- Email: [email protected]
Time–space tradeoffs for integer multiplication on various types of input oblivious sequential machines
ELSEVIER
Time-space
Information
Processing
Letters
Information Processing Letters
51 (1994) 265-269
tradeoffs for integer multiplication on various types of input oblivious sequential machines Jordan
Gergov
’
Fachbereich IV - Informatik, University of Trier, D-54286 Trier, Germany Communicated
Key words: Data processing; Time-space tradeoffs
Data management;
by D. Dolev; received
Algorithms;
2 February
1994
Integer multiplication;
Branching
programs;
BDDs;
1. Introduction Lower bounds on the branching program size imply lower bounds on the space complexity of computations. Similarly, tradeoff results for branching program depth and size imply timespace tradeoffs on sequential machines (see e.g. [4]). However, for single output Boolean functions, nontrivial tradeoff results of this type have been established only for restricted classes of branching programs. We mention here the two least restricted branching program models for which the above problem could be solved: namely, oblivious branching programs [1,2,11], and syntactic read-k-times branching programs [5,10,13]. Using results of Alon and Maass [ 11 and Bryant [6], we show that any oblivious branching program of linear length for the nth bit of the product of two n-bit integers must have exponential width. Note that we consider the multiplication as decision problem. This lower bound can be extended to nondeterministic, co-nondeterministic, or MOD, oblivious branching pro-
’ Email: [email protected]. 0020-0190/94/$07.00 0 1994 Elsevier SSDI 0020-0190(94)00094-F
Science
grams of linear length, too. That is why integer multiplication cannot be computed by determinisnondeterministic, co-nondeterministic, or tic, MOD, oblivious Turing machines simultaneously in linear time and logarithmic space (a MOD,, Turing machine is a nondeterministic TM which accepts the input if and only if the number of accepting computations is unequal to zero modulo a prime number p>. In the deterministic case, this lower bound is extension of the exponential lower bounds on the size of OBDDs [61 and kOBDDs for integer multiplication [3]. Recently, a restricted type of oblivious branching programs, the so-called ordered binary decision diagrams (OBDDS) has proven to be of great importance in VLSI CAD. For a survey see [7]. Unfortunately, OBDD-representations are not very succinct. There exists a number of important Boolean functions (e.g. integer multiplication) whose OBDDs are definitively of exponential size 161. That is why it is an important open problem to extend the OBDD concept to more general branching programs suitable for efficient solutions of the computational tasks arising in circuit verification, testing, logic synthesis, etc. (see e.g. 17,811 The most OBDD extensions are restricted
B.V. All rights reserved
266
J. Gergor / Information
Processing Letters 51 (1994) 265-269
oblivious branching programs. For example, IBDDs are restricted oblivious branching programs [9]. The above lower bound implies that linear length ordered BDDs which may read the input variables more than once are not powerful enough to allow polynomial size representation of integer multiplication. A second interesting approach are MOD,-OBDDs [8], which are restricted parity oblivious branching programs (parity branching programs are the nonuniform analogue of parity Turing machines). Since any linear time input oblivious parity Turing machine for multiplication requires superlogarithmic space, MOD,-OBDDs and their extensions similar to IBDDs do not provide us with polynomial size data structure for integer multiplication.
2. Preliminaries A branching program P,, is a directed acyclic graph with one source u(,, where each node has outdegree 2 or 0. There are two nodes of outdegree 0, the l-sink u, and O-sink c’_-) which are labeled by the Boolean constants 1 and 0, respectively. The remaining nodes are labeled by Boolean variables from the set X, = or remain unlabeled. One of the [-QJ,,...,~,~,] two successors of an inner node is defined as the left and the other as the right successor. size(P,) is the number of nodes labeled by Boolean variables. P,, is said to be oblivious if it is leveled (i.e., all paths from the source to any one of its nodes are of the same length), and if all nodes on any level are either unlabeled or labeled by one and the same input variable. length( P,>, width( P,) denote the number of levels labeled by Boolean variables and the maximal size of such a level in an oblivious branching program P,, , respectively. Each input assignment to the variables from X,, defines certain computation paths in P,,: Starting at the source u,,, a node u of P, is connected with one of its successor nodes. If u is labeled by a Boolean variable with 0 (l), we have to choose the left (right) successor of c’. A computation path ending at the l-sink u, is called accepting.
P, is deterministic if all nodes of P,, are labelled. A deterministic branching program accepts an input (computes the Boolean value 1) if the uniquely determined computation path ends at the l-sink, i.e. there is one accepting path. A nondeterministic (co-nondeterministic) branching program accepts an input if there is at least one accepting computation path (all computation paths are accepting). A MOD, branching program accepts an input if the number of accepting computation paths is unequal to zero modulo p. The complexity classes defined by polynomial size branching programs of the above types coincide with the complexity classes defined by the corresponding type of nonuniform logarithmic space bounded Turing machines. Since the translation of this relation is straightforward we state it here only in the case of deterministic branching programs and deterministic Turing machines. Recall, that a sequence of branching programs P = (P, I II E N} computes a sequence of Boolean functions f = {fn(xO, x,, . . , x,_ 1) I n E N} if for any II the branching program P,, computes f,; f is computable by polynomial size branching programs if there exists P = (P, ( n E N} computing f with size(P,> = no(‘). Theorem 1 (folklore). A sequence of Boolean functions f (decision problem) is computable by nonuniform logarithmic space bounded Turing machines if and only if f is computable by polynomial size deterministic branching programs. Integer multiplication is one of the hardest functions realized by integrated circuits (ICs), most circuits realize much simpler functions. MULTIPLICATION (mult 1: Input: two integers x, y in binary notation; Output: the product x *y in binary notation. In addition to the notion of deterministic communication complexity [14], we need the notions of nondeterministic, co-nondeterministic, and MODp communication complexity which are defined by means of modified accepting modes, i.e. nondeterministic (co-nondeterministic, MOD,)
J. Gergoc /Information
acceptance corresponds to at least one accepting protocol (all protocols are accepting, the number of accepting protocols is not equal zero modulo PT respectively).
3. Lower bounds for integer multiplication In order to prove lower bounds for mult we use a lower bound argument for mult due to Bryant [6] and a Ramsey-theoretic lemma due to Alon and Maass [l]. Let multi denote the Boolean function representing output i of an n-bit multiplier (suppose the multiplier outputs are numbered, starting with the least significant bit, from 0 to 2n - 1). The claim of the next theorem can easily be extended to mult,: if i is close enough to n. Theorem 2. Any deterministic, nondeterministic, co-nondeterministic, or MOD, (p prime number > oblivious branching program of linear length for m&t,“_, (integer multiplication) is of size 2”‘“’ Proof. For the sake of concreteness we give the proof for parity branching programs. The extension to nondeterministic, co-nondeterministic and MODr, branching programs is straightforward. Let P,, n E N, be a oblivious parity branching program of length 2. k . n, k E N, for mu&,(x,,
x~,...,x,-~;
261
t’rocessing Letters 51 ~1994) 265-269
y,, yl,...,yn-i).
Without loss of generality, we assume that at most k. n of the levels in P,, are labeled by variables from X, = {x,, xi,. . . , x,_~}. Note that any variable of X,, u Y,, appears in the program. Let us denote by A4 = zi, z2,. . . , zzkn, zi E X, u Y,, the sequence of the variables which are labels of the levels in P,, in the same order they appear starting at the source and ending by the sinks. The set of variables X’ cX, which appear at most k times in M has at least n/(k + 1) elements. Further, we can assume without loss of generality that each variable from X’ appears precisely k times in M (in the proof, we only use the fact I X’ I 2 n/(k + 1)). In the following, we use a notation introduced by Alon and Maass [l]. We say that an interval
zi+j of M is a link between S and T, S T are two disjoint subsets of X,, u Y,, if E S u T and if z, E S, zi+j E T or Zi+l,..‘>Zi+,~l Z~E T, z~+~ES. We denote the sequence obtained from M after removing all variables from (X, U Y,) -X’ by M’. Observe that the number of links between any two disjoint subsets of X’ in M is the same as in M’. We need the following slightly modified version of a lemma proven in [l].
z
...,
azd
Lemma 3 (Alon and Maass). Let M = z,, . . . , z, be a sequence in which each element z, E X,, (X = )> appears precisely k times (r = k (x0, XI,...,X,_] . n>, and suppose X’ u X” (=X,,> is a partition of X,, into two (disjoint) non-empty sets. Then there are two subsets S c_X’ and T cX” with I S I > IX’ 1/22k-1, I T I 2 I X” I/2Zk-‘, and such that the number of links between S and T in M is bounded abooe by 2 ’ k - 1. Let XA U Xi be a partition of X’ into two sets of size n/2(k + 1). Further, we assume that the indices of the variables from X,‘, are smaller than the indices of the variables from Xi. Now, the above lemma yields the existence of two sets S c XA and T 2 Xi (S n T = &r),such that (1) I S I 2 n/(k + 1)22k, (2) I T I 2 n/(k + 1)22k, (3) the number of links between S and T in M’ is bounded above by 2. k - 1. Since the cardinality of D = {(xi, xi)1 x, E S, xi E T) is at least n’/(k + 1)224k there exist D’ 2 D, Z G (0, 1,. . . , n - 11 and d E N such that D’ = {(xi, x~+~) I i E I), I D’ I = I I I a n/(k + 1)224k and max(Z) < mm(Z) + d. Consider the following restriction of muft,“_, and denote the resulting function by f: 1 Xi := i0
Yj’=
if P, where P=(i$Z)&(min(Z) if(i-dPZ)&(i@Z)¬(P);
1
ifj=n-max(Z)-1,
1
ifj=n-max(Z)-d-1,
i0
otherwise.
268
J. Gergov /Information
It can be EZ) and significant sented by
verified easily that T’= {x~+~ I i ~11, bit of the sum of the variables in S’
Processing
f(S’; T’), S’ = {xi I i computes the most the numbers repreand T’, i.e.
f(s 0, sl,...,sI; t,, t,,...,t,) = 1 if and only if Cf,,(s, + ti)2’ 2 2’+‘, where 12 n/(k + 1)224k, ~a,.. .,sI (to ,..., tl) denote the variables in S’ CT’) in increasing order of indices, with i, + d - 1.
Letters 51 (1994) 265-269
the above restriction. The computation in PA can be understood in terms of (two-party) communication game. PA can be partitioned in at most 2. k parts starting from the source S,, T,, S,, T 2,. . . , S,, Tk such that, without loss of generality, in Si (?;) appear variables only from Is,,, . . . , sl} (It,, . . . , t,}. That is why PL yields a communication protocol of length (2.k-
1) .log,(width(PL)).
Thus widt,$( P,‘) > 2”/(2k-iXk+1)224k.
Lemma 4. For any prime number p, the deterministic, nondeterministic, co-nondeterministic, or MOD, communication complexity off with respect to Is,, s,, . . . , sJ U It,, t,, . . . , tJ is at least 1. Proof. We denote the communication matrix of f by C, i.e. C is a 2’ x 2’ matrix such that C(s,,,
s,,...,s,;
=f(s,,
r,, t,,...,t,)
s1,...,st;
t,, t,,...,t,).
Obviously, all entries of C on and above the diagonal i + j = 21f1 - 1 are equal to 0 and all entries below this diagonal are equal to 1, i.e. C is a triangular matrix if we remove the first column and row. That is why rank,(C) = 2’ - 1 in all fields F (the rank is high also in the Boolean ring El = (0, 1)). log,(rank,(C)) is a lower bound on the deterministic communication complexity of f for any semiring H [12]. It is well known that log,(rank&C)) (log,(rank, (C))) is a lower bound on the nondeterministic (fiODp) communication complexity of f. The lower bound on the co-nondeterministic communication complexity is straight forward. 0 In order to prove that size(P,,) = 2ncn) we will show that width(PL) = 20cn), where Pi is the program resulting from P, after the above restriction, i.e. Pi computes f. PA can be constructed easily from P,, by reducing all (redundant) levels corresponding to variables set down to 1 or 0 in
References [I] N. Alon and W. Maass, Meanders
and their applications in lower bounds arguments, J. Comput. System .I%. 37 (1988) 118-129. 121L. Babai, N. Nisan and M. Szegedy, Multiparty protocols and logspace-hard pseudorandom sequences, J. Comput. System Sci. 45 (1992) 204-232. D. Sieling and 1. Wegener, [31 B. Bollig, M. Sauerhoff, Read-k times ordered binary decision diagrams. Efficient algorithms in the presence of null chains, Tech. Rept. 474, Universitat Dortumund, 1993. and S. Cook, A time-space tradeoff for 141 A. Borodin sorting on a general sequential model of computation, SMM .I. Comput. 11 (1982) 287-297. and R. Smolensky, On lower [51 A. Borodin, A. Razborov bounds for read-k times branching programs, Comput. Complexity 3 (1993) l-18. [61 R.E. Bryant, On the complexity of VLSI implementations and graph representations of boolean functions with applications to integer multiplication, IEEE Trans. Cornput. 40 (2) (1991) 205-213. manipulation with or[71 R.E. Bryant, Symbolic boolean dered binary decision diagrams, ACM Comput. Surveys 24 (3) (1992) 293-318. of feasible and [81 J. Gergov and Ch. Meinel, Frontiers probabilistic feasible boolean manipulation with branching programs, in: Proc. 10th Ann. Symp. on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 665 (Springer, Berlin, 1993) 576-585. J. Bitner, D.S. Fussell and J.A. [91 J. Jain, M. Abadir, Abraham, IBDD’s: An efficient functional representation for digital circuits, in: Proc. European Design Automation Conf (1992) 440-446. DO1S. Jukna, A note on read-k times branching programs, Tech. Rept. 448, Universitat Dortmund, 1992; RAIRO Theoret. Inform Appl., to appear.
J. Gergou /Information
Processing Letters 51 (1994) 265-269
1111 M. Krause and S. Waack, On oblivious branching programs of linear length, Inform. and Comput. 94 (2) (1991) 232-249. [12] K. Mehlhorn and E.M. Schmidt, Las Vegas is better than determinism in VLSI and distributed computing, in: Proc. 14th ACM Symp. on Theory of Computing (1982) 330-337. [13] E.A. Okolnishikova, Lower bounds on the complexity of
269
realizations of characteristic functions of binary codes by branching programs, Diskretnii Analiz 51 (1991) 61-83 (in Russian). [14] A.C.X. Yao, Some complexity questions related to distributed computing, in: Proc. 11th ACM Symp. on Theory of Computing (1979) 209-213.
Time-space
Information
Processing
Letters
Information Processing Letters
51 (1994) 265-269
tradeoffs for integer multiplication on various types of input oblivious sequential machines Jordan
Gergov
’
Fachbereich IV - Informatik, University of Trier, D-54286 Trier, Germany Communicated
Key words: Data processing; Time-space tradeoffs
Data management;
by D. Dolev; received
Algorithms;
2 February
1994
Integer multiplication;
Branching
programs;
BDDs;
1. Introduction Lower bounds on the branching program size imply lower bounds on the space complexity of computations. Similarly, tradeoff results for branching program depth and size imply timespace tradeoffs on sequential machines (see e.g. [4]). However, for single output Boolean functions, nontrivial tradeoff results of this type have been established only for restricted classes of branching programs. We mention here the two least restricted branching program models for which the above problem could be solved: namely, oblivious branching programs [1,2,11], and syntactic read-k-times branching programs [5,10,13]. Using results of Alon and Maass [ 11 and Bryant [6], we show that any oblivious branching program of linear length for the nth bit of the product of two n-bit integers must have exponential width. Note that we consider the multiplication as decision problem. This lower bound can be extended to nondeterministic, co-nondeterministic, or MOD, oblivious branching pro-
’ Email: [email protected]. 0020-0190/94/$07.00 0 1994 Elsevier SSDI 0020-0190(94)00094-F
Science
grams of linear length, too. That is why integer multiplication cannot be computed by determinisnondeterministic, co-nondeterministic, or tic, MOD, oblivious Turing machines simultaneously in linear time and logarithmic space (a MOD,, Turing machine is a nondeterministic TM which accepts the input if and only if the number of accepting computations is unequal to zero modulo a prime number p>. In the deterministic case, this lower bound is extension of the exponential lower bounds on the size of OBDDs [61 and kOBDDs for integer multiplication [3]. Recently, a restricted type of oblivious branching programs, the so-called ordered binary decision diagrams (OBDDS) has proven to be of great importance in VLSI CAD. For a survey see [7]. Unfortunately, OBDD-representations are not very succinct. There exists a number of important Boolean functions (e.g. integer multiplication) whose OBDDs are definitively of exponential size 161. That is why it is an important open problem to extend the OBDD concept to more general branching programs suitable for efficient solutions of the computational tasks arising in circuit verification, testing, logic synthesis, etc. (see e.g. 17,811 The most OBDD extensions are restricted
B.V. All rights reserved
266
J. Gergor / Information
Processing Letters 51 (1994) 265-269
oblivious branching programs. For example, IBDDs are restricted oblivious branching programs [9]. The above lower bound implies that linear length ordered BDDs which may read the input variables more than once are not powerful enough to allow polynomial size representation of integer multiplication. A second interesting approach are MOD,-OBDDs [8], which are restricted parity oblivious branching programs (parity branching programs are the nonuniform analogue of parity Turing machines). Since any linear time input oblivious parity Turing machine for multiplication requires superlogarithmic space, MOD,-OBDDs and their extensions similar to IBDDs do not provide us with polynomial size data structure for integer multiplication.
2. Preliminaries A branching program P,, is a directed acyclic graph with one source u(,, where each node has outdegree 2 or 0. There are two nodes of outdegree 0, the l-sink u, and O-sink c’_-) which are labeled by the Boolean constants 1 and 0, respectively. The remaining nodes are labeled by Boolean variables from the set X, = or remain unlabeled. One of the [-QJ,,...,~,~,] two successors of an inner node is defined as the left and the other as the right successor. size(P,) is the number of nodes labeled by Boolean variables. P,, is said to be oblivious if it is leveled (i.e., all paths from the source to any one of its nodes are of the same length), and if all nodes on any level are either unlabeled or labeled by one and the same input variable. length( P,>, width( P,) denote the number of levels labeled by Boolean variables and the maximal size of such a level in an oblivious branching program P,, , respectively. Each input assignment to the variables from X,, defines certain computation paths in P,,: Starting at the source u,,, a node u of P, is connected with one of its successor nodes. If u is labeled by a Boolean variable with 0 (l), we have to choose the left (right) successor of c’. A computation path ending at the l-sink u, is called accepting.
P, is deterministic if all nodes of P,, are labelled. A deterministic branching program accepts an input (computes the Boolean value 1) if the uniquely determined computation path ends at the l-sink, i.e. there is one accepting path. A nondeterministic (co-nondeterministic) branching program accepts an input if there is at least one accepting computation path (all computation paths are accepting). A MOD, branching program accepts an input if the number of accepting computation paths is unequal to zero modulo p. The complexity classes defined by polynomial size branching programs of the above types coincide with the complexity classes defined by the corresponding type of nonuniform logarithmic space bounded Turing machines. Since the translation of this relation is straightforward we state it here only in the case of deterministic branching programs and deterministic Turing machines. Recall, that a sequence of branching programs P = (P, I II E N} computes a sequence of Boolean functions f = {fn(xO, x,, . . , x,_ 1) I n E N} if for any II the branching program P,, computes f,; f is computable by polynomial size branching programs if there exists P = (P, ( n E N} computing f with size(P,> = no(‘). Theorem 1 (folklore). A sequence of Boolean functions f (decision problem) is computable by nonuniform logarithmic space bounded Turing machines if and only if f is computable by polynomial size deterministic branching programs. Integer multiplication is one of the hardest functions realized by integrated circuits (ICs), most circuits realize much simpler functions. MULTIPLICATION (mult 1: Input: two integers x, y in binary notation; Output: the product x *y in binary notation. In addition to the notion of deterministic communication complexity [14], we need the notions of nondeterministic, co-nondeterministic, and MODp communication complexity which are defined by means of modified accepting modes, i.e. nondeterministic (co-nondeterministic, MOD,)
J. Gergoc /Information
acceptance corresponds to at least one accepting protocol (all protocols are accepting, the number of accepting protocols is not equal zero modulo PT respectively).
3. Lower bounds for integer multiplication In order to prove lower bounds for mult we use a lower bound argument for mult due to Bryant [6] and a Ramsey-theoretic lemma due to Alon and Maass [l]. Let multi denote the Boolean function representing output i of an n-bit multiplier (suppose the multiplier outputs are numbered, starting with the least significant bit, from 0 to 2n - 1). The claim of the next theorem can easily be extended to mult,: if i is close enough to n. Theorem 2. Any deterministic, nondeterministic, co-nondeterministic, or MOD, (p prime number > oblivious branching program of linear length for m&t,“_, (integer multiplication) is of size 2”‘“’ Proof. For the sake of concreteness we give the proof for parity branching programs. The extension to nondeterministic, co-nondeterministic and MODr, branching programs is straightforward. Let P,, n E N, be a oblivious parity branching program of length 2. k . n, k E N, for mu&,(x,,
x~,...,x,-~;
261
t’rocessing Letters 51 ~1994) 265-269
y,, yl,...,yn-i).
Without loss of generality, we assume that at most k. n of the levels in P,, are labeled by variables from X, = {x,, xi,. . . , x,_~}. Note that any variable of X,, u Y,, appears in the program. Let us denote by A4 = zi, z2,. . . , zzkn, zi E X, u Y,, the sequence of the variables which are labels of the levels in P,, in the same order they appear starting at the source and ending by the sinks. The set of variables X’ cX, which appear at most k times in M has at least n/(k + 1) elements. Further, we can assume without loss of generality that each variable from X’ appears precisely k times in M (in the proof, we only use the fact I X’ I 2 n/(k + 1)). In the following, we use a notation introduced by Alon and Maass [l]. We say that an interval
zi+j of M is a link between S and T, S T are two disjoint subsets of X,, u Y,, if E S u T and if z, E S, zi+j E T or Zi+l,..‘>Zi+,~l Z~E T, z~+~ES. We denote the sequence obtained from M after removing all variables from (X, U Y,) -X’ by M’. Observe that the number of links between any two disjoint subsets of X’ in M is the same as in M’. We need the following slightly modified version of a lemma proven in [l].
z
...,
azd
Lemma 3 (Alon and Maass). Let M = z,, . . . , z, be a sequence in which each element z, E X,, (X = )> appears precisely k times (r = k (x0, XI,...,X,_] . n>, and suppose X’ u X” (=X,,> is a partition of X,, into two (disjoint) non-empty sets. Then there are two subsets S c_X’ and T cX” with I S I > IX’ 1/22k-1, I T I 2 I X” I/2Zk-‘, and such that the number of links between S and T in M is bounded abooe by 2 ’ k - 1. Let XA U Xi be a partition of X’ into two sets of size n/2(k + 1). Further, we assume that the indices of the variables from X,‘, are smaller than the indices of the variables from Xi. Now, the above lemma yields the existence of two sets S c XA and T 2 Xi (S n T = &r),such that (1) I S I 2 n/(k + 1)22k, (2) I T I 2 n/(k + 1)22k, (3) the number of links between S and T in M’ is bounded above by 2. k - 1. Since the cardinality of D = {(xi, xi)1 x, E S, xi E T) is at least n’/(k + 1)224k there exist D’ 2 D, Z G (0, 1,. . . , n - 11 and d E N such that D’ = {(xi, x~+~) I i E I), I D’ I = I I I a n/(k + 1)224k and max(Z) < mm(Z) + d. Consider the following restriction of muft,“_, and denote the resulting function by f: 1 Xi := i0
Yj’=
if P, where P=(i$Z)&(min(Z) if(i-dPZ)&(i@Z)¬(P);
1
ifj=n-max(Z)-1,
1
ifj=n-max(Z)-d-1,
i0
otherwise.
268
J. Gergov /Information
It can be EZ) and significant sented by
verified easily that T’= {x~+~ I i ~11, bit of the sum of the variables in S’
Processing
f(S’; T’), S’ = {xi I i computes the most the numbers repreand T’, i.e.
f(s 0, sl,...,sI; t,, t,,...,t,) = 1 if and only if Cf,,(s, + ti)2’ 2 2’+‘, where 12 n/(k + 1)224k, ~a,.. .,sI (to ,..., tl) denote the variables in S’ CT’) in increasing order of indices, with i,
Letters 51 (1994) 265-269
the above restriction. The computation in PA can be understood in terms of (two-party) communication game. PA can be partitioned in at most 2. k parts starting from the source S,, T,, S,, T 2,. . . , S,, Tk such that, without loss of generality, in Si (?;) appear variables only from Is,,, . . . , sl} (It,, . . . , t,}. That is why PL yields a communication protocol of length (2.k-
1) .log,(width(PL)).
Thus widt,$( P,‘) > 2”/(2k-iXk+1)224k.
Lemma 4. For any prime number p, the deterministic, nondeterministic, co-nondeterministic, or MOD, communication complexity off with respect to Is,, s,, . . . , sJ U It,, t,, . . . , tJ is at least 1. Proof. We denote the communication matrix of f by C, i.e. C is a 2’ x 2’ matrix such that C(s,,,
s,,...,s,;
=f(s,,
r,, t,,...,t,)
s1,...,st;
t,, t,,...,t,).
Obviously, all entries of C on and above the diagonal i + j = 21f1 - 1 are equal to 0 and all entries below this diagonal are equal to 1, i.e. C is a triangular matrix if we remove the first column and row. That is why rank,(C) = 2’ - 1 in all fields F (the rank is high also in the Boolean ring El = (0, 1)). log,(rank,(C)) is a lower bound on the deterministic communication complexity of f for any semiring H [12]. It is well known that log,(rank&C)) (log,(rank, (C))) is a lower bound on the nondeterministic (fiODp) communication complexity of f. The lower bound on the co-nondeterministic communication complexity is straight forward. 0 In order to prove that size(P,,) = 2ncn) we will show that width(PL) = 20cn), where Pi is the program resulting from P, after the above restriction, i.e. Pi computes f. PA can be constructed easily from P,, by reducing all (redundant) levels corresponding to variables set down to 1 or 0 in
References [I] N. Alon and W. Maass, Meanders
and their applications in lower bounds arguments, J. Comput. System .I%. 37 (1988) 118-129. 121L. Babai, N. Nisan and M. Szegedy, Multiparty protocols and logspace-hard pseudorandom sequences, J. Comput. System Sci. 45 (1992) 204-232. D. Sieling and 1. Wegener, [31 B. Bollig, M. Sauerhoff, Read-k times ordered binary decision diagrams. Efficient algorithms in the presence of null chains, Tech. Rept. 474, Universitat Dortumund, 1993. and S. Cook, A time-space tradeoff for 141 A. Borodin sorting on a general sequential model of computation, SMM .I. Comput. 11 (1982) 287-297. and R. Smolensky, On lower [51 A. Borodin, A. Razborov bounds for read-k times branching programs, Comput. Complexity 3 (1993) l-18. [61 R.E. Bryant, On the complexity of VLSI implementations and graph representations of boolean functions with applications to integer multiplication, IEEE Trans. Cornput. 40 (2) (1991) 205-213. manipulation with or[71 R.E. Bryant, Symbolic boolean dered binary decision diagrams, ACM Comput. Surveys 24 (3) (1992) 293-318. of feasible and [81 J. Gergov and Ch. Meinel, Frontiers probabilistic feasible boolean manipulation with branching programs, in: Proc. 10th Ann. Symp. on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 665 (Springer, Berlin, 1993) 576-585. J. Bitner, D.S. Fussell and J.A. [91 J. Jain, M. Abadir, Abraham, IBDD’s: An efficient functional representation for digital circuits, in: Proc. European Design Automation Conf (1992) 440-446. DO1S. Jukna, A note on read-k times branching programs, Tech. Rept. 448, Universitat Dortmund, 1992; RAIRO Theoret. Inform Appl., to appear.
J. Gergou /Information
Processing Letters 51 (1994) 265-269
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