thus not entirely satisfactory. Ron Rivest and S. Bhatt suggested
another method which reuse earlier results on probabilistic construction: randomly
choose a graph in a clever way, then check to see if the graph is a concentrator
or superconcentrator. The main question is obviously that how hard the checking
procedure is. Blum, Karp, Vornberger, Papadimitriou and Yannakakis (1981, [19])
311
gave a negative answer: the check(s) is coNP-complete.
In the meantime, researchers keep working on bounding the best possible size
of the graphs. They also limit attention to graphs with a fixed depth. We will discuss more on this later. In 1983, so much excitement were ignited when expanders
were used to explicitly construct a “parallel sorting network” which sorts numbers in
time using parallel processors. The work was done by M. Ajtai,
J. Komlós, and E. Szemerédi [3], settling a long standing unsolved problem. This
problem looked so impossible that, in fact, Knuth [48] in his previous edition of
“The Art of Computer Programming” Vol. 3, stated it as a exercise to show that
constructing such a sorting network is impossible.
In 1984, an important idea comes into place as Tanner [83] used association
schemes (see [21]), in particular certain class of distance regular graphs arising
from finite geometry, to construct better expanders. The basic idea is to characterize the expansion rate using the graph’s eigenvalue. Immediately after that,
Noga Alon and partially Milman took the idea to its peak by a series of papers
([9, 5, 6, 7, 8]). Alon characterized the expansion rate using the second smallest
of the Laplacian of a graph . Basically, as
gets larger
eigenvalue
the expansion rate is larger. As eigenvalues of real symmetric matrices (as the case
with a graph’s Laplacian) can be computed easily in polynomial time, we now have
another way to “check” if a graph has certain expansion rate. The suggestion by
Rivest and Bhatt can now be used. Alon did just that. He showed how to randomly generate expanders. Notice that since the problem is coNP-complete, quite
a bit of information is lost in the characterization. (See [46], for example, for a
sample research on this loss.) However, this method is fairly practical in certain
applications.
Obviously a very natural question to ask is: “how large can
be?”. Alon
and Boppana (mentioned in [7]) proved that for any fixed and for any infinite
family of graphs with maximum degree ,
~€|8,
p.f
‚8!"
‚8!?
‚8!"
9 9Œ5
~Jƒ…„*†‡‰ˆ0‚8!"0(Š -t‹ 9Œ5 an upper bound that does not have any hidden
Nilli (from Israel, 1991,9 [61]) got
- ‹ is still5 dominating. It is obviously interesting to
constant, but the term 95 of expanders
construct families
which achieve this eigenvalue bound. The special
case when is a prime congruent to modulo Ž was done by Lubotzky, Phillips
9Œ5 (1988, [53]). They conand Sarnak (1988, [51]), and independently by Margulis
structed a family of -regular graphs with  (Š- ‹ , where ,!" is the second
largest eigenvalue of . Formally, we define
9Œ5 -regular graph with vertices
9 is called
9Š5 a Ramanujan graph
Definition 1.5. A
- ‹ .)
if ,!"0(- ‹ . (Or equivalently ‚8!?02Œ
312
Ramanujan graphs are optimal in the sense of the eigenvalue bound. Why
were the graphs named after Ramanujan ? Basically, the construction was the
Cayley graph of certain group, where the generators were chosen to be the solutions
to certain representation of integers in quaternary quadratic form. The general
formula is unknown, but Ramanujan had a conjecture in 1916 about the formula
(well, obviously these kinds of conjectures can only come from Ramanujan), which
was proved in several special cases by Eichler (1954, [30]) and Igusa (1959, [43]).
From this point on, a lot more research papers were published on various aspects of these graphs, several of which will be discussed in later section. One big
open problem (mentioned in [34]) is
Open Problem 1.6. Find an elementary (e.g. purely combinatorial) proof that certain family of graphs is expanders. For example the ones constructed in [34] or in
[51].
Solving this problem would lead to an entirely new wave of research on expanders. The spectral characterization of expansion rate loses a lot of information.
The proofs of known explicit constructions were either too complicated, or used
the spectral characterization which already reduces the power of the graphs. Calculating eigenvalues, although gives the quantitative measure, sheds no light as to
why certain set of graphs are good expanders. A combinatorial proof would be
much more satisfactory and would certainly lead to new development in the area.
It is worth mentioning that these graphs also play important roles in areas other
than switching networks. To mention a few, for example, Noga Alon (1986, [7])
used geometric expanders (expanders constructed from finite geometry) to deduce
a certain strengthening of a theorem of de Bruijn and Erdős on the number of lines
determined by a set of points in a finite projective plane. He also obtained some
results on Hadamard matrices and constructed some graphs relevant to Ramsey
theory. Sipser and Spielman (1996, [81] – see also [82]) constructed asymptotically
good linear codes from expanders.
2 The complexity of checking whether a graph is a superconcentrator or concentrator
As we have mentioned in the introduction, if it is possible to check in polynomial
time if a graph is a (super)concentrator, then it might be possible to devise a randomized algorithm to generate these graphs. The results in this section are from
[19], which says that the checks are both coNP-complete.
’ a tb-BW
W
] ’6K
‘!’ K ’a.“
such that
Definition 2.1. A matcher is a bipartite graph
and that for any -subset of , there exists a matching from
313
’ K L
] into
’a .
We shall show that deciding whether a bipartite graph is a matcher is coNPcomplete, and then deduce as corollaries that checking whether a graph is a concentrator or superconcentrator is also coNP-complete. In this section and throughout the chapter, we will use
to denote the set of integers from to , and
to denote the degree of a vertex in some graph.
5
” •
t–1—˜3
˜
Theorem 2.2 (Blum et al., 1981 [19]). Deciding whether a bipartite graph ‘™
!’ K ’Na. is a matcher is coNP-complete.
Proof. We will reduce the complement of the following NP-complete problem to
an instance of the matcher problem.
Big-clique. Given a graph
( for nodes and for arcs), with
,
does is have a clique of size ?
, since otherwise the answer is clearly NO. Given
We can assume
, we construct a bipartite graph
as follows.
šY›D š
›
šœ-|d
d
›82QžFŸa š¡›D
‘¢r!’ K ’a.“
£ ’ K ¤š¦¥§”-|dt• .
£ ’Na¨¤›*©¡Aª K 1«1«1«HXª,¬ ­c¬®E0©¯ A˜ nK 1«1«1«X˜ ²´n ³ n.µ E , where |,¶·-|d 9 )–¸—, .
n}°|±
£ To get , each vertex XT¹=jº’6K , where ºjhš and T¨jŒ”-|dt• , is connected
to the following vertices in ’ a :
– All arcs ».¼`j¡› which are incident to .
– All the u-nodes ª½K1Xª a 5 1«1«1«¾Xª À¿ÁM u K .
n
– All nodes ˜ ¼ for “
1«1«1«¾|, .
Our proof is complete if we can show that šY›o does not have a clique of
size d iff ‘ is a matcher. Notice that
7 a 9ZÃ
’ a _·- › À-|d n}°|± )–¸—,>Ž}d a r’LKBÄ«
& ’ K H]Æ)(
Moreover, by Hall’s theorem ‘ is a matcher iff />!]8132Å]Æ for all ]
K ’KB}·-|d a . Thus, it suffices
to prove the following claim.
aClaim.
& ’LK such
a
There exists ]
that ]Æx(g-|d and />!]¶1NÇ$]Æ iff š¡›D has
a clique of size d .
!È* . We first make some observation as follows. Let
É bA^j¡š'tXT¹0jq] for some TE_«
314
5 ^Ê Ë É and ̍ͮA¾»Yj§›ZB»Ë K X6a for some K X6aj É E) . Notice that
(*Êq(Š-|d , and that
7
É
/>!]¶1< ®4 A¾»Îj¡9Œ›b5>¾7 » is incident to some \j E)
d
- ;É 9·Ã
-|d6 n}°|Ï t–¸—,
4 d 9Œ5>7 9
-;
-|d)Ê Ì
9i5
Now, />!]81¶Ç™]Æ8(Í-|d)Ê implies ̌eVž Ÿa Ì is exactly the
7 9 making ÌÎ.( However,
(*Êq(Š-|d .
number of edges of a graph with9ÒÊ 5¶vertices,
.
Hence,
À
ž
Ð
5Moreover, if Ê¡eŠd then as ž Ÿav -|d)Ê ž Ðav is a strictlya¸ increasingdÑfunction
in
” S-|dt• , we get a contradiction:
a-|d 2Å]Æ3e 4 d 9Œ587 -|d)Ê 9$4 Ê 2 5>7Í4 d 9Š587 -|d a 94 d ·-|d a
-;
-;
-;
-;
É
Consequently, ʍ·d . This implies ÌÓ͞ Ÿa , so that is a d -clique.
!Ô* . Let É be a clique of size d . Let ]§ É ¥§”-|dt• , then
4 d 9Š5>7 a 9 4 d
/>!]¶1_ - ;
-|d - ; ÇÅ]Æ
Let
It is easy to see that the problems of determining if a graph is a “concentrator”
or “superconcentrator” are in coNP. To prove in polynomial time that a graph is
not a concentrator, we only need to present a vertex cut which separates a set of
more than
inputs to the outputs. Menger’s Theorem ensures that this cut must
exists. The proof that a graph is not a super concentrator is similar. Consequently,
to show that they are coNP-complete, we can just reduce them to the “matcher”
problem.
É
É Corollary 2.3. Deciding whether a graph
!’LK’ a “ ’‰CÆQ-BW
’a
’y W
’y
‘
is a concentrator is coNP-complete
‘ ’6K
Proof. We reduce “matcher” to “concentrator”. Given a bipartite graph
with
, we construct
by orienting all edges from
to
, add a set
of vertices, and then connect all vertices of
to all vertices of
. Clearly is a matcher iff is a concentrator.
315
’a
Corollary 2.4. Deciding whether a graph
is a superconcentrator is coNP-complete.
’ y to be -BW .
Corollary 2.5. Deciding whether a graph is an Nf} -expander is coNPProof. In the previous proof, change the size of
complete.
3
Expanders
The main objective of this section is to present several results on the eigenvalue
characterization of the expansion rate of expanders. Moreover, we shall give a
partial answer to the question: “how large can the second smallest eigenvalue of the
Laplacian of a graph be?”, and give some pointers to results on direct constructions
of expanders.
3.1 Algebraic graph theory
We first fix notations and terminologies from algebraic graph theory needed for the
rest of the section. The reader is referred to the standard books by Biggs (1993,
[18]), Godsil (1993, [36]), Cvetković, Doob, and Sachs (1995, [27]), and Chung
(1997, [25]) for more information.
be a simple graph with vertices, we will always assume
Let
the eigenvalues of
are
. Those are the eigenvalues of
the adjacency matrix
of . Moreover,
will be used to denote . Let
be the
diagonal matrix indexed by vertices of with
;
then, the matrix
is called the Laplacian matrix of . We shall use
to denote the eigenvalues of . In contrast to the ’s, we
use
to denote
. The reason for this is that when
is -regular then
.
Let
be the incident matrix of any orientation
of
. Let
(
) be the space of real valued functions on ( ), with the usual inner product
and the usual norm
. Note that
is isomorphic to
and the Rayleigh quotient for is
. Also note that
ÕÖ!’¶
LK^2¦ a 2Õ×1×1×`2' n
› ,!"
a
Ø
œ¥Y Ø 9
Ø ´ÙSÙÎ$t–¸—˜3
Ú$ÛÄ ›
‚UK2‚ 9 a 2Ü×1×1×02‚ n
Ú

‚8!?
‚ ntu K
 C ¤ ‚Lntu C€R,K
Ý Ó!’¶ Ú a !’“
aÚ “ š
’ a
áß>á?ãâ ÞÀßSß6à
Ú !’
ä n ÞÀß´—3à
ì
ì
ß å ­ Àß6æèç3é é¸ê é¾Á ë
ÞÚ>ßSß6àí ޚ î½Ã š\ßSß6àU¢Þš\9 ߚqß6aà
³Äï Ù µ°Bð ³òñ µ ÀߪL ߘ3X
Ãê 9 a
ïHó Ù ÀߪL ߘ3X
316
ªhôZ˜
Ú
‚n
ª½X˜‰Djõ
‚½C,2Šf36öxT
f
‚ ntu K>¤f ‚ n}u K
a
Proposition 3.1. Let ßqjYÚ !’? such that ÷ ٠ߘ3·f , then
Ã
9 a
Àßà ª ߘ3X
B
ï
ó
Þ
>
Ú
ß
S
6
ß
à
Ù
‚8!?>( áß>á ßÙ a ˜3
Here
is used in lieu of
for convenience. The previous equation
implies that is non-negative definite, thus
. Moreover, it is not hard to
see that
is always and that
iff is not connected.
The following bounds on
will be used quite often.
In fact, a stronger statement holds
‚8!"·„Óé}Iù ƒJøs ÞÚ>áßß>Sá ß6à
with the „ƒJø runs over all ß satisfying ÷ ٠ߘ38f .
Ã
9 a
Note.
ïHó Ù ÀߪL ߘ3X is sometime called the Dirichlet sum of .
Proof. Let ª n Zú)+ ‹ be a unit ‚ n -eigenvector of Ú , then the variational characterization of the eigenvalues (see [37], e.g.) gives
‚Ln}u K s Iù „Óé ƒJ°Bï ø ûHü å ç Àß6 s Iù „Óé ƒ…°Hø û.ü ÞÚ0áßß>Sá ß6à
éý ü
éýcþ
ï
ߪ¤f .
The condition ß ÿZú is the same as ÷
9 j^Ú R a !’ is a ‚8!a " -eigenvector.
Proposition 3.2. Suppose is connected, and ß^
R
u
Let ’
ÛÄ$A˜¡jœ’$.ߘ‰#egf)E and ’ ÛÄ¢’ ’ . Let —¡j§Ú !’“ be defined
by
˜jq’ R
—˜‰8 fߘ3 ifotherwise.
Then, we have
Ã
9 a
—Ã ª —˜3X
H
ï
ó
Ù
‚^2
—Ù a ˜3
317
is connected, ‚¤f , making ߤf . Hence, ’ R . By
Ú>ß6¸˜‰¶‚½ßØ3¹ö˜ jq’ . Thus,
Ú>ß6¸˜‰´ß˜3
Ù
°
‚Y Ã ß a ˜3
Ù °
à a à a
Ù ° ß ˜3¶ Ù ° — ˜3
Proof. Note that since
definition, we have
But,
and,
Ã
Ã
9 Ã
a
Ù °
Ú>ß6¸˜‰´ß˜3 Ù °
à x˜‰´ß ˜39 ï ° ³ Ù7 µ ߘ‰´ßà ª
9
a
ï Ù °.ð ³ µ ÀߪL ߘ3X ï Ù °.ð ³ 3µ ߪ¸ÀߪL ߘ3X
Ã
9 a
ê
2 ïHó Ù —ª —˜3X
which completes our proof.
3.2 The eigenvalue characterization of expansion rate for regular strong
expanders
The results in this section are from Tanner (1984, [83]), Alon and Milman (1985,
[9]), and Alon (1986, [6]). As the title of the section indicated, our goal is to show
that the larger the expansion rate of a regular strong expander is, the larger
has to be, and vice versa.
We again need to define several types of graphs closely related to expanders:
enlargers, magnifiers, and bounded concentrators.
‚8!"
N -enlarger is a graph on vertices with maximum
‚ ntu K¾!">2 .
9 !% ’¶ on% vertices, !?
Definition 3.4. An N -magnifier is a graph g¢
%
&
%
%
and for every ’ with 0(Ë,+.- , /> >2Ü_ holds. The extended double cover of a graph ã™!’¶ with ’'” • is the bipartite graph
Ý on the sets of inputs % <AÊ K 1«1«1«XÊNnxE and outputs Í$AÌ K 1«1«1«¾XÌ|nNE so that
ÊNC´X̾¼¾0jY Ý^ iff TUÒ or TSFÂt:jY !" .
Definition 3.3. An
degree and
318
5
is a bipartite graph
‰> -bounded strong concentrator
Ö % @ % inputs,5 _ outputs,% ·Ç ,% and at most _ edges, such
#(, , then /> 1#2[_ . An ‰ -bounded
> -bounded strong concentrator.
For the purpose of this section, enlargers were introduced just to shorten the
sentence: “let be a graph with maximum degree and ‚8!" large enough.”
Definition 3.5. An
with
that if
with
concentrator is an
On the other hand, we need magnifiers because their extended double covers are
expanders. Magnifiers are, in a sense, the non-bipartite version of expanders. It
is intuitively clear, and will be made precise later, that bounded concentrators are
closely related to expanders. The proof of the following lemma is straightforward,
hence omitted.
Lemma 3.6. The double cover of an
7“5
N1 -magnifier is an -expander.
The following theorem, which is an improved version of Theorem 3.4 in [6],
makes our goal precise.
9 be a -regular bipartite graph, where =
í
| ‚Y‚8!? (· ,!" ). The following hold:
²Á
(i) If is an strong expander then ‚^2 u z! ² R a ² Á .
a#"$F" u%Á $ Á .
(ii) If ‚^2 , then is an -expander, where ¨
We shall prove this theorem using a sequence of lemmas, with the following
plan in mind. To show T´ , we will show
is a strong expander OV is a magnifier OV is an enlarger (i.e. ‚8!? is large)
Similarly, to prove T!T¹ we shall show the reverse sequence
is an enlarger OQ is a strong expander
Theorem 3.7. Let
and
Let us now proceed to show that a strong expander is a magnifier.
be an N1 -strong-expander.
²
À-BN K avu ² % & satisfies
Proof. Be definition, every K
465>7 5 9 % K %
%
/> Kv1)2 | ; KBÄ«
Lemma 3.8. Let
-magnifier.
319
Then
is a
(1)
In particular,
(2)
/> % K 1)2Z % K %
% @ , setting
When 9 K Ʀ,+.- , (1) implies œ(¦- . Moreover, for every a
% K0¤ /> % a in (2) yields /0 % a :2Å % a .
% & "©\
with size at most , it must be the
We are to show that for every
case that
9 % 5 9 %
%
/> 32 - % %'& and % aD %'& . The intuition is that when % K is very small
Let K % 9 there will be a lot of neighbors of % a lying outside of % K
comparing to a , then
% % large. On the other hand, when % KB is relatively large
in , making /0 %
%
%
comparing to a , then there will be “many” neighbors of K not in a . We now
5Æ9 ² rigor.
turn this intuition into mathematical
%
%
, we have5 9
When KB)(Å a ò
% )( - ( % a}Ä«
9%
9 % % 9 % % 5 9 %
%
%
/> 3 2Å /> a 1 KBt2Å a KBt2 ( a t2 - Ä«
59 ² % n
%
>ÇÅ K )( a ,5 we get 5 9
When a}ò
% )Ç 4 587 5 9 ² % KB_ ( - 9 % KBÄ«
;
This relation, the fact that #(Š- , and (1) yield
9%
9 %
%
%
/> Ü2 />5> 7 Kv1 9 5Æ 9 5 a %
2 9 -
² 1 K Ž 9 - ( 9 % KB
e Ž - 5 - 9 % 2 5 - 9 % Hence,
320
% K )2 na it follows that
9%
9 % 5>7 9 % 5 9 %
%
%
/> )2Å /0 Kv1 a )2b - - - Ž ^2 Ž 50732 5¨- 9 Ä«
| nÐ XMÊ is
Here, we use the fact that (i- and that the function ßÊL:Ë
increasing when ^2Š- .
Next, to complete part T¹ of Theorem 3.7 we show that every magnifier has
“large” ‚ .
Lemma 3.9. Let !Á ’“ be an -magnifier, then is an N ²
enlarger where 0 a R a ³ò² R,K µ Á .
Proof. We apply
and use notations of Proposition 3.2. Since we have to show that
‚^20 a R a ³è² ² Á R,K µ Á , it suffices to show that
Ã
9 a
ïBó Ù —à ª —˜‰X 2 7 a 7·5 a
- -‰ —Ù a ˜3
This is done by using the maxflow-mincut theorem. Consider a network š with
R *Y’*\A-,SE , where ) is the source, , is the sink, and * denotes
vertex set A)_E+* ’
the disjoint union. The edges and their capacities are defined as follows.
5>7
R
(a) For each ªYjq’ , .)}Xª has capacity ¸»0/c.)_XªL¶
5 .
R ¥Y’ , v»0/cª,X˜381 if ªx˜ jY or ª˜
(b) For each pair ª½X˜‰:jq’
f otherwise.
5
(c) For each ˜ÎjY’ , ¸»-/c˜x#,X> .
5=7 R
É
We claim that the min-cut of š has capacity 1
’
. Consider any cut .
É
%
%
R x.)}Xªqj + E . As% has magnifying
5#7 % rate and /c±? “
Let
% ©\/2:ÛÄ % A ª·, itj¤is’ easy
1 . 5NMoreover,
to see that / ± 182
É for9 every
7
7
must be at least one edge with capacity one
in incident
to it.
˜qjõ/ ± % there
É
%
R
7 R thus the capacity of is at leastR ¸’ All these edges5Æare
5 / Ʊ 7 % 1,2ÍR disjoint,
1’ . Lastly, the cut At.)}Xª#|ªÒjŒ’ E has capacity exactly
1’ , proving the claim. By the maxflow-mincut
theorem, there exists an
ä
orientation 3 of edges in and a function 4¡Û5Z
5 3 O such that
f“(4½ª,X˜3:( for all ª,X˜3 j 3
Lastly, when
321
Ã
4½ª,X˜3> f
7Ù 6 ³Äï ê Ù µ°9ð 8
507
Ã
ï 6 ³èï ê Ù µ° ð 8 4,ª½X˜3 (
5
ªYjY’ R
if
otherwise.
for all
˜ jq’
Now, the following is straightforward:
Ã
a4 ª,X˜3 ž —ª 7 —˜3 a ( - à 4 a ª½X˜3 ž — a ªL 7 — a ˜‰ ³Äï ê Ù µ° ð 8
è³Ã ï ê Ù µ° ð 8 4 Ã
7 Ã a
a
a
- Ù °: — ˜3 ï 6 ³Äï Ù µ°9ð 8 4 ª½X˜‰ ï 6 ³ Ù ï µÀ°;ð 8 4 ˜NXªL ;
5>7 5>7 a à ê a
ê
( -‰ Ù ° — ˜‰
and
Ã
9 a
à a 4 Ã
9 Ã
a
³Äï ê Ù µ°9ð 8 4½ª,X˜3 ž — ª — ˜3 ï :° à — ªL Ù76 ³èï ê Ù µÀ°<ð 8 4½ª½X˜‰ Ù76 ³ Ù ê ï µÀ°<ð 8 4½˜NXªL ;
2 Ù °: — a ˜‰
Now, combining all inequalities above along with Cauchy-Schwarz inequality we
322
get
‚ 2
2
2
2
Ã
9 a
Hï ó Ù žÀ—à ª —˜3 —Ù a ˜‰
Ã
9 a à a
7 a
ïHó Ù žÀ—à ª —˜3 à ³Äï ê Ù µ° ð 8 4 ª,X˜3.ž!—7 ª —˜3 —Ù a ˜‰ ³Äï Ù µÀ°+ð 8 4 a ª,X˜3.ž!—ª —˜3 a
ê 9
4 Ã
a ª — a ˜31 a
,
4
½
ª
X
3
˜
1
—
;
³èï ê Ù µ50°97 ð 8 7¤5 Ã
-‰ a .ž Ù ° — a ˜‰ a
Ã
9 a a
a
5
4,ª½X˜3¸ — ªL — ˜3X 7>
7 7¤5 a = ³èï ê Ù µÀ° ð 8 à a
>>
- -‰ ==
— ˜‰
Ù
:
°
7 a 7¤5 a
- -‰ T¹
T!T¹
V ' 5 'W
5
)–¸—T¹o»N¹öT#jœ )–¸—@?_DBA öC?j*
D
œ¥qWÜf
W¡CFE TG?ÅjÍ WCFEœíf
‘ HDID î
‘
)K 2 a 2ã×1×1×:21 n
‘ ª½KXª a 1«1«1«¾Xª n
‘Ó C ¼
T Â
T
A
÷ ¼ ‘MC ¼o¤»JA¾¹öT
ú3+ ‹ ‘
»JA
‘ – –¾C
–
à ‘–H¹C½Kt–1C Ã
x –CX|r ¼ ‘ÓMC¼H–S¼)t(Å –1CS ¼ ‘ÓMC ¼=¤»JA| –1CS
The previous two lemmas trivially imply part
of Theorem 3.7. Now we
are ready to complete part
of Theorem 3.7. We first need a lemma from [83].
Let
be a bipartite graph such that
,
, and that
,
,
. Let
be an
-matrix whose
rows are indexed by and whose columns are indexed by such that
if
and
otherwise. Let
, then clearly
is real,
symmetric, and non-negative definite. As usual, we let
be
the eigenvalues of
and
be a set of corresponding orthonormal
eigenvectors. Notice that
is the number of common neighbors of and ,
. (Each neighbor of is counted times in the sum.) This
hence
implies
is an eigenvector of with corresponding eigenvalue . Suppose
is any eigenvalue of and is a -eigenvector. Let be a coordinate of with
highest absolute value, then
implies
323
»JA is also the largest eigenvalue of ‘ , i.e. K <»JA . We may thus
ª½K>Zú3+ ‹ .
Lemma 3.10. If K eLBa , then is an XWY+H»N>|@UX -bounded strong concentrator, where
|@:2 :»JA 9 » a a 7 a
% @ with
Proof. Let M be a positive
number not exceeding . For any
% LBM6 , let Ê*jA¾f3 5 E n real
%
a
be ’s characteristic vector. Clearly ÊNÊ î [áÊ8á %
M6 . Similarly, let b/> and Ì be% its characteristic vector. As for any ?Ójh
,
ÊND¢E is the number of vertices in adjacent to ? , the sum of entries in ÊCD is
exactly M66» . Hence,
Ã
4 ÷ E °:O ÊND¢E a M a a » a
a
a
(3)
áÊCDÍá E °:O ÊND¢ E 2
| 
Ñ
;
7 7 7
P a ª a 7 ×1×1× P n ª 7 n we7 get
Now, write ʍPxKª6K
ʑ¢QP K K ª K P)a7Bavªxa ×1×1× PtnRHn_ªxnN«
Thus, by orthonormality we have
7 a7 7 a
a
a
áÊNDÜá ¢Ê‘ÓMÊ î K}K P K a P a ×1×1× n P n
(4)
Consequently,
assume that
Now,
PxK0ÊNªNîK ÷ ‹ C ÊNC QM ‹ Hence, (4) gives
7 Ãn a
a
a
áÊCDÍá _K P K ¼I a ¸¼SP ¼
7
9 a
a
a
( »TAUa M 9 a áʶ7 á P K M 8»JA a a M6
Lastly, combining (3) and (5) yield
/>% % 1  2 9 » a 7
M6 M>»JA Ba Ba
As this inequality is true for all M§(V , the proof is completed.
324
(5)
T!T¹ of Theorem 3.7 could now be derived as a corollary of this lemma.
Corollary 3.11. If Zr
is a -regular bipartite graph with t$
)
and ‚Y¤‚8!" , then
(i) is an N -strong expander, where
9 a
¨ -._‚ a ‚
(ii) is an N1 -expander, where
9 a
9
¨ a -.__‚ ‚ 7 ‚ ‚ a +.Proof. Let D and ‘ be the matrices for as in Lemma 3.10, and let › be ’s
adjacency matrix. Also let tKÆ2i×1×1×N2Q n be the eigenvalues of ‘ and 6K¨2i×1×1×x2
Na´n be the eigenvalues of › as usual. Sincea ‘Ó C ¼ ¢@DWD î C¼ counts the number
of common neighbors of T and  , while › MC ¼ counts the number of length- - paths
from T to  , it is obvious that
4‘ f a
f ‘ ; ¤›
9 is standard that as is bipartite, whenever
9
 is an eigenvalue of › , then so is
It
9
 . Moreover,a LK#¢ and  %'a r@ ‚ since % is -regular. Thus, tKo¢ a and
a ¢ ‚, . Now, for any with ^r +H , applying Lemma 3.10 gives
/> % 1Í2 ž a 9 9 ‚, 9 a a 7 5Æ 9 9 ‚, a % 4½5>7 À9 -._‚ ‚ 9 a ¸ 5ÆU9 %
4 5>7 À-.a _‚ À9 -.}‚ ‚ a 4 ‚ 5 a 9¸ % U ; %
2
a
;¨; %
When 3(*,+.- , we get
9 a 459 %
>
5
7
4
/> % 1‰2
a À9 -._}‚ ‚ 7 ‚ ‚ a +.- ;=; % Part
325
3.3 On the second eigenvalue of a graph
‚ ntu K
As we have seen, the expanding rate of a graph increases as its second smallest
Laplacian eigenvalue increases. Thus, it makes sense to study how large
can be. Through out this section, we shall consider only -regular graphs and let
. Alon and Boppana (mentioned in [7]) proved
that for any fixed and for any infinite family of graphs with maximum degree
,
9
‚Y‚8!"¤‚ n}u K¾!"8¤  a !"
9 9Œ5
~Jƒ…„*9†5‡‰ˆ0‚8!"0(Š -t‹ 5
The bound is sharp when is a prime congruent to modulo Ž , as shown by
the explicit constructions of Lubotzky, Phillips and Sarnak [51], and independently
by Margulis [53]. Alon [7] conjectured the following
9 9Œ5.9
5 (Alon [7]). As YOQP , the probability that ‚8!">(Œ - ‹ 5Conjecture 3.12
?‰ goes to .
9 In other9Š5 words, as gets large, with very high probability we have ‚8!"(
- ‹ . The conjecture is still open as far as we know. Friedman, Kahn and
Szemeédi (1989, [33]) showed that
9 9Š5:9 9 5
‚8!?>2Œ - ‹ ~€|> ?‰ as ŒO P . Nilli (1991, [61]) got an upper bound that does not have any hidden
constant:
7 [61]). Let be a graph in which there are two edges
Theorem 3.13 (Nilli, 1991
with distance at least -|d
degree of . Then,
- , and let be the maximum
9 9Š587 - ‹ 9Œ7·5:5 9Œ5
‚8!?0(Œ - ‹ d
7
Proof. Let ª K Xªxa¾ and YX K #XÆa¾ be two edges with distance at least -|d
- . Let
5
Ú be the Laplacian matrix of as usual. Let Z¶s#ÅAª½KXª a E and [\s=¢A-XoK#X a E .
For ([T(¦d , let ZcC (resp. [qC ) be the set of vertices of distance T from Z¶s
(resp. [\s ). Clearly ZbÛÄg© sS\CY\ Z,C has distance at least - from [ ÛÄi© sS\CY\ [qC ,
9Œ5 is no edge joining Ÿ the 9Štwo5 unions. Moreover, it is easy to seeŸ that
so that there
]Z C t(b 1]Z C u K ä and ^[ C t(b 1^[ C u K .
be defined as
Let ßqÛt’ !?O
9Œ5 u Cdc a
a » 9Š5 u CFc a for ˜ jeZcC´f(*T¶(Šd
ߘ‰¶ _` A. for ˜ jf[\CMf(ÒT8(Šd
otherwise.
`b f
326
»ŒeÍf AYÇÜf
÷ ٠ߘ3f . The variational
‚8!?¶Š‚Ln}u K ( ÞÚ>ÞÀßßSSß6ß6à à
We also have
à a 7 à a
ÞÀßSß6àí ï °:g ß ª h °:i ß YXo
ß
a5 7 ß
a5
*
9
Œ
9
»
A
CJIs 7 ]Z,C C C€Is ^[qC C
›`K ‘?K
where › K and ‘ K are the first and second sum, respectively. Moreover, as we
9^5
Z 5 and [ ; (b) there are no edges
have mentioned: (a) there are9 no edges joining 9^
connecting Z,C or [\C to ’!? ZY©j[ if T8(Šd
; and (c) there are at most edges joining a vertex of ZcC ( [qC ) to a vertex of ZcCJR,K ( [qCJR,K ), we have
Ã
9 a
ÞÚ0ßSß6àQ ïHó Ù ÀߪL ߘ3X
Ã
9 a7 Ã
9 a
k ïHó Ù Àߪ ߘ‰X k ïHó Ù ÀߪL ߘ3X
ï ê ÙSl m g Ioù n
ï ê Ùpl m i Ioù n
Ãu K Ã
9 a 7 ŸÃ u K Ã
9 a
Ÿ
C€Is ïHó Ù ÀߪL ߘ‰X CJIs ïHó Ù Àߪ ߘ3X
Ã ï °:iur ê Ù °:iur Jt a
7 ï °qgqr Ãê Ù °:gsr Jt
7a
ÀߪLX
ÀߪLX
B
ï
ó
B
ï
ó
Ù
Ù
ï °qg ¿ ê Ù °uCgovsi
ï °:i ¿ ê Ù °uCgovsi 5
5
9Š5 7
Ã
Œ
9
5
9
7
4
K
u
a
Ÿ
Œ
9
( » axw CJIs ]Z,CSò 9Œ5 Cdc a 9Œ5 ³ C€R,K µ c a ; ]Z Ÿ 5 ŸCy
5 9
5
9Œ5
Ã
Œ
9
5
7
4
K
u
a
A axw Ÿ C€Is ^[qCò 9Œ5 Cdc a 9Œ5 ³ CJR,K µ c a ; ^[ Ÿ 9Œ5 Ÿ y
à Ÿ ]9ŒZcC5 9 9Œ5 7 9Š5 9Œ5 ]9ŒZ 5 7
x
a
w
» CJIs C - ‹ À- ‹ Ÿ Ÿ y
aA xw à Ÿ ^[q9ŒC5 C 9 - ‹ 9Œ5 7 À- ‹ 9Œ5 9Œ5 ^[ 9ŒŸ 5 Ÿ y
7 C€Is ›a ‘a
where
,
are real numbers such that
characterization yields
327
way. To this end, we are left to show that
›=a and ‘#a defined
7 in the obvious
9Œ5 9*5
9
Œ
9
8
5
7
7
› a ‘ a (Œ - ‹ - ‹ 7¤5 “Û É
›K ‘K
d
É and ‘ a +B‘?K( É instead. Notice that ³ "¸¬ ugsrK ¬ µ r 2
We shall show that › a +v›"K(
³ "¸¬ ugsrK zµ r t z¬ t , clearly
ß
a 5 a 7¤5 ]9ŒZ Ÿ 5 Œ
9
»
›`K> CJIs ]Z,C C 2*» Àd Ÿ
hence,
9Š5 9*5
9
Œ
9
>
5
7
› a ¤ - ‹ » a - ‹ ^[ 9ŒŸ 5 ( É
›"K
›"K Ÿ
‘ a +B‘"KÆ( É is proved similarly.
with
Corollary 3.14. Let
regular, then
,
and
d
be defined as in the previous theorem, If
9Œ5 4 5 9 ¤7 5 5 7 ¤7 5 5
 a !?>2Š- ‹ d ; d
is 3.4 Explicit Constructions of Expanders
Many practical applications require explicit constructions of expander graphs. Explicit constructions turn out to be a lot more difficult than showing existence. In
1973, Margulis [52] gave the first explicit construction of (strong) expanders. He
for
,
,
explicitly constructed a family of bipartite graphs
and show that there is a constant
such that for each
,
,
is an
-strong expander. This construction was certainly undesirable
as the constant was not known. Moreover, his proof used deep results from
Representation Theory. Angluin (1979, [12]) pointed out that Margulis’ method
could be used to construct
-strong expanders but the constant
is also
not known. Gabber and Galil (1981, [34]) slightly modified Margulis’ construction and used relatively elementary Taylor analysis to show how to construct a
family of
-strong expanders for each
. They also constructed a family of
-strong expanders. Let us mention here
their first construction. For each
, construct an
bipartite graph
om n
Sp)Sd
d
dÅe™f
{)SdJ| 9
a
W Sp)¾À- ‹ a {}X+HŽt 9
W {t¾À- ‹ {_X+.-_
W j¤l
328
5
a
A m n E ¡W Wa S -)1«1«1«
$¦W W rj l
dJ|
Wkjhl
W a ¥§W a
~3 } Á ¢ } } where } } i
}  } ¥€ } and each vertex TFÂtÆjY } is
connected to p vertices in defined by the following permutations:
 KTSFÂt TFÂt 7
 a TSFÂt TXT 7 Ât 7¤5
 y TSFÂt TX7 T   z TSFÂt T 7 Â_F7¤Ât 5
 TSFÂt T  FÂ)
Here, the additions are done modulo W . As we have mentioned, there is no known
elementary proof that this family are expanders with the prescribed expansion rate.
This construction was later modified slightly by Jimbo and Maruoka (1987, [44])
and Alon, Galil and Milman (1987, [8]) to obtain better superconcentrators.
As we have discussed in the introduction, after the eigenvalue characterization
of expansion rate, the main problem was to construct Ramanujan graphs, which are
is a prime
optimal expanders in the eigenvalue sense. The special case where
congruent to modulo was “solved” by Lubotzky, Phillips, and Sarnak (1988,
[51]), and independently by Margulis (1988, [53]). Later, in 1994 Morgenstern
[58, 57] constructed for every prime power many families of
-regular
Ramanujan graphs. All these constructions were Cayley graphs of certain groups.
Other works and information on expanders could be found in [4, 50, 11, 2, 46,
79, 17, 25].
5
Ž
@‚
‚
9^5
7$5
4 Concentrators and Superconcentrators
4.1 Linear Concentrators and Superconcentrators
Ó~€| a ,
-:{:ƒB
Valiant (1975, [84]) showed that there exists -superconcentrators of size at most
and depth
. Valiant’s proof was based on a recursive construction using Pinsker’s concentrator (1973, [65]). Pinsker showed that there exist
-concentrators with at most
edges. A somewhat weaker version of this result
was also obtained independently by Pippenger (1973, [66]). In 1977, Pippenger
[67] gave a simpler construction of -superconcentrators with size at most
,
maximum degree , and depth
, while Valiant and Pinsker’s graphs did
not have
degree bound. This was certainly a big improvement. On the same
line of reasoning, with finer analysis Fan Chung (1979, [24]) showed that there
exists -concentrators of size at most
and -superconcentrators of size at most
. Bassalygo (1981, [13]) improved the -concentrator bound to
and superconcentrator bound to
.
5
{:ƒ)«®pB
Ó 5(
-:„B
{(
Ó~€|>,
-_{H
329
Ž_f.
-.f.
We present here the results of Pippenger [67] and a small generalization given
by Gabber and Galil [34], since although it is not the best result, it is fairly intuitive.
(W
W
(
Ž|W
„
d§(L{BW
] d
]5
d9¢5
(
5 ( on9·5… . Let
Proof. Let
, and † be any permutation
Ä
Û
Ö
¾
A
3
f
1
1
«
1
«
B
«
{
W
E
5 obtained from † by taking A¾f3 1«1«1«H( W E as inÓ@†½ be the5 bipartite9rgraph
E as outputs, and Î!"§ÛÄ AtXʍ„J‡ WY¾@†¶Ê8„J‡
puts, A¾f3 1«1«1«.XŽ|W
Ž|WYX:1ÊYjˆ…'E .
We way that @†½ is good if there are no dY({BW , a d -subset › of the inputs,
@ ‘ . We would like to look for a
and a d -subset ‘ of the outputs such that /0›o
good graph, which will satisfy our criteria. The existence of good graphs is shown
5 that the probability of @†½ being bad (i.e. not good) is strictly less
by proving
than .
(
Any d -subset › (‘ ) of the inputs (outputs) corresponds uniquely to a d -subset
@
9 ( />›D ‘ iff †¶Êj
( „|d -subset) ‰ ( Š ) of … (but not vice versa.) Moreover,
(
Š¨¹öÊ¡j€‰ . For fixed ‰ and Š , there are .„|d Ÿ .{ W dU‹ permutations † which
satisfy this condition. Hence, let Œ } be the probability of @†½ being bad, we have
à y } 4 ( W 4 Ž|W .„|d .{ ( W 9 ( dU‹
Œ } ( I,K dõ; dõ; Ÿ .{ ( WYU‹
ŸÃ y } ž } ž z } žŽ Ÿ
I,K Ÿ ž y Ÿ } Ÿ “۝Ÿ } 5 Ÿ
What we want to show is } Ç . As } has {BW terms, we first want to bound the
largest term. Combinatorially, it’s easy to see that
4 { ( W 4 ( W 4 Ž|W 4 - ( W
( d ; 2 d ; d ; Ž}d ;
Hence,
à y } ž  Ÿ à y }
} ( I,K ž a z Ÿ} I,K Ú Ÿ where Ú Ÿ ÛÄ ž ža z ŸŸ} Ÿ
Ÿ Ÿ Ÿ 5
It is not difficult to check that the sequence +BÚ is unimodal, thus the largest Ú
Ÿ
Ÿ
is either ÚÆK or Ú y } .
Lemma 4.1 (Pippenger, 1977). For every , there is a bipartite graph with
inputs and
outputs, in which every input has outdegree at most , every output
has indegree at most , and, for every
and every set of inputs, there
exists a matching from into some -subset of the outputs.
330
As
{BW Ú K Ç
5
Ú>y } eŒÚ K , so that }
ž aK ! }} À-_5 {HWYU‹… 5 -BWYU‹… 5 Ž|WYU‹
B{ W Ú y } {BW ž aK a }} ƒBWYU‹….„BWYU‹…À- ( WYU‹
trivially, we can assume
is at most
To this end, we use the following two inequalities. The first one is a good version
of Stirling’s formula (see Robbins [78]).
‹ -†,– un n (*‹N(Œ– t Á t ü ‹ -†½– un n
and
5
:
5
9
–Ð ( Ê 5
Applying these inequalities yields {BW Ú y } Ç
for all W 2{ . The cases where
W(Š- are trivial.
(
Corollary 4.2. For every W , there is a bipartite graph with Ž|W inputs and W
outputs, in which every input has outdegree at most „ , every output has indegree at
(
most , and, for every dÎ({BW and every set ] of d outputs, there exists a matching
from ] into some d -subset of the inputs.
n
Let )t, be the minimum size of an -superconcentrator, and N,DÛÄ$Ž‘ T’ .
We first obtain a recursive inequality for )), .
5 7
{B )t@x,X .
Lemma 4.3. For any , )t,>(
n
Proof. Let W“‘ R’ , and and | be the graphs of Lemma 4.1 and Corollary 4.2
5 7
respectively. Let ]”| be a x, -superconcentrator with ))@N,X edges. Construct
an -superconcentrator as shown in Figure 1. Clearly ] has size at most {B
) ž N, .
Now we are ready for the main result:
Ž_f.
Theorem 4.4 (Pippenger, 1977). We have
.
7
)t,>(Q{:„B Ó~€|>, . In fact, )),>(
9Œ5
)) ,>({B8À-•‘~€| y ’ 5 it is certainly a superconcentrator.
because the Beneš network is rearrangeable,
This gives )t,>({:„B for ^({ ·- ƒ_{ .
Proof. The ternary Beneš network (see, e.g. [16]) gives
331
n edges
G
S’
to be deleted
Figure 1: Recursive construction of an
superconcentrator
G’
-superconcentrator from a N, -
s
values of , we shall use the previous
lemma. Define ,"Ë ,
R:– ,K For7,¨5 larger
,
R
K
—N@:–X,X . Pick , such that q–X,DeL{ 2L:– , , then apply Lemma
times, we get
4.3 ,
5 s 7 K 7 7
7 R,K
)),>( {‰@ , , ×1×1× – ,X ))@ – ,X
7
a
ƒ , which implies
It is easy to show by induction on , that – ,>( ž y_ – 7·5 7
)),0({:„B f.ŽNY, {_
Moreover, as x,h
(##y z z##y !´5 zK , we have { ǘ:–,§( ž z#y#!´ zK – . Hence, ,\
z
´
!
Ó~€|>, because K Ç . Finer analysis on , shows that )),>(*Ž_f. .
5
(
(
Notice that the graph of Lemma 4.1 is nothing but a W^S-_+{) +.-_ bounded concentrator. The above construction can be straightforwardly generalized as follows.
a K Ÿ u%R,™ K
5
‰Sd +.-_
Lemma 4.5 (Gabber-Galil, 1981 [34]). A family of -superconcentrator of density
can be constructed if for each an
bounded concentrator
is given.
332
-.f.
As we have already mentioned, Bassalygo (1981, [13]) improved this bound
further to
, however we shall not discuss that result here. One might wonder what is known about the lower bound of
. Lev and Valiant (1983, [49])
provided the best known so far, whose proof is omitted here.
)t,
f
f
9
pB ?‰,
Theorem 4.6 (Lev and Valiant, 1983). An -superconcentrator whose inputs have
indegree and outputs have outdegree has size at least
.
The proof of this lower bound is quite involved, and certainly more difficult
than the upper bound proof. It is fairly disturbing that the gap between the lower
and upper bounds remain quite large. Let us put it as an open problem.
Open Problem 4.7. Close the gap of
9
7
pB ?‰,>()),>(Š-.f. ?3, .
4.2 Superconcentrators with a given depth
The following functions often show up in probabilistic arguments concerning the
problem.
Definition 4.8. Let
›š.Àß,X denote ß,M E ³ K µ .
Definition 4.9. Let
5
~€| š ^ÛĤ„Óƒ…øNA0œU2Šf“½~€|¨«1žp«1¡Ÿ«~€|8 ( E
where the logarithms are to base - . By induction on d , define
Ÿ ¤ ¿×1 pT×1 t× ž¤ Ÿ ¤ ¿×1 pR×1 t× ž¤ 5
Ÿ ¤ ×1 p¿×1 × ž¤
~€| šU¢ ¿ £ ^ÛĤ~€| ^ÛĤ„ÓƒJøA0œc2Œf½~€| «1«1žp¡Ÿ«v~€| ( E
The question of finding the minimum size of an -superconcentrator with a
5 Pippenger
given depth d was raised by
(1982, [71]). Let us denote this function
a
by )tSd . Clearly )) hí . In the same paper, Pippenger showed that
¥ `~J|¶,“¦)tS-_
`~€| a , . Dolev, Dwork, Pippenger and Wigderson
(1983, [29]) found that for even depth at least Ž ,
(6)
)tS-|d¶¨§ž`~€| š ¢ ¿ Tt £ «
Rather surprisingly, Pudlák (1994, [76]) showed that for dÎ2Š- , it is also true that
7¤5
)tS-|d ¶©§ ž `~€| š ¢ ¿ Rt £ «
(7)
333
Ž
Hence, when the depth is at least the extra “odd” level does not help improve
the superconcentrator size. Alon and Pudlák (1994, [10]) filled part of the gap by
determining the minimum size for depth , proving
{
)){_¶©§`~J|¶`~€|¶,
)tS-_> ¥ ž 8~€|8,«ªÁ -
(8)
))Sd
They also improved Pippenger’s lower bound for depth superconcentrators to
. The only one case left where
has not been determined (up to a constant factor) is when
, rather weird. One would think that
the depth- case would be easier than larger depth cases. Finally, Radhakrishnan
and Ta-Shma (2000, [77]) have determined the last value:
d b-
4 ` ~€| a )tS-_8©§ ~J|8~€|8 ;
(9)
-5
)tS-_
We describe here only the initial results of Pippenger on
to get a feel
of what is going on. In superconcentrators of depth , every path has length at
most . By adding a vertex into every path of length , we increase the size of
the superconcentrator by at most a factor of , which is irrelevant for our purposes.
Hence, we may assume that the set of vertices of our -superconcentrator can be
partitioned into three disjoint subset
, where is the set of middle
vertices called links.
Let
be an -superconcentrator of size
and depth . Let
( ) be a
random set of inputs (outputs) where each
(
) appears independently
with probability . Let the random variable (resp. ) denote the cardinality of
(resp. ). As is a superconcentrator, there exists a set of
vertex
disjoint paths joining and . We first get a lower bound for
.
-
®
®
/
%
’
’b·¬*ˆZ­*
Z
š%
- % ˜Yj ˜¡j
%
Ê Ì
W¤„ƒJøAʽXÌxE
” Wѕ
7
Lemma 4.10. With the notations just introduced, we have
Δ Wѕ62*J/ ž J/ ªÁ 5
Proof. Applying Markov’s inequality, with jõ” f3 • to be chosen, we get
5:9
5Æ9
” WѕÕ2 J/c 5:9 vŒ” W2*J/c 5 9 F•
59
J/c 5:9 vŒa” Ê¡2ŒJ/c 5:9 vXÌ 2*J/U F•
J/c vŒ ” ÊY2ŒJ/c F•
334
(10)
Ê
59
” ÊN•6gJ/ and ’`»R¯” ʕLgJ/c / .
5:9
9
Œ” ÊqÇ*J/c vF•¦( Œ”J Ê J/Ut2ŒJ/C´•
( ’"J5»s/C¯v” ÊNa •
( J/C a
(11)
a t
Combining inequalities (10) and (11), then set >ܞ nS° ª gives
5 9 465:9 5 a
” WѕÕ2 J/c v J/o a ;
5 9 4 5:9 2 J/c v J/o a ;
4 5:9 - t a
J/ J/ ª ;
465:9 - t
2 J/ 7 -‰ J/ ª ;
J/ L±_J/L ª:Á ²
As is the sum of random indicators,
Now, Chebyshev’s inequality gives
ßï
ª
ª\j³Z
For each
, let
and
respectively. Then clearly
´
ª
—ï
be the number of edges directed into and out of
Ã
7
ï
š ï Àß — ï µã^´
%
We say that a link is useful if it is on a path from an input of to an output of
. Let be the set of useful links and
. The useful links are “useful” in
the proof of the following theorem.
Theorem 4.11.
7
š 2 { `~J|8 Ó
,
™
335
5Æ9 5Æ9
%
ï/.
Proof. The probability that a link ª is connected from is
L
/
·
(
ß
é
¶
Similarly, the probability that ª is connected to is (җ ï / . Hence,
5 a
5 7 a a
ï
ï
Œ” ªYje´ •6(Š„ÓƒJøxA Sß — / E`(Œ„ÓƒJøA ¾Àß ï — ï / +HŽ3E
This implies
Ã
Ã
5 7 aa
Δ µ.• ï °:g Œ” ªYj·´Æ•6( ï °:g „ƒJøA ¾Àß ï — ï / +HŽ3E
As it is trivial that ” µ.•c2Œ ” Wѕ , we have
Ã
7
5 7 a a
Á
J/ ¡žJ/ ª (ŒÎ” Wѕ½(ŒÎ” µ.•,( ï °:g „ÓƒJøxA ¾Àß ï — ï / +HŽ3E
uC
To this end, let dŒ¹¸~€|>»º , set5 /g[- , multiply both sides of the above
C
equation by - , and then sum it over (ŒT¶(Šd , we get
7
ß Ã
7 a uC
C
`~€|> Ó,0( CJI,K ï „Óƒ…øNAH- ¾Àß ï — ï - +HŽ3E
(12)
7
7 9
5
7
7
ï
ï
ï
ï Â , then
and ,?Ü~€|Àß
For a fixed ª·j¨Z , let Âh¼¸~€|Àß
—
½
º
—
v
¼
R
K
u
f(¾,0Ç , and ß ï — ï ·- – . As the function - – - – is convex, we get
ß
¼Ã C 7 à Ÿ a ¼vR a u C
7 a uC
C
ï
ï
C€I,K „Óƒ…øAH- ¾Àß — - +HŽ3E ( CJI,K -7 CJI‰¼KR,u K - 7 –
( Àß ï 7 — ï ¸À- – - – ( {‰Àß ï — ï (13)
Consequently, equations (12) and (13) yield
à { 7
7
`~€|¶ Ó,>( ï - Àß ï — ï 8 {- š
4.3 Explicit Constructions and other results
We have seen how to use bounded concentrators to construct superconcentrators.
Moreover, it is possible to construct bounded concentrators from expanders, as the
following lemma shows. Consequently, explicit constructions of expanders induce
explicit constructions of superconcentrators.
336
5
/ie
õ
°°R,K
§
\
j
l
|
@|Sd °|ua K 7 7¤587
À-|d {_¿/ >OÕf 7^5 YOVP
7^5
§
7
5
5
Proof. Firstly, for each divisible by À/
such that J/+3À/ is even, we con +.-_ -bounded concentrator g¢N
“ . 7¢By5 definition,
struct an ‰¾Àd
has inputs and | outputs. The inputs are partitioned into
a large
9·,5 +3twoÀ/ parts: vertices.
part Ú containing | vertices and a small part ] containing
X -expander.
Each
The large part is connected to the output by a @|SdS-_+3À/
b
7
5
5
vertex in ] is connected to exactly / consecutive outputs, so that the neighbors of
‰% & +.-_ vertices in ] are completely disjoint. To show that is an ‰¾Àd
%
%
bounded concentrator, let be any subset of with 3(Å +.- . Let œ6r
%& ]Æ . We need to show that /> % 1:2Õ % . When )h2Õ % +/ , Úo] and )qÖ
%
is connected to at least outputs, so that the inequality holds trivially. When
)"ÇÅ % +/ , we must9Œhave
5 œc2¾¯"“9*‘ °.5 u° K % ’ . Sincea 9Œ % 5 )(Å +.5 - , we have
¯"ÂÁ / / % Ãh(ÄÁ / -U/ ÃYÂÁ / -U/ a Ú# Ã^( - Úo
9Z5 of Ú with ¯ elements we can now
Thus, let å be a subset
use the fact that Ú¨
"
is an @_SdS-_+3À/
X -expander to get />å"1=2Ö % . The simple details are
omitted.
Secondly, given any , let7\ 5 | be 5 the smallest integer eÕ such that _ | is
even and construct an | ‰¾Àd
‰ +.-_ -bounded9 concentrator as above. Delete5
7
Z
7
5
5
vertices, turning it into an
from the small part of this concentrator’s inputs Å|
n ¾Àd ‰ +.-_ -bounded concentrator, where n ã
nÆ n un Ç?‰ .
Lemma 4.12 (Gabber-Galil, 1981 [34]). Let
be a fixed integer, and
. Suppose for each
such that
is an even integer we can construct an
-expander, then we can construct a family of superconcentrators with
density
, where
as
.
Lemma 4.5 now completes the proof.
(5
5
5-_{ «]ƒ
„.f 5
Gabber and Galil used this lemma and their explicitly constructed expanders
to construct families of superconcentrators with density
. This was improved
and later by Buck (1986, [22]) to
. Jimbo and Maruoka
by Chung to
(1987, [44]) and Alon, Galil and Milman (1987, [8]) improved Gabber-Galil expanders slightly to obtained superconcentrators of density
. The Ramanujan graphs allow Lubotzky et al. to reduce this to . Pippenger pointed out
that density
is possible using double covers of -regular Ramanujan graphs,
whose explicit constructions could be found in Morgenstern (1994, [57]). Morgenstern (1995, [59]) also explicitly constructed a family of bounded concentrators not using expanders. This construction yields density . All linear superconcentrators constructed above have logarithmic depths. Wigderson and Zuckerman (1999, [88]) constructed a linear-sized superconcentrator with sub-logarithmic
depth:
.
- «®p
(Ž
ƒ
{ƒ
-|-)«Ä{HŽ
(:(
~€|8, a c y RoE ³ K µ
337
( Ž|
-.f.
Open Problem 4.13. The probabilistic bound of
remains quite far apart from
the best explicit construction bound of
. Thus, an open question is to construct
-superconcentrators of size
, and as close to
as possible.
Ç ( Ž|
-.f.
As for limited depth,7 Meshulam
[55]) constructed depth - supercony c a Ó K cK a ,(1984,
centrators of size {B
while Wigderson and Zuckerman [88] constructed one with size š , . Since connectors (defined in the next section) are
also superconcentrators, the explicit constructions of -connectors of depths d
yield explicit
For depth
75 constructions of -superconcentrators of the sameKMR depth.
t
J
t
d¢Ö-1Â , -connectors were
constructed with size Ó
£ , and depth
¢
È
Á
M
K
R
T
t
d¡Z-1 , ¡2b- with size Ó ª È . More information can be found in the next
section.
It is worthwhile to mention that there are several variations of concentrators
and superconcentrators, which are also models for different types of switching networks. These include self-routing superconcentrators [73], partial concentrators
[41, 42, 38], and natural bounded concentrators [59].
5
Connectors
In this section, We discuss the graphical models for rearrangeable, strictly nonblocking and wide-sense nonblocking networks.
É
T ³ÀˏÉ:ÌUÍT¹
T"jŠ
É
T Ÿ OÊ? Ÿ R‰¼ n|µ R,K
 Definition 5.2. An generalized -connector is an -network so that: given any
one-to-many correspondence Î between inputs and disjoint sets of outputs, there
exists a set of vertex disjoint trees joining T to the set ÎUT¹ for all T with ÎUT¹ defined.
Definition 5.3. An generalized -concentrator is an -network satisfying the condition that given any correspondence between inputs a nonnegative integers summing up to at most , there exists a set of vertex disjoint trees that join each input
Definition 5.1. An -connector is an -network with inputs and outputs so
that for any one-to-one correspondence between and , there exist a set of
vertex disjoint paths joining to
for each input
. When is restricted
to those of the form
, for some , the -connector is called an
-shifter.
to the corresponding number of distinct outputs.
5
Generalized concentrators used to be widely referred to as generalizer. Here
we adopt this terminology from [31] because it is consistent with our overall use
of terminologies. Clearly when each corresponding integer is or , generalized
concentrators are concentrators.
f
338
The -connectors are the graphical version of a rearrangeable networks, while
generalized -connectors are the version for multicast rearrangeable networks. We
shall also use the term rearrangeable -connector for -connector, to emphasize the difference of this network with strictly nonblocking connector and widesense nonblocking connector, which are to be defined shortly. Rearrangeable connector used to be called rearrangeable -network. As the names suggested,
these connectors are graphical versions of strictly nonblocking networks and widesense nonblocking networks, respectively.
9 9
ATXE A0?tE
T=jh ?Îjõ
Definition 5.4. A strictly non-blocking -connector (SNB -connector) is an ,
and a
connector with input set and output set , such that for any
set of vertex disjoint paths from
to
, there is a path from to
which is vertex disjoint from .
Ï
Ï
T ?
To formally define wide-sense nonblockingness (WSNBness), we need to settle
several other technical concepts, whose counter parts in switching network are
clear from the names.
Let be a network with input set and output set . A route in is a directed
path from an input to an output. Two routes are compatible if they share only an
initial segment, which could be empty. A state of is a set of pairwise compatible
routes. The set of all states of
could be partially ordered by inclusion. Hence,
we can speak of a state
being contained in a state . A vertex or an edge in a
state is busy if it is in some route of the state, and idle otherwise.
in
. A connection request
is
A connection request is an element
said to be satisfied by a route if originates from and ends at . A generalized
connection assignment (GCA) is a set of connection requests, whose outputs are
disjoint. A GCA is realized by a state if each request of the GCA is satisfied by
some route of the state.
A GCA is
-limited if it contains at most requests, of which at most
have a common input. A state is
-limited if the maximal GCA it realizes is
-limited. Often we speak of the maximal GCA realized by a state as the
GCA realized by . A connection request
is
-limited in a state if
is idle, and the GCA obtained by adjoining
into the GCA realized by is
-limited.
Now we are ready to place the formal definition of WSNBness.
®
]cK
»xSß6
»xSß6
Ð
T
T?_
?
»
»xSß6
]
®
]a
T?_ c¥0
å å
»xSß6
®
®
»xSß6
T?_ »xSß6
T?_
]
ß
] ?
]
Definition 5.5. A WSNB
-limited generalized connector is a network for
which there exists a collection of states called the safe states, such that:
(i) The empty state
(ii) If
Ð
is in in .
]œjÑÐ , then any state contained in ]
339
Ð
is also in .
]jIÐ and any »xSß6 -limited connection request T?_ in ] , there
]
|j€Ð such that ]”|oҐ] and that 5 ]
| has a route satisfying T?_ .
A WSNB » -limited connector is a WSNB »x -limited generalized connector. A
WSBN connector is a P -limited connector. The prefixes “ XWY -” and “ -” could
(iii) Given
exists
be appended with the obvious meaning.
5.1 Rearrangeable connectors
As we have mentioned, connectors are models for rearrangeable networks. We
put “rearrangeable” in front of “connectors” to emphasize the difference with SNB
connectors and WSNB connectors. Let
denote the minimum size of an be the minimum size of -connectors with depth . Pipconnector, and
penger and Valiant (1976, [74]) showed that
|,
|Sd
d
7
(
{B`~€| y ^(Œ|,0( `~€| y ,
7
(
Pippenger (1980, [70]) improved the lower bound of |, to be |,02
~€| Ó, , and adopted a comment from the referee7 James Shearer of his paper to get
(14)
.,:2 Ž}{ p `~€| ,
5 a
which is the best lower bound known so far for ., .
In the case of connectors with a given depth, it is clear that | > . When
dÎb- , de Bruijn, Erdős and Spencer (1974, [28]), while solving a problemÁ of van
Lint (1973, [86]), used a probabilistic argument to show |S-_ Z
Ó¬ª ‹ ~€|¶, .
Pippenger and Yao (1982, [75]) used an argument from Pippenger and Valiant
(1976, [74]) to show
|Sd¶ ¥ KMR ¿t and another probabilistic argument to prove
|Sd¶g
KMR ¿t ~€|>, ¿t (15)
(16)
which implies the results by de Bruijn et al.
In this section, we shall present the proofs of the two relations (14) and (15).
First, we need a famous result which was conjectured by Minc (1963, [56]) and
proved by Brègman (1973, [20]). Let be a -square matrix of order , and
the symmetric group of order as usual. The permanent of : per is defined to
be
›
f
5
à Õn
per ·
› Ó °Ô CJI,K ›D C Ó ³ C µ
ü
340
›
›
]n
5 Theorem, 1973). Let ›
Theorem 5.6 (Brègman-Minc
1«1«1«HX . Then
where row T sums to ¯¾C , TU
n Ötr
Õ
per ݁(
CJI,K Y¯C½‹Ä
be an
i¥Š¤f
5
-matrix
7
}
Ž
p
|,>2 { `~€| Ó,
Proof. Let Q N
“’¶ be an -connector with input set T , output set ,
vertex set ’ and edge set . We can assume each ˜jŒ has indegree at least
- and outdegree at least - . Now, if some vertex ˜ has two inarcs ª K ˜ , ªxa¸˜ and outarcs ˜RXoK and ˜RX a , then we could delete ˜ and add edges ª,KXoK , ª6K#X a , ª a XoK and
ª a X a without changing the rearrangeability of the graph and without9 increasing
¨©Î
has
the number of edges. Consequently, we may assume each vertex in ’
total degree at least p . Let † be any one to one correspondence between and .
Let | ¢!’ | | be the graph obtained from by9 gluing together ˜ and †¶˜3 for
every ˜Îj¡ , and add a loop to every vertex ˜ÎjY’
=©
. Let J2 ˜‰ be the total
degree of vertex ˜ in , then
5 Ã
7 Ã
w
Î_ - Ù °×vsO T2 ˜3 Ù:°c ×UvsO R2 ˜‰ y
and,
5 Ã
7 Ã
7 9
w
|  - Ù °«×vRO R2 ˜‰ Ùq°:c ×vRO R2:˜3 -’ #© y
5 Ã
5 Ã
7
Ã
7
( - w Ù °«×vRO R2 ˜‰ Ùq°:c ×vRO R2:˜3 y p Ùq°:c ×vRO R2 ˜3
Ã
7 Ã
5
{
w
( f Ù °«×vRO T20˜‰ Ùq°:c ×vRO R2 ˜3 y
p{ Ñ
É
Let be the set of cycle decompositions of | , then since is rearrangeable,
É
É
each to matching results in a different cycle decomposition in . Hence t2
Theorem 5.7 (Pippenger, 1980 [70]).
341
5
É
ï
‹ . Clearly, per › where › ï is the adjacency matrix of | , i.e. › Ù iff
ª½X˜‰:j¡›| and f otherwise. Let ¯ be thea sum of row ª of › , then ÷ Ù ¯Ùor 5|F .
It is easy to see that the function ~J|ÊŋÄX+HÊ over the positive integers is decreasing
a
if Êr2{ and increasing when Êr({ , namely ~€|xÊŋÄX+HÊ gets its maximum at
ÊÑ{ . Theorem 5.6 gives
Ã
à ~€|xY¯ Ù ‹Ä
a
{
|  Ù ¯Ù"2 ~J| ( Ù °: Æ ¯Ù
à ~€| Y¯Ùq‹Ä
„ Ù ° Æ ¯¾Ù
2 „U~J| per ›o«
Thus,
ÑÍ2 p{ | 2 Ž}{ p ~J| per ›D
Ž}{ p ~J| É 2 Ž}{ p ~J| ‹Ä 7
2 Ž}{ p `~J| Ó,
7
It should be noted that using the same trick for the case where
gives a lower bound for the size of an undirected -connector:
5
is undirected
Ñ)2 p ƒ ` ~€| , Ó
,
Theorem 5.8 (Pippenger-Yao, 1982 [75]). An -shifter of depth d has at least
d) KMR ¿t edges.
Proof. Let Ø , be a directed rooted tree with leaves and depth d where all
Ÿ
edges directed to the direction of the leaves. Let Œ>K1«1«1«¾Œ n be the paths from
the root to the leaves of Ø , . Let
Ÿ
Ãn Ã
ÙØ Ÿ ,X>ÛÄ ¼vI,K Ù °«Ú ?Bª%,´t–1—˜3
È
342
ÙØ a Ÿ ,X02Šd) KMR ¿t by5 induction on d .
As ¡ÙØ K ,X"Ë , the case d§
is trivial. For dœ2- , supposed the root
C
has degree 5 , which is connected to subtrees Ø u K LC! , where the tree ØLC has LC
Ÿ
KMR ¿ Rt t is convex in Ê , we have
leaves, for (*T¶(Œ . Then, since the function Ê
We first show that
Ù Ø Ÿ ,Xí } 2 }
2 }
2 }
7 Ã" C
C7 ÀI," K Ù9ŠØ Ÿ 5 u K LC!X
KMR ¿ Tt t
À
d
M
C
C€I,K
7 9Œ5 w ÷ C€" I,K LC KMR ¿ Tt t
Àd ¹ y
7 9Œ5 KMR Tt t
xÀd Û± ² ¿ «
Lastly, straightforward calculus completes the induction:
7
9Œ5 KMR Rt t KMR t
_ Àd Û± ² ¿ 2 d) ¿ «
5
Now, let be an -shifter. By definition, for each Âh
1«1«1«¾X there are7 g
7
5
5
vertex disjoint paths Œ C¼ joining each input T8j¡ to output ?“jq
, where ?`T Â
„J‡, . Fix T , vary  from to and assemble the ŒcC ¼ into a tree, keeping
only the initial common segments. Call the resulting tree ؽC , then ØC has leaves
KMR ¿t . Let
and depth d . We thus have ÙØC!02Šd)
5
– is an arc from a node of ŒcC ¼
‚8TFÂ_S–HÆÛÄ f ifotherwise
then
Ã
Ã
Ü °.ð ‚8TFÂ|S–B 2 Ù °«Ú:r È î rS˜3
343
˜ along Œ C ¼ got split in Ø C . Consequently,
Ãn Ãn Ã
Ãn Ãn Ã
CJI,K ¼vI,K Ü °.ð ‚8TFÂ|S–B 2 ÃCJI,n K ¼I,K Ù °Ú:r È î r ˜3
CJI,K ÙØ C 2 d) a R ¿t
with strict inequality when some
Lastly, as
This implies
Œ>K ¼_1«1«1«¾Œ n ¼ are vertex disjoint. Thus,
Ãn
5
CJI,K ‚8TFÂ_S–H:(
is rearrangeable,
ad) R ¿t ( à n à n à ‚8TSFÂ|S–HÆ( à n à 5 (*> Î
€C I,K ¼vI,K Ü °Bð
¼v,I K Ü °.ð
which completes the proof.
Corollary 5.9.
|Sd¶ ¥ KMR ¿t Proof. Any -connector is an -shifter
5.2 Non-blocking connectors
d
XSd
,¸Sd
In this section, we discuss results on SNB and WSNB connectors. Let
be
the minimum size of a SNB -connector of depth . Let
be the minimum
size of a WSNB -connector of depth . When there is no limitation on the depth,
we use
and
respectively.
The multistage Clos network (see [16]) is strictly nonblocking, and thus it gives
an upper bound of
for
and
,
. In other words,
the extended Clos network gives
,1,
d
X“,
9Œ5 5
t
M
K
R
Ó ¿ 1, S-|d ,1S-|d d2
,¸Sd¶i
Ó KMR,K½c«Ý ¿ JÁßt Þ «
344
(17)
,1Sd"2r.SdD ¥ KMR ¿t d
,1Sd¶ ¥ KMR ¿ Tt t (18)
closing the gap when d\-){ . Concerning ,¸, , Pippenger (1978, [69]) showed
by a probabilistic argument that
,¸,:(„.f.`~€| y «
(19)
:
(
(
improving a previous bound of
~€|8 by Bassalygo¥ and Pinsker (1973, [14]).
`~€|8, . The best lower
Shannon (1950, [80]) was the first to show that |,¶
bound for ., was shown in Theorem 5.7, which implies
7
}
Ž
p
,¸, { `~€| Ó,
(20)
¥ KMR ¿ Rt t and upOpen Problem 5.10. Close the gap between the lower bound KMR,K½c«Ý ¿ zÁ t Þ of ,1Sd when dÑ2ŒŽ .
per bound Ó
A SNB network is also a WSNB network, hence X,=(,1, and X“Sd#(
,1Sd . A WSNB network is rearrangeable, hence X“,“2$|, and X“Sd2
|Sd . Thus, we already had
(21)
X“Sd¶ ¥ KMR ¿t «
Feldman, Friedman, and Pippenger (1988, [31]) showed that a WSNB generalKMR,K½c Ÿ ~€|>, K u K½c Ÿ ,
ized connector with a fixed depth d exists, whose size is Ó
namely
X“Sd8g
KMR ¿t ~J|8, K u ¿t (22)
¥ KMR ¿t and upper
Open Problem 5.11. Close the gap between the lower bound KMR ¿t ~€|>, K u ¿t of X“Sd .
bound Moreover, a SNB -connector is certainly a rearrangeable -connector, thus the
result of Pippenger and Yao mentioned in Corollary 5.9 of the previous section
implies that
. Friedman (1988, [32]) gave the only
other known lower bound on SNB -connector of a given depth :
In this section, we give an improved version of the result by Friedman, who
showed relation (18).
Let
be a SNB -connector. Assume that can be partitioned
into stages
, by adding more edges if necessary, as doing so
would not increase the lower bound. For any vertex
, let
(
) be
the set of vertices in
(
) which are connected to . Also, define
and
.
Let us first gives a lower bound of
when
.
b¢
’“
’xs ·’LK1«1«1«¾’ Ÿ g
’NC u K ’CJR,K
Ø ç ˜31 à ˜3:ÛÄ Øáà ˜‰1
Ñ
345
’
˜Îj\’ C Ø ç ˜3 Øáà ˜‰
˜
ç ˜3¨ÛÄ
dÓg-
T¶Ã j¡ , 4 ??j\5 , let7 ’ CFE 5 ÛÄ Øáà T¹58 7 &ÑØ5 ç @?_ , then
Ù °:7rãâ ç ˜3 à ˜3.; 2 5

}
Proof. Assume ’CFEÆbA˜)K1«1«1«X˜ E . Let and ä be two rearrangements of A 1«1«1«HXW\E
such that
ç ˜:å t >(Œ ç ˜:å Á 0(g×1×1×(Š ç ˜å7æ and
à ˜ç t >(Œ à ˜ç Á >(g×1×1×x(Œ à ˜ç æ «
5 ç à ˜:å È 0(õ or 7r à 5 ˜ç È 0(
We first claim that there is a ÂÓ(ŒW such that 7r
either
5 for contradiction that ç ˜ å È 2Í or ˜ ç È 2Í for all
 . Assume
¡ 1«1«1«¾XW . As the request TS?| must be routed7Z5through one of ˜‰K17Z«1«15«¾X˜ } ,} if
we could find a state of not involving T and ? which uses all vertices ˜K1«1«1«X˜ ,
5 reach a contradiction. As ç ˜:å È 2 or à ˜ç È Ó2r for all
then we
º9 9 1«1«1«¾XW , it is easy to find a matching from some subset ATå t 1«1«1«XTGå7æ=E of
ATXE onto ’CFE and another matching from ’NCFE onto some subset A0?:å t 1«1«1«¾?å æ E
of A0?}E . The set of paths Œ¼ÎTGå È O ˜å È Oè?å È use up all vertices in ’NCéE ,
contradicting the fact that is SNB.
5 let  be5 a number5 (¢à W such5 that ç ˜:å È ?(i5  or à ˜ç È `(i . Notice
Now,
that +B ˜.CÀ05 2
à ç 4 7+H and5 +B ˜.CF>24 +H 5 , for7 all T 7 1«1«15 «¾X . Thus,
7 W 9Â7
Ù °7r]â ç ˜‰ à ˜3 ; 2 4 ç 5 ˜å t 7 ×1×1× 7 ç 5 ˜:å È ; 7 9
à ˜ç t ×1×1× à ˜ç .9 ; W Â
7 Â 7 -BW È -1Â
Â
2 ˜:å à 9 ˜ç 50ç 7 Â È 7 -BW -1 È
2 507 5 (23)
2 Theorem 5.13. Let ¦QN
“’¶ 7 be a strictly non-blocking -connector of
a . Moreover,7 there exists a strictly nondepth - , then has at size at least a .
blocking -connector of depth - and size exactly Lemma 5.12. For any
346
Proof. Given the lemma above, the proof of this theorem is straight forward, as
shown below.
à à 4 5 7 5
4a 587 5
¡
; ( C à °× ê E °4 O Ù °:5 7rãâ 7 ç ˜35 à ˜3 ;
Ù ° 5 ˜3 à 5 ˜‰.; ®AtTS?| ˜Îjq’ CFE E)
à t4 ç 7
Ù ° ˜3 à ˜‰.; ç ˜‰¹ à ˜3
ç
t
Ñ
7
a
We could construct a SNB -connector of depth - and size by setting
’Ns_}’LKB_$’ a _· , connecting ’Ns to ’LK by a perfect matching
using edges,
a
and connecting every vertex in ’LK to every vertex in ’ a using more edges. The
resulting graph is clearly a SNB -connector.
This line of reasoning could be extended to find a lower bound for ,1Sd with
dÎ2{ .
Lemma 5.14. For any pair T>j¡ and ?jq
, let Ï be the set of all paths from T to
? , and let
› CFE ÛÄ A˜Îjq’ K & ’& YÏ?SE
‘=CFE ÛÄ A˜Îjq’ Ÿ u K ’ YÏ?SE
Then,
à 5 7 à 5 507 5
Ù ° ­ rãâ ç ˜3 Ù °êNrãâ à ˜3 2 «
Proof. Let WV › CFE and W | 㠑 CFE . Suppose › CFE 5 ÍA˜ K 1«1«1«HX˜ } E 5 and ‘ CFE Aª½K1«1«1«Xª } Æ E . Let  and ä be two permutations onà A 1«1«1«BXW\E and A 1«1«1«HXWˆ|…E ,
respectively, such that the arrays A¾ ˜:å r SE and A¾ ªNç r SE are weakly increasing.
ç
Similar to Lemma 5.12, we claim that there is either a Â(ŒW such that ˜:å 0(
à
È
 , or a  | (·W | such that@ 9 ˜ ç È Æ =(ŒÂ | . Assume otherwise, then there areç distinct
9 be matched one to one onto › CFE ,
vertices AT K 1«1«1«HXT } E
could
A}TE which
@
A0?tE which could be matched one to
and distinct vertices A0?)K1«1«1«¾? Æ E
one onto ‘ CFE . Now, connect › CFE to ‘ CFE by a maximal set of vertex disjoint paths
ë . Make ë a set of vertex disjoint paths from to by adjoining ATK1«1«1«¾XT } E
a7
347
A0? K 1«1«1«¾? ë } Æ E as possible. Clearly, there does not exist a new path from T to ?
. Without loss of generality, we assume there is a ÂÎ(W such that
ç ˜:5 å È >(õ . We have5
Ã
4 5 7 7 5 7 WÑ|
7 Ã
Ù ° ­ r]â ç ˜3 ï °êNrãâ à ªL 2 ç ˜:å t 7 ×1×1× ç ˜å È B; 2  ˜:5 å WÑ |
58ç 7 È
2 (24)
and
disjoint from
such that
dÎ2{
Theorem 5.15. Let
depth
, then
¦QN
“’¶ be a strictly non-blocking -connector of
5 7
Î32 ƒ 8 -_ ¢ ¿ Tt t £
Proof. Let
› ÄÛ A˜Îjq’r¾ ç ˜302 Žx Ñ E
‘ ÛÄ A˜Îjq’r¾ à ˜302 Žx Ñ E , and
’ | ÛÄ ›*©Î‘
à
n
n
Since ÷ Ù ˜3 [÷ Ù ˜‰V Ñ , ›0( z and ‘:( z , which imply
’›|Àt( na . ç
Let Ï be a maximal set of vertex disjoint paths from to such that each path
Œ¢j€Ï hits at least9 one member of ’›| .
Let ’Ö
3 ÛÄ'’ ’ | ©õ’YÏ? , then the induced subgraph of on ’ 3 : 8 «N3 ;
3 C’3 3 is strictly non-blocking because any path from to which is vertex
disjoint from Ï does not hit ’5| due to the maximality of Ï . Also note that 3 z and
3 each has at least na vertices, and that ç and à of each ˜Ñjì’ 3 are at most ¬n ð ¬ .
We assume 8 and 8 both have size exactly ,+.- , removing some vertices in 3
&
and/or 3 if necessary. Let ’‰3 C½g’C ’ 3 , and for any pair T8j© 3 and ?“j1
3 , let
›í3 E ÛÄ set of vertices in ’L3 K which can reach ?
‘#3 CVÛÄ set of vertices in ’ 3 Ÿ u K reachable from T
348
s› 3 E ½(܎x Î+H, Ÿ u K ‘ 3 C ½(܎x Î+H, Ÿ u K . We also define › 3 CFE
‘#3 CFE
5 7 Ã 5
0
5
7
Ã
Ã
a
± - ² - (
C ° × 8 ê E ° O 8 Ù ° ­ 8 rãâ ç ˜‰ Ù ° ê%8 r]â à ˜35 à Ã
7
E ° O 8 Ù ° ­ 8 â # T ’s connected to ˜3 ˜3 ç
5
à Ã
Cà °C× 8 Ù °9ê%87*r #à ? ’s connected to ˜‰ à ˜3 E ° O 8 q›¬3 E_ C ° × 8 ‘#3 CS
( Žx Î+H, Ÿ u K
This inequality and dÎ2Q{ imply
545 t 7
5 7
R
t
Ñ)2 Ž Ž ; ¿ 8 -_ ¿ Tt t 2 ƒ 8 -_ ¿ Tt t
Then, clearly
and
and
in the sense of Lemma 5.14, then by Lemma 5.14
5
Î)2 {|- KMR ¢ ¿ Tt t £
Remark 5.16. The original theorem from Friedman gives
‘
Ž
›
There are room for improvement. Firstly the constant in the definition of
and might still be optimized. Secondly, we could do something better than just
considering the second and the next-to-last stage of .
5.3 Generalized connectors and generalized concentrators
—t|, —s,1, —sX“, —,
—Sd
Let
,
,
and
be the minimum sizes of generalized -connectors,
generalized SNB -connectors, generalized WSNB -connectors, and generalized
-concentrators, respectively. Naturally, we also use
,
,
,
and
to denote the minimum size of the graphs have depth- .
As we have mentioned in the previous section, Pippenger and Valiant (1976,
[74]) showed that
—).Sd —s,1Sd —sX“Sd
d
7
(
{B`~€| y ^(Œ|,0( `~€| y ,
349
Y·- Ÿ , Ofman (1965, [62]) implicitly
5 showed
7
—t|,0( f.`~J| a Ó,
While if Y{ Ÿ , Thompson (1977, in
5 a tech report
7 at CMU) showed
—t|,0( -B`~J| y Ó,
Pippenger (1978, [68]) connect the two functions
7 by showing
—t|,¶·., Ó,
5 -concentrators and showed
In this proof, Pippenger needed a bound for generalized
|
5
5
that generalized -concentrators with at most -.f. edges exist. Fan Chung [24]
improved this bound to ƒ)«®pB , more
5|5 precisely
7
(25)
—,:( ƒ)«®pB ~€|¶,
Dolev, Dwork, Pippenger and Wigderson (1983, [29]) probabilistically showed
—t|Sd8g
X`~€|8, KMR ¿t (26)
and constructively proved
9
—t|{1 -_¶g
KMR È t (27)
For
Masson and Jordan (1972, [54]), and Nassimi and Sahni (1982, [60]) gave two
different constructions which proves
—t|{_8g
ïªî (28)
Kirkpatrick, Klawe and Pippenger (1985, [47]) explicitly constructed generalized -connectors to show
—).{_0g
Áª ~J|8, Át (29)
and extended this to
9Œ5
—t|S-|9Åd 5 ¶g
KMR ¿t ~€|>, ¿ RÁ t (30)
9* 5 differs very little from that of the best explicit
The upper bound for —t|À-|d
(see next section).
construction bound for |S-|d
It is easy to see that —t|Sd:(җsX“Sd (*—s,¸Sd . Feldman, Friedman, and
Pippenger (1988, [31]) showed that
—RXSd0·
Ó KMR ¿t ~€|8, K u ¿t (31)
which in effect gives the best upper bound so far for —t|Sd also. There is no
known upper bound for —s,¸Sd .
350
—s,1Sd .
Note also that the corresponding lower bounds of |Sd , X“Sd and ,1Sd
Open Problem 5.17. Find an upper bound for
yield lower bounds for their generalized versions.
5.4 Explicit Constructions
`~€|8,
Rearrangeable -connectors with size
were constructed by Beizer [15],
and rediscovered by Beneš (1965, [16]), Joel (1968, [45]) and Waksman (1968,
[87]). Pippenger (1978, [69]) slightly generalized these constructions to get the
. In the same paper, Pippenger also showed how the clasbound of
sical construction by Slepian, Duguid and LeCorre can be used to construct connectors with depth
and size
. Another construction based
on combinatorial designs by Richards and Hwang (1985, []) gives -connectors of
depth and size
(see also [31]). This construction can be used with the previous construction to construct -connectors of depth and size
.
Clos (1953, [26]), Cantor (1971, [23]) and Pippenger (1978, [69]) showed that
by explicit constructions. That
was
shown by explicit constructions of Cantor [23] and its generalization by Pippenger
[69].
Masson and Jordan (1972, [54]) constructed WSNB generalized -connector
of depth and size
. Pippenger (1973, [66]) and Nassimi and Sahni (1982,
[60]) constructed WSNB generalized connectors with depth
and size
, implying the results by Masson and Jordan. Note that any construction
of SNB network yields also a WSNB network.
Dolev, Dwork, Pippenger and Wigderson (1983, [29]) constructed generalized
-connectors of depth
and size
. Kirkpatrick, Klawe and Pippenger (1985, [47]) explicitly constructed generalized -connectors of depth and
size
, and of depth
and size
.
Feldman, Friedman, and Pippenger (1988, [31]) constructed WSNB generalized -connectors with depth and size
, and with depth and size
. Wigderson and Zuckerman (1999, []) constructed depth WSNB generalized connectors of size
, which is within a factor of
to the optimal
bound of equation (22)).
( `~€| y -
9h5
7g5
Ó c y
À-1Â ,1S-|d ¶g
KMR ¿t {
KMR a cÀ¼ Ó KMR È zt t -1Â
Ó
KMR a c ³ y ¼ u K µ ,1,¶g
8~€|¶, a 9a 7  {1 {
c y Ó ªÁ ~€|>, Át X
K
½
K
c
9
.{|d -_
-
-|d
KMR ¿t RoE ³ K µ
KMR ¿t Ó
Œ9 5
{
MK R ¿t ~€|¶, ¿ TÁ t c y {
d
E ³Kµ
6 Conclusions
In this chapter, we have surveyed studies on the complexity of switching networks,
mostly on the tradeoff between the size and the depth of various types of switching
351
networks. The graph models of different switching networks were investigated: expanders, concentrators, superconcentrators, expanders, rearrangeable, strictly nonblocking and wide-sense non-blocking connectors, plus their generalizations.
Researches on these graphs were thoughroughly collected, with more intuitive
results presented. Many open questions were also specified, which hopefully will
help practitioners new to this field identify research problems.
This research area is certainly very interesting, has deep connection to many
different areas of Mathematics and Computer Science. Algebraic Graph Theory
and Probabilistic Method are the two popular tools which are used to deal with
questions arising from switching networks. Obviously, researches on switching
networks have also enriched techniques and problems in the former two fields.
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