38 research outputs found
An In-Place Sorting with O(n log n) Comparisons and O(n) Moves
We present the first in-place algorithm for sorting an array of size n that
performs, in the worst case, at most O(n log n) element comparisons and O(n)
element transports.
This solves a long-standing open problem, stated explicitly, e.g., in [J.I.
Munro and V. Raman, Sorting with minimum data movement, J. Algorithms, 13,
374-93, 1992], of whether there exists a sorting algorithm that matches the
asymptotic lower bounds on all computational resources simultaneously
Two-Way Automata Making Choices Only at the Endmarkers
The question of the state-size cost for simulation of two-way
nondeterministic automata (2NFAs) by two-way deterministic automata (2DFAs) was
raised in 1978 and, despite many attempts, it is still open. Subsequently, the
problem was attacked by restricting the power of 2DFAs (e.g., using a
restricted input head movement) to the degree for which it was already possible
to derive some exponential gaps between the weaker model and the standard
2NFAs. Here we use an opposite approach, increasing the power of 2DFAs to the
degree for which it is still possible to obtain a subexponential conversion
from the stronger model to the standard 2DFAs. In particular, it turns out that
subexponential conversion is possible for two-way automata that make
nondeterministic choices only when the input head scans one of the input tape
endmarkers. However, there is no restriction on the input head movement. This
implies that an exponential gap between 2NFAs and 2DFAs can be obtained only
for unrestricted 2NFAs using capabilities beyond the proposed new model. As an
additional bonus, conversion into a machine for the complement of the original
language is polynomial in this model. The same holds for making such machines
self-verifying, halting, or unambiguous. Finally, any superpolynomial lower
bound for the simulation of such machines by standard 2DFAs would imply LNL.
In the same way, the alternating version of these machines is related to L =?
NL =? P, the classical computational complexity problems.Comment: 23 page
Refinement of the Alternating Space Hierarchy
We refine the alternating space hierarchy by separating the classes break sgak spa(s(n)) and piak spa(s(n)) from deak spa(s(n)) as well as from break deak+1 spa(s(n)), for each s(n)in Omega(Ćlgn) cap o(Ćgn), and k geq 2. We also present unary (tally) sets separating sga2 spa(s(n)) and pia2 spa(s(n)) from break dea2 spa(s(n)) as well as from dea3 spa(s(n))
Converting two-way nondeterministic unary automata into simpler automata
AbstractWe show that, on inputs of length exceeding 5n2, any n-state unary two-way nondeterministic finite automaton (2nfa) can be simulated by a (2n+2)-state quasi-sweeping 2nfa. Such a result, besides providing a ânormal formâ for 2nfa's, enables us to get a subexponential simulation of unary 2nfa's by two-way deterministic finite automata (2dfa's). In fact, we prove that any n-state unary 2nfa can be simulated by a sweeping 2dfa with O(nâlog2(n+1)+3â) states
Translation from Classical Two-Way Automata to Pebble Two-Way Automata
We study the relation between the standard two-way automata and more powerful
devices, namely, two-way finite automata with an additional "pebble" movable
along the input tape. Similarly as in the case of the classical two-way
machines, it is not known whether there exists a polynomial trade-off, in the
number of states, between the nondeterministic and deterministic pebble two-way
automata. However, we show that these two machine models are not independent:
if there exists a polynomial trade-off for the classical two-way automata, then
there must also exist a polynomial trade-off for the pebble two-way automata.
Thus, we have an upward collapse (or a downward separation) from the classical
two-way automata to more powerful pebble automata, still staying within the
class of regular languages. The same upward collapse holds for complementation
of nondeterministic two-way machines.
These results are obtained by showing that each pebble machine can be, by
using suitable inputs, simulated by a classical two-way automaton with a linear
number of states (and vice versa), despite the existing exponential blow-up
between the classical and pebble two-way machines
Graph-Controlled Insertion-Deletion Systems
In this article, we consider the operations of insertion and deletion working
in a graph-controlled manner. We show that like in the case of context-free
productions, the computational power is strictly increased when using a control
graph: computational completeness can be obtained by systems with insertion or
deletion rules involving at most two symbols in a contextual or in a
context-free manner and with the control graph having only four nodes.Comment: In Proceedings DCFS 2010, arXiv:1008.127
A representation of recursively enumerable languages by two homomorphisms and a quotient
AbstractA new representation for recursively enumerable languages is presented. It uses a pair of homomorphisms and the left (or right) quotient: For each recursively enumerable language L one can find homomorphisms h1, h2: ââA â ââB, such that w â ââL is a word in L if and only if w =h1(α)h2(α) for some αââ+A. (Or, each recursively enumerable language can be given by L = O(h1h2) â© ââL, where O(h1h2) is the so-called right overflow languaged defined as O(h1h2) = {h1(x)h2(x); x â ââA}.