power) which have access to input data from di erent nite sets S1 and S2 to compute ..... claim immediately follows from Lemma 2 and Fermat's Little Theorem. 2.
The \Log Rank" Conjecture for Modular Communication Complexity Christoph Meinel FB IV - Informatik Universitat Trier D{54286 Trier
Stephan Waack Institut fur Numerische und Angewandte Mathematik Georg{August{Universitat Gottingen D{37027 Gottingen Abstract
The \log rank" conjecture consists in the question how exact the deterministic communication complexity of a problem can be determinied in terms of algebraic invarants of the communication matrix of this problem. In the following, we answer this question in the context of modular communication complexity. We show that the modular communication complexity can be exactly characterised in terms of the logarithm of a certain rigidity function of the communication matrix. Thus, we are able to exactly determine the modular communication complexity of several problems, such as, e.g., set disjointness, comparability, and undirected graph connectivity. From the obtained bounds for the modular communication complexity we can conclude exponential lower bounds on the size of depth two circuits having arbitary symmetric gates at the bottom level and a MODm {gate at the top. Area: Computational Complexity Keywords: Communication Protocols, Modular Acception Modes, Depth Two Circuits Having Arbitary Symmetric Gates in the Bottom Level, and a MODm {Gate at the Top, Invertible Incidence Functions, Undirected Graph Connectivity.
Introduction In the basic model of communication complexity the number of bits is investigated which have to be exchanged in order to enable two processors P1 and P2 (with unlimited computational power) which have access to input data from di erent nite sets S1 and S2 to compute functions f : S1 S2 ! f0; 1g. The deterministic communication complexity CC(f ) of such an f is the minimum number of bits the processors have to exchange for the worst case input. This model, rst introduced by Yao [19], has many applications in di erent branches of complexity theory and has been studied in many papers. (See [6] for a survey.) 1
For the communication matrix Mfs ;s = f (s1; s2) of the problem f given in distributed form we have the following inequality, log2 rk Mf CC(f ); 1
2
IR
due to Mehlhorn and Schmidt [8]. The question, which is called the \log rank" conjecture (see [7]), is whether this lower bound is sharp. Considering this conjecture several authors interesting separation results which show gaps between CC (f ) and have fobtained log2 rk M . Recently, Raz and Spieker [12] proved that there is a nonconstant gap. (We review this result shortly in Section 4.) We shall study an analogous problem in the case of modular communication complexity. The investigation of the modular communication complexity is motivated by circuits of nite depth having MODm {gates (see [20], and Section 1, Theorem 1). We succeed in characterizing the modular communication complexity in terms of the logarithm of a certain rigidity function of the communication matrix. The consideration of such regidity functions came into prominence in the last few years. Lower bounds on rigidity functions of explicit matrices proved useful in algebraic circuit complexity theory (see [9], [15]), for branching programs (see [1]) and threshold circuits (see [4]), and for communication complexity (see [11]). Lokam [5] uni es and strengthens many of these results. We adopt and weaken a rigidity function of [4] which we call then rigidity rank of a matrix, abbreviated rrkm M . First, we show that the modular communication complexity MODm -CC(f ) is equal to (log2 rrkmMf ) (see Section 2). In contrast to the variation rank of [4], which could only be computed for the identity matrix, we determine the rigidity rank in many more cases (see Sections 3 and 4). Thus, we can determine exactly the modular communication complexity of the underlying problems (e.g., set disjointness, comparability, undirected graph connectivity). Due to the close connection between modular communication complexity and the size of depth two circuits having arbitary symmetric gates in the bottom level, and a MODm {gate at the top (see Theorem 1), nally, we can conclude some exponential lower bounds on the circuit size for these problems. IR
1 The Computational Model and its Motivation Assume that we are given a function f : S1 S2 ! f0; 1g in distributed form, where the Si are nite sets. A communication protocol P of two processors P1 and P2 of length L computing a function f is, as usual, de ned as follows. In order to compute f (s1; s2), the processor Pi has the element si 2 Si as input, for i = 1; 2. The one processor is not allowed to read the input of the other directly. They have to communicate via a common communication tape. The computation of the whole structure is going on in rounds. Starting with P1, the processors write alternatingly bits on the communication tape. These bits depend on the input available to the processor which is to move and on the bits already written on the communication tape before. If the last bit written onto the communication tape is \1" or \0", the computation is called accepting or rejecting, respectively. Under the usual assumption of pre x{freeness 2
for messages, the co-operative computation can be thought of as to be a Boolean string, the string of the bits communicated. The length of the string is the communication complexity of the computation. Since we consider the worst{case complexity in this paper, we assume without loss of generality that all computations of a protocol which in uence the nal output are of equal length, say L. If the processors are nondeterministic, the outcome of one computation is, of course, not identical to the nal vote. The output of a protocol P , for a given input (s1; s2), depends P;rej only on the numbers MP;acc s ;s and Ms ;s of accepting and rejecting computations performed by the protocol accessing this input. How are these numbers, considered as matrices, related to the length of the protocol? The answer is given by the following 1
2
1
2
Lemma 1 If P is a protocol of length L, and if R is any semiring, then we have the inequalities
rkR(MP;acc) 2L?1; rkR(MP;rej ) 2L?1: The proof can be done in a straightforward way (see, e.g., [2]). As usual, the R{rank of a N1 N2{matrix A over R, which we denote by rkRA, is de ned to be the minimal number K such that A = B C , where B is a N1 K {matrix and C is a K N2{matrix over R. This dependency between the matrices MP;acc and MP;rej and the nal outcome of the protocol is formally given by a function : 2 ! f0; 1g, the counting acception mode. More precisely, if Mfs ;s := f (s1 ; s2) is the matrix associated with the problem, the communication matrix, then P;rej : ; M Mfs ;s = MP;acc s ;s s ;s IN
1
2
1
1
2
2
1
2
A protocol P equipped with a counting acception mode is called a -protocol. The function f : S1 S2 ! f0; 1g computed by a -protocol P is called to be {computed. If a protocol {computing a function f is chosen in such a way that its length is minimal, we de ne the number -CC(f ) := L to be the {communication complexity of the function f . In this paper, we discuss in particular the modular acception modes MODm in which the protocol accepts an input if the number of accepting computations is not equal to 0 modulo m: MODm(n1; n2) = 1 () n1 6 0 (mod m): We study the asymptotic modular communication MODm-CC(f2n) complexity of the sequence (f2n : n n ! f0; 1g)n2 . We shall see later on that it is possible to settle down in a rst step to modes MODm , where m is square free. In order to do so, we need some constructions which are in some sense polynomials of protocols. IN
De nition 1 Let S1 S2 be as before, and let P1; : : :; Pm , and P be protocols of length L1; : : : ; Lm, and L.
3
1. We de ne the product P1 P2 of length L1 + L2 as follows. Given an input (s1; s2) 2 S1 S2, the protocol P1P2 proceeds in the same way as P1 does while the rst L1 bits are being exchanged, and according to P2 then. But we make the following restriction in the last round: For i = 1; 2, if (i) 2 f0; 1gLi are ri{round computations of the (i) 2 f0; 1g are the last bits of (i), and if (1) 6= (2) , protocols Pi on (s1; s2), if last last last then the computation is stopped without result before the round r1 + r2 . (This means, each computation of length L1 + L2 is either a concatenation of two accepting or of two rejecting computations.) 2. For ai 2 , P1a : : : Pmam is de ned on the basis of item 1 in the canonical way. (We assume Pi0 to be the unique protocol of length 0.) IN
1
3. The protocol Pk , for k 2 , of length kL on an input (s1; s2 ) 2 S1 S2 works as follows. It proceeds as P k does but with the following modi cation in the last round: Let f0; 1gL be arbitrarily but xed totally ordered. If (i) 2 f0; 1gL are ri {round computations of the protocol Pi on (s1; s2) such that (1) : : : (k) is a r-round computation of P k , where r = r1 + : : : + rk , then the computation is stopped without result unless (1) < : : : < (k). 4. Let a1; : : :; am bePpositive natural numbers. The sum (a1 P1 + : : : + amPm ) is the protocol of length dlog2 ( mi=1 ai)e +maxfLi j i = 1; : : : ; mg which is de ned on (s1; s2 ) 2 S1 S2 by the following rules: First, the protocols are assumed to be prolonged to the length maxfLi j i = P1; : : : ; mg. Second, processor P1 sheds nondeterministically a number P m m P1 proceed j 2 f1; 2; : : : ; i=1 aig as a message of length dlog2 ( i=1 ai)e. Third, P2 and in theP same way as P1 and P2 do according to protocol Pi , provided that Pi?=11 a + 1 j i=1 a . IN
A straightforward calculation reveals the next lemma.
Lemma 2 The situation is assumed to be as in De nition 1. Then, for all (s1; s2) 2 S1 S2, a a MPs ;s :::Pmm ;acc = Qmi=1 MPs i;s;acc ai ; P;acc P Ms( k;s);acc = Ms ;s ; 1 1 1 2
1
1
1
k
2
2
2
M(sa;sP +:::+amPm );acc = Pmi=1 aiMPs i;s;acc: 1
1 1 2
1
2
We observe that analogous relations for the numbers Ms;rej;s are valid. Now we have to de ne what we mean by reductions. Fortunately, this is much easier here than in machine{based complexity theory. 1
2
De nition 2 Let F = (f2n : n n ! f0; 1g)n2 and G = (g2n : ?n ?n ! f0; 1g)n2 be two decision problems. We say that F is rectangular reducible to G with respect to q, where q : ! is a nondecreasing function, i there are two transformations ln; rn : n ! ?q(n) such that for all n and for all ~x; ~y 2 n we have f2n(~x; ~y) = g2q(n) (ln (~x); rn(~y)). We write F qrec G. IN
IN
IN
4
IN
Clearly, we can utilize rectangular reductions for proving lower bounds on communication complexity in the usual way. One ecient way to get rectangular reductions is to work with projection reductions. In accordance with Skyum and Valiant (see [14]) we de ne.
De nition 3 Let F = (f2n (x1; : : :; x2n))n2 and G = (g2n (y1; : : : ; y2n))n2 be two p sequences of functions given in distributed form f2n; g2n : f0; 1g2n ! f0; 1g. We write F G IN
IN
if there is a 2p(n){projection reduction
= (2n : fy1; : : :; y2p(n)g ! fx1; : : :; x2n; :x1; : : :; :x2n; 0; 1g)n2
IN
(see [14]) which respects the distribution of the variables, 2n(fy1; : : :; yp(n)g) fx1; : : : ; xn; :x1; : : : ; :xn; 0; 1g; 2n(fyp(n)+1; : : :; y2p(n)g) fxn+1; : : : ; x2n; :xn+1 ; : : :; :x2n; 0; 1g :
On balance of this section, let us motivate the modular communication complexity. In fact, we do more. We motivate all counting modes. In the eighties, AC0-circuits with MODm{gates adjoined came into prominence. Let us [ m] denote by AC0 the class of all sequences of Boolean functions computable by polynomially size bounded circuits of this type. We know, due to Razborov [10] and Smolensky [16], that AC[0p] and AC[0q], for p; q di erent prime numbers, are incomparable to each other with respect to inclusion. In contrast to that case, the Sclasses AC[0m], m a composite number of more than one prime divisor, and the class ACC := 1m=2 AC[0m] are not well-understood yet. The only nontrivial bound for the whole class, an upper one, is due to Yao [20] : ACC TC0;3 where TC0;3 is the class of all Boolean functions computable by quasipolynomially bounded (i.e. 2(log n)O ) threshold circuits of depth 3. The aim is to separate ACC from classes like NC1. In order to make one step in this direction, in [4] depth-two circuits with arbitrary symmetric gates in the bottom level and MODm {gates at the top, called (SYMM; MODm){circuits, are considered. An exponential lower bound for the sequence equality function is shown. The following theorem supplies a general lower bound for (SYMM; ){circuits in terms of counting communication complexity. Thereby, a (SYMM; ){circuit is a depth two circuit having arbitrary symmetric gates in the bottom level and a {gate at the top, where : 2 ! f0; 1g. Recall, a {gate for the gate function g (z ; : : : ; z 00 ) is de ned by (#fj j z = j 1 1g; #fj j zj = 0g). (1)
IN
Theorem 1 Let f (x1; : : :; xn; y1; : : :; yn) be a boolean function. Let C2n be a (SYMM; ){ circuit computing f2n . Then
log2(SIZE(C2n)) = (-CC(f2n)): 5
Proof. Let := SIZE(C2n), and let (e1; : : : ; e00 ) be the sequence of wires of the circuit C2n
leading to the top gate. Then we de ne the sequence (g1(z11; : : : ; z10 ); : : : ; g1(z001; : : :; z000 00 )) to be the sequence of gate functions of the source gates of (e1; : : :; e00 ). Let 2(in) : fzij j j = 1; : : : ; i0g ! fxei; yie; 0; 1 j i = 1; : : :; ng, for i = 1; : : : ; 00, be projections resulting from the wires of C2n which lead from the input nodes (to the bottom gates. Then i ) t the functions g~i(x1; : : :; xn; y1; : : : ; yn) are de ned to be gi (2n ) , for i = 1; : : : ; 00 (see De nition 3). We describe a protocol P of length 2dlog2 e + 1, where processor P1 has (x1; : : :; xn) as inputs and P2 the vector (y1; : : : ; yn). Round 1. P1 chooses nondeterministically a number i 2 f1; : : : ; 00g and sheds it, encoded as a message of length dlog2 e. Round 2. P2 sheds the number P1e=0 Pnk=1 yke j(2(in))?1 (yke )j; again encoded as a message of length dlog2 e. Round 3. Since the functions g~i are symmetric and g~i in gi , processor P1 is able to compute g~i(x1; : : : ; xn; y1; : : : ; yn) now. It sheds the result. It is easy to see that the protocol P {computes the function f if and only if the top gate is a {gate. 2 1
( ) 2
2 A Partial Solution of the \Log Rank" Conjecture for Modular Communication Complexity Throughout this section, let f2n denote a function f2n : n n ! f0; 1g, where is a nite alphabet. If m = pl1 : : : plrr is the unique decomposition of the natural number m into powers of pairwise di erent prime numbers, the so{called primary decomposition, then (m), the so called radical, is de ned to be the number p1 : : : pr . This section is aimed at proving Theorem 2 and Corollary 1. This theorem characterizes the modular communication complexity of a function sequence f2n in terms of a rigidity function depending on the communication matrix Mf n . To do so, we adopt and weaken in De nition 4 the concept of variation ranks of communication matrices developed in [4]. (In Section 4 and Section 3, it turns out, that this extends and simpli es the possibilities to calculate the modular communication complexity. This will justify the headline of this section.) Our explanation is going on in two global steps. The rst one is devoted to prove Proposition 1 saying that MODm{protocols and MOD (m){protocols are of equal power. In the second one we introduce our rigidity function \rigidity rank", justify it in Lemma 6, and formulate and prove Theorem 2. Let us turn to the rst global step. We prove rst that MOD (m){protocols can be eciently simulated by MODm {protocols. 1
2
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Lemma 3 If m1jm2, then MODm -CC(f ) MODm -CC(f ) + dlog2 mm e for each function 2
f.
2
1
1
Proof. Let P be the of length L for f . De ne the protocol P 0 of length mMOD m {protocol 0 m L + dlog2 m e to be m P . Then MPs ;s;acc = mm MP;acc s ;s . Consequently, MPs 0;s;acc 0 (mod m2) () MP;acc s ;s 0 (mod m1): 1
2
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2 Next, we show that prime powers are not more powerful than primes.
Lemma 4 Let p be a prime number and l 2 a natural number. Then MODp-CC(f2n) = MODpl -CC(f2n)
for all f2n.
Proof. Again we write f instead of f2n.
The lower bound for MODp-CC(f2n) follows directly from Lemma 3. Let us turn to prove the upper bound. If a 2 , then it is well{known that ! a l a 0 (mod p ) () p 0 (mod pl?1) and a 0 (mod p): Consequently, using the above notations, a 0 (mod pl ) !(i) a () p 0 (mod p); for all i = 0; : : :; l ? 1 0 0 !(i)1p?11 lY ?1 () 1 + (p ? 1) B@1 + (p ? 1) @ ap A CA 0 (mod p): IN
i=0
Let P be a MODpl {protocol of length L for f . Let P~ be the protocol of length 1, where processor P1 sheds 1. De ne P 0 to be 0 0 0 !(l?1)1p?111 0 !(0)1p?11 0 B@P~ + (p ? 1) B@P~ + (p ? 1) @ P A CA : : : B@P~ + (p ? 1) @ P A CACA : p p Then
MsP 0;s;acc 1
2
0 0 P;acc!(i)1p?11 = 1 + (p ? 1) B @1 + (p ? 1) @ Mps ;s A CA i=0 lY ?1
1
7
2
Thus
l P 0 ;acc MP;acc s ;s 0 (mod p ) () Ms ;s 0 (mod p): 1
2
The length of the protocol
P0
1
is as desired.
2
2
The following lemma ensures that Lemma 4 can be applied to prove Proposition 1 in the following way. First, dissect the modulus m into its primary components by Lemma 5, Claim 1. Second, apply Lemma 4 to each of these components. Third, combine the protocols resulting from Lemma 4 by Lemma 5, Claim 2.
Lemma 5
1. If m = m1 : : : mr, where the numbers mi 2 are pairwise coprime, then each f2n has a representation f2n = f2n;1 _ : : : _ f2n;r , such that IN
max fMODmi -CC(f2n;i) j i = 1; : : : ; rg MODm-CC(f2n): 2. Assume that f2n is represented by f2n = f2n;1 _ : : : _ f2n;r . Assume, moreover, that m = m1 : : : mr , where the numbers mi 2 are pairwise coprime. Then IN
MODm -CC(f2n ) = O (max fMODmi -CC(f2n;i) j i = 1; : : :; rg)
Proof. Throughout this proof, we write f instead of f2n and fi instead of f2n;i.
Claim 1. Let P be the protocol that MODm {computes the function f . De ne fi to be the functions MODmi {computed by the protocol P . Clearly, f = f1 _ : : : _ fr . Claim 2. Let Pi , for i = 1; : : : ; r, be MODmi {protocols for fi of length Li. Let e1 ; : : :; er 2 f1; : : :; m ? 1g be representatives of the orthogonal idempotents in =m1 : : : =mr , which are, of course, pairwise di erent. Remember, that for all x; x1; : : :; xr 2 , Z Z
Z Z
Z Z
Z Z
xi x (mod mi), for i = 1; : : :; r () x
r X i=1
xiei (mod m)
We consider the protocol P := (e1P1 + : : : + erPr ) of length & X r ' log2 ei + maxfLi j i = 1; : : : ; rg; i=1
which is, of course, less than
!' & r + 2 max fLi j i = 1; : : :; rg + log2(rm ? 2 ) : It follows from Lemma 2 that, for all (s1; s2) 2 S1 S2, MPs i;s;acc 0 (mod mi) for all i = 1; : : : ; k () fi(s1; s2) = 0 for all i = 1; : : : ; k () MP;acc s ;s 0 (mod m): 1
1
2
2
8
Z Z
Consequently, the protocol P MODm{computes f . 2 Before formulating the next lemma, let us introduce the following notations for a; r 2 . ! a (0) := a r a!(i) !(i+1) a := rr r (i) If P is a protocol of length L, then, of course, we can recursively de ne protocol Pr of length piL such that P;acc !(i) P ) i ;acc M ( s ;s r ; = Ms ;s r IN
( )
1
1
2
2
by De nition 1 and Lemma 2. Proposition 1 follows now.
Proposition 1 Let m > 1 be a natural number. Then
MODm-CC(f2n) = MOD (m)-CC(f2n)
Before de ning our rigidity function, let us justify it.
Lemma 6 Let m = p1 : : : pr , where the pi are pairwise di erent prime numbers. If P is a MODQm{protocol of length L computing the function f , then there is a protocol P 0 of length L ri=1 (pi ? 1) computing the same function which ful lls the following property: For all inputs ; s2) 2 f ?1(1), there index set ; 6= I f1; : : : ; rg such that is a nonempty P 0 ;acc (Ps01;acc 8 2 I Ms ;s 1 (mod p ) and 8 62 I Ms ;s 0 (mod p ) : 1
1
2
2
Qr
of length L Qri=1 (pi ? 1). Then the claim immediately follows from Lemma 2 and Fermat's Little Theorem. 2 Now we are prepared for our crucial de nition.
Proof. We consider the protocol P 0 := P
i=1 (pi?1)
De nition 4
1. Let m be a product p1 : : : pr , where the pi are pairwise di erent prime numbers. Two N N {matrices A and B over the ring of integers are de ned to be modm{equivalent, if and only if, for all entry indices i, j , and all prime number indices k = 1; : : : ; r, aij bij (mod pk ) or aij bij 0 (mod pk ) and
aij 0 (mod m)
() bij 0 (mod m): 9
2. Let A be an integer matrix, and let m > 0 be an arbitrary natural number. We de ne the rigidity rank rrkm (A) to be the minimum of all numbers rk = (m) (B mod (m)) where B is an integer matrix which is mod (m) {equivalent to A. Z Z
Z Z
Theorem 2 Let m > 1 be a natural number. Then
MODm-CC(f2n) = log2 rrkm(M f n ) : 2
Proof. Referring to Proposition 1, we assume without loss of generality that m = (m).
The lower bound is obvious, if we choose the optimal MODm{protocol P 0 in such a way that the conditions of Lemma 6 are ful lled. The upper bound. We choose an integer matrix B which is modm{equivalent to M f n , such that r = rk =m (B mod m) = rrkm (M f n ). Then B = B (1) + : : : + B (r), where the B (k) have =m {rank 1. This is equivalent to Bij(k) Ui(k) Vj(k) (mod m), for Ui(k); Vj(k) 2 f1; : : : ; mg, and for i; j = 1; : : : N . Now we can describe the following protocol P . Assume that the input is (i; j ) 2 S1 S2. First, processor P1 chooses nondeterministically some indices k, 1 k r, and l1, 1 l1 Ui(k), and sheds (k; l1). Second, processor P2 chooses nondeterministically some index l2, 1 l2 Vj(k), and sheds (l2; 1). Clearly, there are Prk=1 Ui(k) Vj(k) Bij (mod m) many accepting computations assigned to the input (i; j ). It follows that the -protocol P computes the function f . Obviously, the length of the protocol is bounded above by log2 r + 2 log2 m + 1. 2 2
Z Z
Z Z
Z Z
2
Z Z
In the case of m being a prime number, we can even do better.
Corollary 1 If m = p is a prime number, we have
MODp-CC(f2n) = log2 rk
Z Z
=pZZ
(M f n ) : 2
3 Application 1: Invertible Incidence Functions A function f : S1 S2 ! f0; 1g, where jS1j = jS2j, is called an invertible incidence function if and only if its communication matrix can be transformed by interchanging rows and columns into a triangular form with the main diagonal elements all being 1. Clearly, the next lemma is crucial.
Lemma 7 Let DN denote an upper or lower triangular N N {matrix over f0; 1g with 1's in the main diagonal, and let m = p1 : : : pr , where the pi are pairwise di erent prime numbers. Then rrkm (DN ) dN=re: 10
Proof. Let A = (aij ) be an integer matrix such that A is modm {equivalent to DN and
rrkm (DN ) = rk =m (A mod m): By de nition we have, for all i, aii 6 0 (mod m); and aij 0 (mod m) for all j < i: Moreover, for all k = 1; : : :; r, and all i = 1; : : :; N , aii 1 (mod pk ) or aii 0 (mod pk ). We conclude that there is an index k0 and a set of indices I f1; 2; : : : ; N g; #I def = N0 dN=re; such that aii 1 (mod pk ) for all i 2 I . After deleting all rows and columns of A whose indices do not belong to I , we get an integer N 0 N 0{matrix D0 which is upper triangular and modpk with 1's on the main diagonal. Obviously, we have rk =m (A mod m) rk =pk (A mod pk ) rk =pk (D0 mod pk ) = N 0 2 The main result of this section, the next theorem, immediately follows from Lemma 7 and Theorem 2. The following corollary makes use of Theorem 1. Z Z
Z Z
0
0
Z Z
Z Z
Z Z
0
0
Z Z
Z Z
0
0
Z Z
Theorem 3 Let m > 1 be a natural number, and let f2n be an invertible incidence function. Then
MODm-CC(f2n) = (n) :
Corollary 2 Let C2n be a (SYMM; MODm){circuit computing an invertible incidence func-
tion f2n . Then SIZE(C2n ) = 2 (n) :
The following functions are, for example, invertible incidence functions. (See Section 4, Remark 1 for more.) - The set disjointness test SDT = (SDT2n : f0; 1g2n ! f0; 1g)n2 , de ned by _n SDT2n (x1; : : :xn ; y1; : : :; yn) def = 1 ? (xi ^ yi) IN
i=1
(see e.g. [6]), - The sequence equality function SEQ = (SEQ2n : f0; 1g2n ! f0; 1g)n2 , de ned by ^n SEQ2n(x1; : : : ; xn; y1; : : : ; yn) = (1 ? ((xi + yi) mod 2)); IN
i=1
- The order function ORDER = (ORDER2n : f0; 1g2n ! f0; 1g)n2 , de ned by n def X (xi ? yi)2i?1 0: ORDER2n(x1; : : : xn; y1; : : : ; yn) = 1 () IN
i=1
Corollary 3 For arbitrary m, we have that MODm -CC(SEQ2n) = MODm-CC(SDT2n) = MODm -CC(ORDER2n ) = (n): 11
4 Application 2: Undirected Graph Connectivity The graph connectivity problem for undirected graphs UCON = (UCONn(n?1))n2 in distributed form can be formulated as follows. Assume that we are given two not necessarily edge-disjoint undirected graphs G1 = (V; E1) and G2 = (V; E2) on a common n n{set of vertices V , where both graphs are represented as Boolean vectors of length 2 . The question is whether or not the graph G def = G1 [ G2 = (V; E1 [ E2) is connected, i.e. each pair of vertices in G is connected. In [17], the major developments in understanding the complexity of the graph connectivity problem in several computational models are surveyed. We pursue the aim to prove the following theorem in this section: IN
Theorem 4 Let m be arbitrary. Then MODm-CC(UCONn(n?1) ) = (n). But rst we review some results and methods which are strongly related to ours. Hajnal, Maass, and Turan proved in [3] the following theorem:
Theorem A CC(UCONn(n?1)) = (n log n). Their method involves the use of the Mobius function for the lattice of partitions of an n{set. Lovasz and Saks extended in [6] and [7] this idea to a large class of problems, the so-called meet problems for nite lattices, which can be formulated as follows. Let S be a nite lattice, and let both processors P1 and P2 be given an element x and y, respectively. Then they have to decide whether x ^ y = 0. More formally, MEETS : S S ! f0; 1g is de ned by MEETS (x; y) = (0; x ^ y).
Theorem B Let MEETS be the meet problem of a nite lattice. Let S have a atoms and b Mobius elements (i.e., elements x such that (0; x) = 6 0). Then log b CC(MEETS ) (log a)(log b): Recently, Raz and Spieker [12] proved
Theorem C If processor P1 as well as processor P2 have a bipartite perfect matching on 2n vertices with two colors of size n as an input, and if their goal is to determine whether the union of the two matchings forms a Hamiltonian cycle, the nondeterministic communication complexity of the problem is (n log log n). Since the problem of Theorem C is a subproblem of UCON , it follows
Corollary D N-CC(UCONn(n?1) ) = (n log log n). 12
Thus the modular acception modes are better than ordinary nondeterminism for detecting undirected graph connectivity. Let us turn to the proof of Theorem 4 now. We proceed as follows. We transform the Mobius function method for proving lower bounds on the length of deterministic protocols to the case of MODm {protocols. Thus we prove the upper bound of Proposition 3. We get the lower bound of Proposition 4 when we reduce the sequence equality function (see Section 3, Corollary 3) to undirected graph connectivity via a polynomial projection reduction. We can only give a very brief treatment on Mobius functions. For more see, e.g., [13]. Let S be a nite partially ordered set. The incidence algebra A(S ) is de ned as follows: Consider the set of functions of two variables f (x; y), for x and y ranging in S , having values in , the eld of real numbers, and with the property that f (x; y) = 0 whenever x 6 y. The sum and the multiplication by scalars are de ned pointwise. The product of f and g is de ned as follows. X (fg)(x; y) def = f (x; z)g(z; y) IR
z2S
Clearly, Kronecker's {function is the 1 of A(S ). The zeta function (x; y) 2 A(S ) is de ned by (x; y) = 1 if x y and (x; y) = 0 otherwise. It is easy to see that the zeta function has in A(S ) a multiplicative inverse, the so{called Mobius function, which has in fact values in . The Mobius Inversion Formula is well{known: Let f (x) be an {valued function, for x ranging in the nite poset S , and let g(x) = Py f (y) (y; x). Then f (x) = Py g(y)(y; x). It is easy to see that (x; y) only depends on the the structure of the interval [x; y], and not on the whole poset P . Moreover, we know that if is the Mobius function of the dual poset S , then (x; y) = (y; x). Let us motivate the notation \invertible incidence function" of Section 3 at this point. Z Z
IR
Remark 1 Let f : S1 S2 ! f0; 1g be an invertible incidence function in the sense of Section 3. Then, of course, there is a poset S , and there are bijections j : Sj ! S , for j = 1; 2, such that f (1 2) is a unit in A(S ). The other way round, let f 2 A(S ) be a {valued unit, and let, moreover, f be naturally represented as a function from S S to f0; 1gk , for some k 2 , by encoding f (S ) as a set of strings in f0; 1gk . If we de ne -CC(f) to be max1k (-CC(f )), where f is the th coordinate function of f , then it results from Theorem 3 that MODm-CC(f ) = (jS j). Z Z
IN
Let us assume from now on that the poset S is a lattice. Let M be a 0{1 matrix. Check whether there are two equal rows or colomns in M and if this is the case, then delete one of them. Do that as long as possible. The resulting matrix M~ is called the core of M . Clearly, any communication complexity of two problems whose communication matrices have the same core is the same. Now it is not dicult to see that the core of MUCONn n? equals the core of MMEETP n , where P (n) is the lattice dual to the lattice of partitions of an n{set. Thus, in order to prove the upper bound of Proposition 3, we can proceed as follows. (
( )
13
1)
First, we show that the MODp{communication complexity of the meet problem for a nite lattice S is equal to the logarithm of a number which might be called the number of Mobius elements mod p of the lattice S (see Proposition 2). Second, we compute this number in the case of S = P (n) (see Lemma 8). We are done with Lemma 3 then.
Proposition 2 Let p be a prime number. Let MEETS be the meet problem of a nite lattice S . Let S have c elements x such that (0; x) 6 0 (mod p). Then MODp-CC(MEETS ) = (log c):
Proof. Let M be the communication matrix of the meet problem assigned to the nite ~ be the diagonal matrix diag((0; x))x2S , which has, as we have already lattice S . Let M stressed, coecients in the ring of integers, and let = ( (x; y))x;y2S be the matrix ~ = M. The claim associated with the zeta function. Wilf observed in [18], that T M Z Z
follows from the Mobius Inversion Formula and from Corollary 1.
2
Lemma 8 Let P (n) be the lattice dual to the lattice P (n) of partitions, let p < n be a prime number, and let be the Mobius function of P (n) . Then #fx 2 P (n)j (0; x) 6 0 (mod p)g pn : Proof. The following three facts can be found in any standard book of combinatorics. Fact 1. If x 2 P (n), and if b(x) is the number of blocks of the partition x, then [x; 1] = P (b(x)). Fact 2. If is the Mobius function of P (n), then (0; 1) = (0; 1) = (?1)n?1(n ? 1)!. of partitions of an n{set into exactly k blocks (Stirling Fact 3. Let S (n; k) denote the number P n numbers of the second kind), then k=0 S (n; k)[X ]k = X n ; where X is an indeterminant and [X ]k = X (X ? 1) : : : (X ? k + 1) is the falling factorial. We get the following sequence of equations. #fx 2 P (n)j (0; x) 6 0 (mod p)g = #fx 2 P (n) j (x; 1) 6 0 (mod p)g p X #fx 2 P (n) j (x; 1) 6 0 (mod p)g = S (n; k) k=0 p p X X S (n; k) S (n; k)[p]k = pn ; k=0
k=0
where the rst one is clear, the second one follows from Fact 1 and Fact 2, and the third one from Fact 3. 2
Proposition 3 Let m be arbitrary. Then MODm-CC(UCONn(n?1) ) = O(n). 14
Lemma 9 SEQ = (SEQ2n)n2 is reducible to UCON = (UCONn(n?1) )n2 given in distributed form via an O(n2 ){projection reduction with respect to the partition of the variables. IN
IN
Proof. Consider an input (t1; : : : ; tn; u1; : : :; un ) of SEQ2n. The projection reduction n(n?1) : fxij ; yij j i; j = 1; : : : ; n; i < j g ! f0; 1; t ; u ; :t ; :u j = 1; : : : ; ng;
where the values of the Boolean variables xij and yij de ne the graphs G1 and G2 accessible to the processors P1 and P2, is de ned as shown in Figure 1 and Figure 2. We visualize this graph in such a way that the edges which are not constant are labelled with the corresponding literals. The meaning is that such an edge belongs to the graph if and only if the labelling literal is true. Clearly, this graph is connected if and only if SEQ2n(t1; : : :; tn; u1; : : :; un) = 1:
2
The lower bound easily results from Lemma 9, and Corollary 3: Proposition 4 Let m be arbitrary. Then MODm-CC(UCONn(n?1) ) = (n). Corollary 4 Let C2n be a (SYMM; MODm){circuit computing (UCONn(n?1))n2 . Then SIZE(C2n) = 2 (n) :
ee
IN
G (t1; u1) K2;2
...
...
K2;2
t
G (tn; un) SS
u
Figure 1. The graph shows the construction
nQQ n QQ n bb n b n n n t
: t
: u
u
: t : u
Figure 2. The graphs G (t; u)
of the projection of SEQ2n . (K2;2 denotes the full bipartite graph having 2 2 nodes, G (t; u) is de ned in Figure 2.)
of Figure 1.
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References [1] A. Borodin, A. Razborov, R. Smolensky, On Lower Bounds for Read{k{times Branching Programs, Computational Complexity 3(1993), pp. 1{18. [2] C. Damm, M. Krause, Ch. Meinel, St. Waack, Separating Counting Communication Complexity Classes, in: Proc. 9th STACS, Lecture Notes in Computer Science 577, Springer Verlag 1992, pp. 281{293. [3] A. Hajnal, W. Maass, G. Turan, On the Communication Complexity of Graph Problems, in: Proc. 20th ACM STOC 1988, pp. 186{191. [4] M. Krause, St. Waack, Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan{in, in: Proc. 32nd IEEE FOCS 1991, pp. 777{782. [5] S. Lokam, Spectral methods for matrix rigidity with applications to size{depth tradeo s and communication complexity, in: Proc. 36th IEEE FOCS 1995. [6] L. Lovasz, Communication complexity: A survey, in: Paths, ows and VLSI{layouts, Springer{Verlag 1990, pp. 235{266. [7] L. Lovasz, M. Saks, Communication complexity and combinatorial lattice theory, Journal of Computer and System Sciences 47(1993), pp. 322-349. [8] K. Mehlhorn, E. M. Schmidt, Las Vegas is better than determinism in VLSI and distributed computing, in: Proc. 14th ACM STOC 1982, pp. 330{337. [9] P. Pudlak, Large Communication in Constant Depth Circuits, Combinatorica 14(2)(1994), pp. 203{216. [10] A.A. Razborov, Lower bounds for the size of circuits of bounded depth with basis f^; g, Journ. Math. Zametki 41(1987), pp. 598{607. [11] A.A. Razborov, On rigid matrices, Manuscript. [12] R. Raz, B. Spieker, On the \log rank"{ conjecture in communication complexity, in: Proc. 34th IEEE FOCS 1993, pp. 168{176. [13] G.-C. Rota, On the foundation of combinatorial theory: I. Theory of Mobius functions, Z. Wahrscheinlichkeitstheorie 2(1964), pp. 340{368. [14] L. Skyum, L. V. Valiant, A complexity theory based on Boolean algebra, in: Proc. 22th IEEE FOCS, pp. 244-253. [15] V. Shoup, R. Smolensky, Lower Bounds for Polynomial Evaluation and Interpolation, in: Proc. 32nd IEEE FOCS (1991), pp. 378{383. 16
[16] R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity, in: Proc. 19th ACM STOC (1987), pp. 77{82. [17] A. Wigderson, The complexity of graph connectivity, TR 92-19, Leibnitz Center for Research in Computer Science, Institute of Computer Science, Hebrew University, Jerusalem. [18] S. Wilf, Hadamard determinants, Mobius functions and the cromatic number of graphs, Bull./Amer. Math. Soc. 74(1968), pp. 960{964. [19] A. C.-C. Yao, Some Complexity Questions Related to Distributed Computing, in: Proc. 11st ACM STOC 1979, pp. 209{213. [20] A. C.-C. Yao, On ACC and threshold circuits, in: Proc. 31st IEEE FOCS 1990, pp. 619{ 627.
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