A nondeterministic space-time tradeoff for linear codes

We are interested in proving exponential lower bounds on the size of nondeterministic D-way branching programs computing functions in linear time, that is, in time at most kn for a constant k. Ajtai has proved such lower bounds for explicit functions over domains D of size about n, and Beame, Saks and Thathachar for functions over domains of size about k²². We prove an exponential lower bound ²Ω(n/ck) for an explicit function over substantially smaller domain D of size about k². Our function is a universal function of linear codes.     

A lower bound on the size of ε-free NFA corresponding to a regular expression

Hromkovic? et al. showed how to transform a regular expression of size n into an ε-free nondeterministic finite automaton (which defines the same language as the expression) with O(n) states and O(nlog²(n)) transitions. They also established a lower bound on the number of transitions. We improve the lower bound to .     

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(n²,4) enjoys several attractive topological properties. Let max_?G(m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that , where p₀>p₁>?>pr are nonnegative integers defined by . We then apply this formula to find the bisection width of RC(n²,4). The conclusion shows that, as n-dimensional cube, RC(n²,4) enjoys a linear bisection width.     

New instability results for high-dimensional nearest neighbor search

Consider a dataset of n(d) points generated independently from Rd according to a common p.d.f. fd with support(fd)=d[0,1] and sup{fd(Rd)} growing sub-exponentially in d. We prove that: (i) if n(d) grows sub-exponentially in d, then, for any query point and any ?>0, the ratio of the distance between any two dataset points and is less that 1+? with probability →1 as d→∞; (ii) if n(d)>d[4(1+?)] for large d, then for all (except a small subset) and any ?>0, the distance ratio is less than 1+? with limiting probability strictly bounded away from one. Moreover, we provide preliminary results along the lines of (i) when .     

The k-fundamental group of a closed k-surface

In this paper, we deal with the problem of computing the digital fundamental group of a closed k-surface by using various properties of both a (simple) closed k-surface and a digital covering map. To be specific, let be a simple closed k_i-curve with l_i elements in Z^n_i, i∈{1,2}. Then, the Cartesian product is not always a closed k-surface with some k-adjacency of Z^n₁+n₂. Thus, we provide a condition for to be a (simple) closed k-surface with some k-adjacency depending on the k_i-adjacency, i∈{1,2}. Besides, even if is not a closed k-surface, we show that the k-fundamental group of can be calculated by both a k-homotopic thinning and a strong k-deformation retract.     

Improved upper bounds on synchronizing nondeterministic automata

We show that i-directable nondeterministic automata can be i-directed with a word of length O(2ⁿ) for i=1,2, where n stands for the number of states. Since for i=1,2 there exist i-directable automata having i-directing words of length Ω(2ⁿ), these upper bounds are asymptotically optimal. We also show that a 3-directable nondeterministic automaton with n states can be 3-directed with a word of length , improving the previously known upper bound O(2ⁿ). Here the best known lower bound is .     

A tighter upper bound for random MAX 2-SAT

Given a conjunctive normal form F with n variables and m=cn 2-clauses, it is interesting to study the maximum number of clauses satisfied by all the assignments of the variables (MAX 2-SAT). When c is sufficiently large, the upper bound of of random MAX 2-SAT had been derived by the first-moment argument. In this paper, we provide a tighter upper bound (3/4)cn+g(c)cn also using the first-moment argument but correcting the error items for f(n,cn), and when considering the ε³ error item. Furthermore, we extrapolate the region of the validity of the first-moment method is c>2.4094 by analyzing the higher order error items.     

Testing juntas

We show that a boolean valued function over n variables, where each variable ranges in an arbitrary probability space, can be tested for the property of depending on only J of them using a number of queries that depends only polynomially on J and the approximation parameter ε. We present several tests that require a number of queries that is polynomial in J and linear in ε⁻¹. We show a non-adaptive test that has one-sided error, an adaptive version of it that requires fewer queries, and a non-adaptive two-sided version of the test that requires the least number of queries. We also show a two-sided non-adaptive test that applies to functions over n boolean variables, and has a more compact analysis.We then provide a lower bound of on the number of queries required for the non-adaptive testing of the above property; a lower bound of for adaptive algorithms naturally follows from this. In establishing this lower bound we also prove a result about random walks on the group Z^q₂ that may be interesting in its own right. We show that for some , the distributions of the random walk at times t and t+2 are close to each other, independently of the step distribution of the walk.We also discuss related questions. In particular, when given in advance a known J-junta function , we show how to test a function for the property of being identical to up to a permutation of the variables, in a number of queries that is polynomial in J and ε⁻¹.