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On the Robustness of Functional Equations
 SIAM Journal on Computing
, 1994
"... In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for ..."
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Cited by 22 (2 self)
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In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for robustness. We then study a general class of functional equations, which are of the form 8x; y F [f(x \Gamma y); f(x + y); f(x); f(y)] = 0, where F is an algebraic function. We give conditions on such functional equations that imply robustness. Our results have applications to the area of selftesting/correcting programs. We show that selftesters and selfcorrectors can be found for many functions satisfying robust functional equations, including algebraic functions of trigonometric functions such as tan x; 1 1+cotx ; Ax 1\GammaAx ; cosh x. 1 Introduction The mathematical field of functional equations is concerned with the following prototypical problem: Given a set of properties (fun...
Semidirect product in groups and zigzag product in graphs: connections and applications
 In Proceedings of the 42nd FOCS
, 2001
"... We consider the standard semidirect product of finite groups. We show that with certain choices of generators for these three groups, the Cayley graph of is (essentially) the zigzag product of the Cayley graphs of and. Thus, using the results of [RVW00], the new Cayley graph is an expander if and o ..."
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Cited by 19 (6 self)
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We consider the standard semidirect product of finite groups. We show that with certain choices of generators for these three groups, the Cayley graph of is (essentially) the zigzag product of the Cayley graphs of and. Thus, using the results of [RVW00], the new Cayley graph is an expander if and only if its two components are. We develop some general ways of using this construction to obtain large constantdegree expanding Cayley graphs from small ones. In [LW93], Lubotzky and Weiss asked whether expansion is a group property; namely, is being expander for (a Cayley graph of) a group depend solely on and not on the choice of generators. We use the above construction to answer the question in negative, by showing an infinite family of groups which are expanders with one choice of (constantsize) set of generators and are not with another such choice. It is interesting to note that this problem is still open, though, for “natural” families of groups, like the symmetric groups or the simple groups.
A new family of Cayley expanders
 Proc. 36th STOC, 2004
"... We assume that for some fixed large enough integer d, the symmetric group Sd can be generated as an expander using d 1/30 generators. Under this assumption, we explicitly construct an infinite family of groups Gn, and explicit sets of generators Yn ⊂ Gn, such that all generating sets have bounded si ..."
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Cited by 8 (3 self)
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We assume that for some fixed large enough integer d, the symmetric group Sd can be generated as an expander using d 1/30 generators. Under this assumption, we explicitly construct an infinite family of groups Gn, and explicit sets of generators Yn ⊂ Gn, such that all generating sets have bounded size (at most d 1/7), and the associated Cayley graphs are all expanders. The groups Gn above are very simple, and completely different from previous known examples of expanding groups. Indeed, Gn is (essentially) all symmetries of the dregular tree of depth n. The proof is completely elementary, using only simple combinatorics and linear algebra. The recursive structure of the groups Gn (iterated wreath products of the alternating group Ad) allows for an inductive proof of expansion, using the group theoretic analogue [4] of the zigzag graph product of [37]. The explicit construction of the generating sets Yn uses an efficient algorithm for solving certain equations over these groups, which relies on the work of [32] on the commutator width of perfect groups. We stress that our assumption above on weak expansion in the symmetric group is an open problem. We conjecture that it holds for all d. We discuss known results related to its likelihood in the paper.
The Power of the Queue
 M B
, 1992
"... Queues, stacks, and tapes are basic concepts which have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or lastinfirstout storage) have been thoroughly investigated and are well understood, this is much less the case for queues (firstin ..."
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Cited by 6 (0 self)
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Queues, stacks, and tapes are basic concepts which have direct applications in compiler design and the general design of algorithms. Whereas stacks (pushdown store or lastinfirstout storage) have been thoroughly investigated and are well understood, this is much less the case for queues (firstinfirst out storage). In this paper we present a comprehensive study comparing queues to stacks and tapes (offline and with oneway input). The techniques we use rely on Kolmogorov complexity. In particular, 1 queue and 1 tape (or stack) are not comparable: (1) Simulating 1 stack (and hence 1 tape) by 1 queue requires\Omega\Gamma n 4=3 = log n) time in both the deterministic and the nondeterministic cases. (2) Simulating 1 queue by 1 tape requires\Omega\Gamma n 2 ) time in the deterministic case, and\Omega\Gamma n 4=3 =(log n) 2=3 ) in the nondeterministic case; We further compare the relative power between different numbers of queues: (3) Nondeterministically simulating 2 queues...
Iterative construction of cayley expander graphs
 Theory of Computing
"... Abstract We construct a sequence of groups Gn, and explicit sets of generators Yn ae Gn, such that allgenerating sets have bounded size, and the associated Cayley graphs are all expanders. The group G1 isthe alternating group Ad, the set of even permutations on the elements {1, 2,..., d}. The group ..."
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Cited by 4 (0 self)
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Abstract We construct a sequence of groups Gn, and explicit sets of generators Yn ae Gn, such that allgenerating sets have bounded size, and the associated Cayley graphs are all expanders. The group G1 isthe alternating group Ad, the set of even permutations on the elements {1, 2,..., d}. The group Gn isthe group of all even symmetries of the rooted dregular tree of depth n. Our results hold for any largeenough d.We also describe a finitelygenerated infinite group G1 with generating set Y1, given with a mapping fn from G1 to Gn for every n, which sends Y1 to Yn. In particular, under the assumption describedabove, G1 has property (o / ) with respect to the family of subgroups ker(fn).The proof is elementary, using only simple combinatorics and linear algebra. The recursive structure of the groups Gn (iterated wreath products of the alternating group Ad) allows for an inductive proof ofexpansion, using the group theoretic analogue [4] of the zigzag graph product of [42]. The basis of the inductive proof is a recent result by Kassabov [22] on expanding generating sets for the group Ad.Essential use is made of the fact that our groups have the commutator property: every element is a commutator. We prove that direct products of such groups are expanding even with highly correlatedtuples of generators. Equivalently, highly dependent random walks on several copies of these groups converge to stationarity on all of them essentially as quickly as independent random walks. Moreover,our explicit construction of the generating sets
Quantum Expanders: Motivation and Constructions
 THEORY OF COMPUTING
, 2009
"... We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Philips and Sarna ..."
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We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Philips and Sarnak [29]. The second construction is combinatorial, and is based on a quantum variant of the ZigZag product introduced by Reingold, Vadhan and Wigderson [37]. Both constructions are of constant degree, and the second one is explicit. Using another construction of quantum expanders by Ambainis and Smith [6], we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK–complete (where QSZK is the class of languages having a zeroknowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof follows the classical proof structure that the entropy difference problem is SZK–complete, but crucially depends on the use of quantum expanders.
Monotone expanders constructions and applications
"... The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree ..."
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The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree dimension expanders in finite fields, resolving a question of [BISW04]. 2. O(1)page and O(1)pushdown expanders, resolving a question of [GKS86], and leading to tight lower bounds on simulation time for certain Turing Machines. Bourgain [Bou09] gave recently an ingenious construction of such constant degree monotone expanders. The first application (1) above follows from a reduction in [DS08]. We give a short exposition of both construction and reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that a variant of the zigzag graph product preserves monotonicity, and use it to give a simple alternative construction of monotone expanders, with nearconstant degree. 1
www.theoryofcomputing.org Quantum Expanders: Motivation and Constructions
, 2008
"... Abstract: We define quantum expanders in a natural way and give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Phillips ..."
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Abstract: We define quantum expanders in a natural way and give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2,q) given by Lubotzky, Phillips, and Sarnak (1988). The second construction is combinatorial, and is based on a quantum variant of the ZigZag product introduced by Reingold, Vadhan, and Wigderson (2000). Both constructions are of constant degree, and the second one is explicit. Using another construction of quantum expanders by Ambainis and Smith (2004), we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK–complete (where QSZK is the class of languages having a zeroknowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof follows the classical proof structure that the entropy difference problem is SZK– complete, but crucially depends on the use of quantum expanders.