Alon, N., and Maass, W. Meanders and their applications in lower bounds arguments. J. Comput. System Sci. 37, 2 (1988), 118-129.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of a large class of symmetric Boolean functions## n log n log log n) lower bound was obtained in [2] but for this they could not use the help of Ramsey type methods. It was further improved to ## n log n) in [24] and [12] independently, the later again using Ramsey methods. Alon and Maass [12, 13] have several results about lower bounds for the length of branching programs of various symmetric functions, or lower bounds for the time space complexity trado# in a general input oblivious sequential model of computation of certain functions, using the following Ramsey theoretic lemma stated ....

N. Alon and W. Maass, Meanders and their applications in lower bounds arguments, J. Computer and System Sciences 37 (1988), 118--129.

....proves lower bounds for other functions. We will have more to say about these lower bounds in Sections 2.4.2, 2.5, and 2.6. Of course, all other lower bounds mentioned below for stronger models imply a fortiori equally strong lower bounds for OBDD s. Also, in a very different vein, Alon and Maass [AM88] prove lower bounds for arbitrary oblivious programs of linear length, which do not obey any restriction on the number of times a variable is read. Their lower bound is discussed in Section 2.5.3. In similar spirit, Krause and Waack [KW91] show that any oblivious program of linear length for the ....

....in Section 2.5.3. In similar spirit, Krause and Waack [KW91] show that any oblivious program of linear length for the problem of directed s t connectivity requires exponential size; in [KMW92] similar lower bounds are proved for such programs with nondeterminism added. Using a lemma from [AM88], and the communication complexity arguments outlined in Section 2.5.1, Gergov [Ge94] proves that computing MULT requires size for arbitrary oblivious programs of linear length, even with nondeterministic AND, OR, or PARITY nodes. There has also been great success in proving lower bounds on ....

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N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37, 1988, pp. 118-- 129.

....programs which test the same variable at each time step along any path. For oblivious branching programs, linear length and read k for some constant k are essentially the same and several size length tradeo# lower bounds for oblivious branching programs have been shown using this connection [AM88, BNS92] Oblivious read once branching programs, known as OBDD s, have been very useful as representations of functions used in verification [Bry86, BCL 94] and so have generated significant independent interest. Borodin, Razborov, and Smolensky [BRS93] observed that read k branching ....

Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....programs are nearly as good as the best lower bounds known even for the much simpler oblivious deterministic branching programs. The best lower bounds in the oblivious case have all been obtained using some form of communication complexity. Using two party communication complexity, Alon and Maass [AM88] derived lower bounds of the form T = n log(n=S) and using multi party communication complexity, Babai, Nisan, and Szegedy [BNS92] derived the best current lower bounds which are of the form T = n log (n=S) The use of rectangles in our results as well as all those referenced in Table ....

....as all those referenced in Table 1 is related to 2 party communication complexity (see e.g. KN97] and most of the difficulty in these arguments is in extending the bounds from the oblivious to the general case. In fact, the basic approach provides an alternate way to obtain the same bounds as [AM88] for oblivious branching programs (see the discussion prior to Lemma 4.3) Recently, these methods have been extended [BV02] to include multi party communication complexity ideas which yield an alternate way to obtain the bounds of [BNS92] for oblivious branching programs. These results also ....

Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....trees to directed graphs that provide elegant models of both non uniform time T and space S simultaneously. The key ideas in these recent papers extend notions from 2 party communication complexity previously used in the study of restricted branching programs, such as oblivious branching programs [3] or read k branching programs [9] to general branching programs. In this paper we extend and improve these results in several directions. We develop a new lower bound criterion, based on extending 2 party communication complexity ideas to multiparty communication complexity, that applies to ....

....set N and a partition P of N N , the number of alternations of s with respect to P is the minimal r such that s can be written as s = s1s2 sr and each s i contains no elements from at least one class in P . Each s i is called an alternation of s with respect to P . PROPOSITION 3.1. [10, 3, 4] Let f : D f0; 1g, let be a partial assignment to the variables of f , and let P be a partition of unset( If there is an oblivious branching program B computing f that has width W and whose query sequence has at most r alternations with respect to P , then (r 1) log W 1 CP (f ....

Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....and other notions from communication complexity used here. In the second part of the proof, one then derives a lower bound on the number of rectangles in a rectangle cover for f . The respective technique for oblivious BPs following this pattern goes back to papers of Jukna [12] Alon and Maass [3], and Krause [13, 14] The logarithm of the size of oblivious BPs can be directly lower bounded in terms of (two party) communication complexity, i.e. we (implicitly) work with the usual notion of combinatorial rectangles here. We give a concrete description only for the special case of OBDDs. ....

N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....Page 1 is almost n, as bad as possible. Despite the above, we can give some weak lower bounds on time space trade offs, for the same bilinear forms used to exhibit the gap above. We show that they require 2 T=n TS = Omega (n 2 ) This follows a simple adaptation of the Alon Maass technique ([AM88]) originally devised for oblivious branching programs. In section 3 we switch gears and move to discuss circuits over the real or complex fields, which can use only bounded constants in the computation. This may be a severe restriction. However, considering this model has several motivations. ....

....F ) computing b M , with T (A) T and S(A) S. Then 2 T=n TS = Omega0 n 2 ) Proof: Let T = nk=4. The straight line program A defines a sequence oe of variables from fx 1 ; 1 1 1 ; x n ; y 1 ; 1 1 1 ; y n g of length at most nk=4, in the order they appear in A. By the main lemma in [AM88], there are subsets X 0 ; Y 0 of the x and y variables, respectively, such that jX 0 j = jY 0 j = n= 2 k ) satisfying the following property: if we remove all other variables from oe, the resulting subsequence has at most k alternations between x and y variables. Setting the remaining ....

N. Alon and W. Maass, "Meanders and their applications in lower bounds arguments ", JCSS, 37, pp. 118--129, 1988.

....programs which test the same variable at each time step along any path. For oblivious branching programs, linear length and read k for some constant k are essentially the same and several size length tradeoff lower bounds for oblivious branching programs have been shown using this connection [AM88, BNS92] Oblivious read once branching programs, known as OBDD s, have been very useful as representations of functions used in verification [Bry86, BCL 94] and so have generated significant independent interest. Borodin, Razborov, and Smolensky [BRS93] observed that read k branching ....

Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....have been 1 Since the publication of a preliminary version of our results, Ajtai [Ajt99a] see also [Ajt99b] using related techniques, has exhibited an explicit family of boolean functions for which any linear size branching program must have exponential size. 3 shown using this connection [AM88, BNS92] Oblivious read once branching programs, known as OBDD s, have been very useful as representations of functions used in verification [Bry86, BCL 94] and so have generated significant independent interest. Borodin, Razborov, and Smolensky [BRS93] observed that read k branching ....

Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....# nodes) Me89, SDG94, Ge94, and others] Recently proposed alternative models include graph driven BDDs [SW95] and binary moment diagrams [BC94] The latter are not branching programs and do not compute a function, but they do allow polynomial size representation of multiplication. Also, in [AM88] lower bounds are proved for any oblivious programs of linear length, regardless of the order in which variables are read. From the proof of Bryant s lower bound for OBDDs [Br91] it follows by a simple communication complexity argument that MULT cannot be computed in polynomialsize by k OBDDs ....

....the order in which variables are read. From the proof of Bryant s lower bound for OBDDs [Br91] it follows by a simple communication complexity argument that MULT cannot be computed in polynomialsize by k OBDDs [Kr91, BSSW93] or the various nondeterministic OBDDs [Ge94] Incorporating results from [AM88], Ge94] extends the lower bound to arbitrary linearlength oblivious programs. Indeed, all of these oblivious models have been found too weak to compute MULT in polynomial size. It is therefore natural to consider nonoblivious programs, the simplest of these being read once programs. 1.4. ....

N. Alon and W. Maass, Meanders and their applications in lower bounds arguments, J. Comput. System Sci., 37 (1988), pp. 118--129.

....which includes OBDDs and k IBDDs. It is well known how lower bounds on communication complexity can be used to obtain lower bounds on the size of deterministic, nondeterministic or randomized oblivious BDDs with bounded length. This technique goes back to papers of Jukna [7] Alon and Maass [1], and Krause [8, 9] One shows that an arbitrary oblivious BDD for the considered function can be turned into a BDD of the form described in the following lemma by cleverly setting variables to constants. Lemma 1. Let G be a BDD for a function f defined on the variable set X , and let # = X 1 ....

N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....(see [4] that if G is solvable, then exponential length programs are required to compute the AND function, whereas there is a polynomial upper bound for nonsolvable groups. Theorem 1. 3(a) improves on a Omega Gamma n log log n= log log log n) lower bound due to Pudl ak [13] Alon and Maass [1] establish Omega Gamma n log n) lower bounds on program length for L T (g) where g(n) is of the form n ffi ; and Babai, et al. 2] find Omega Gamma n log n) lower bounds for asymptotically almost all threshold functions. Our result in 1.3(b) while smaller than the bounds established in ....

....establish Omega Gamma n log n) lower bounds on program length for L T (g) where g(n) is of the form n ffi ; and Babai, et al. 2] find Omega Gamma n log n) lower bounds for asymptotically almost all threshold functions. Our result in 1. 3(b) while smaller than the bounds established in [1] and [2] is the best one known to apply to all nonconstant threshold functions (that is, threshold g(n) where neither g(n) nor n Gamma g(n) is bounded above by a constant) 4 Theorem 1.4 is an important step in our program to prove that ACC 0 is strictly contained in NC 1 : Of course, ....

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N. Alon and W. Maass, Meanders and their applications in lower bounds arguments, J. Comp. Sys. Sci. 37, 118-129 (1988). 14

....of familiar complexity classes defined via circuit complexity in terms of branching programs. The most important result of this kind is closely related to the history of results on width restricted branching programs. We do not report this history here (for references, see, e.g. [10] or the monograph [112] We first give a definition of the width of a branching program and the relevant complexity classes. Definition 1.8: For a node v of a branching program, define l(v) as the number of edges on the longest path from the source to v. For i 0, let the ith level of G be the ....

....in polynomial time. An exponential lower bound on the size of MOD 2 OBDDs (and hence, also POBDDs) for the middle bit of multiplication has been proven by Gergov [41] For this, Gergov has used the rank method of communication complexity theory and Ramsey theoretic arguments of Alon and Maass [10] as tools. 28 2.2 Randomized Branching Programs Analogously to the last section, we introduce randomized variants of general as well as restricted branching programs (Section 2.2.1 and Section 2.2.2, resp. 2.2.1 Randomized General Branching Programs In order to define randomized branching ....

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N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988. 133

....is almost n, as bad as possible. Despite the above, we can give some weak lower bounds on time space trade offs, for the same bilinear forms used to exhibit the gap above. We show that they require 2 T=n TS = Omega Gamma n 2 ) This follows a simple adaptation of the Alon Maass technique ([AM88]) originally devised for oblivious branching programs. In section 3 we switch gears and move to discuss circuits over the real or complex fields, which can use only bounded constants in the computation. This may be a severe restriction. However, considering this model has several motivations. ....

....= T and S(A) S. Then 2 T=n TS = Omega Gamma n 2 ) 8 Proof: Let T = nk=4. The straight line program A defines a sequence oe of variables from fx 1 ; Delta Delta Delta ; x n ; y 1 ; Delta Delta Delta ; y n g of length at most nk=4, in the order they appear in A. By the main lemma in [AM88], there are subsets X 0 ; Y 0 of the x and y variables, respectively, such that jX 0 j = jY 0 j = n= 2 k ) satisfying the following property: if we remove all other variables from oe, the resulting subsequence has at most k alternations between x and y variables. Setting the remaining variables ....

N. Alon and W. Maass, "Meanders and their applications in lower bounds arguments", JCSS, 37, pp. 118--129, 1988.

....n Delta W (n) where W (n) is the inverse of van der Waerden function. Ajtai, Babai, Hajnal, Komlos, Pudl ak, Rodl, Szemeredi and Tur an [1] established the bound RSw (f n ) Omega Gamma n log n= log log n) for some symmetric f n . The following bound was independently proved by Alon, Maass [4] and Babai, Pudl ak, Rodl and Szemeredi [6] Theorem 13 ( 4, 6] RSw (MAJ n ) Omega Gamma n log n) An interesting general algebraic technique for obtaining superlinear lower bounds on RSw (f n ) f n symmetric, was developed by Barrington and Straubing [8] It allows to obtain bounds of ....

....function. Ajtai, Babai, Hajnal, Komlos, Pudl ak, Rodl, Szemeredi and Tur an [1] established the bound RSw (f n ) Omega Gamma n log n= log log n) for some symmetric f n . The following bound was independently proved by Alon, Maass [4] and Babai, Pudl ak, Rodl and Szemeredi [6] Theorem 13 ([4, 6]) RSw (MAJ n ) Omega Gamma n log n) An interesting general algebraic technique for obtaining superlinear lower bounds on RSw (f n ) f n symmetric, was developed by Barrington and Straubing [8] It allows to obtain bounds of order Omega Gamma n log log n) only but can be applied to a wider ....

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N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....stronger rectifier switching model which is one of the natural nondeterministic extensions of the branching program model. For the class of non constant threshold functions, Barrington and Straubing [BS91] using algebraic techniques, improved this bound to Omega0 n log log n) Alon and Maass [AM88] and Babai et al. BPRS90] independently proved that any oblivious branching program of width w p n for majority has length Omega0 n log n log w ) Their bounds apply to all but a vanishingly small fraction of symmetric functions and are currently the best known length lower bounds for ....

....fraction of symmetric functions, any constant width branching program requires length Omega0 n log n) We will give some 23 intuition on this result in the next subsection. 3.1. 2 Communication Complexity Technique Some of the best known lower bound results on branching programs (including [AM88] and [BPRS90] are based on communication complexity. Informally, communication complexity measures the amount of communication needed to compute the given function when two parties are each given half of the input. We assume that two processors (conveniently named Alice and Bob) with unlimited ....

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Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

....and Straubing [BS95] using algebraic techniques, improved this bound to Omega Gamma n log log n) It can be easily seen that any constant width branching program can be transformed into an equivalent oblivious branching program with a constant blow up in the length and the width. Alon and Maass [AM88] and Babai et al. BPRS90] independently proved that any oblivious branching program of width w p n for majority has length Omega Gamma n log n log w ) Their bounds are based on a communication complexity argument and imply that for computing all but a vanishingly small fraction of ....

....Our method also yields a spectrum of branching programs, one for each width greater than log n. For width w n, the length of the resulting branching program is O i n log 2 n log w log log log n j , which is within O i log n log log log n j of the length lower bound of Alon and Maass [AM88] Our constructions have other nice properties. For example, for any k = O i log 2 n log log n log log log n j they can be modified to give efficient syntactic read k branching programs. See Borodin et al. BRS93] for definitions, motivations, and a survey of results. Moreover, for any ....

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Noga Alon and Wolfgang Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

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Alon, N., and Maass, W. Meanders and their applications in lower bounds arguments. J. Comput. System Sci. 37, 2 (1988), 118-129.

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N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37(2):118--129, 1988.

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N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. JCSS, 37:118-129, 1988.

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N. Alon and W. Maass. Meanders and their applications in lower bounds arguments. Journal of Computer and System Sciences, 37:118--129, 1988.

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Alon, N. and Maass, W. (1988). Meanders and their applications in lower bound arguments. Journal of Computer and System Sciences 37, 118--129.

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N. Alon, and W. Maass, "Meanders and their Applications in Lower Bound Arguments", JCSS, Vol. 37, No. 2, pp. 118-129, 1988. (Early version in Proc. of 27th FOCS, pp. 410-417, 1986.) 271

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