Lower Bounds for Synchronizing Word Lengths in Partial Automata

Lower Bounds for Synchronizing Word Lengths in Partial Automata

It was conjectured by \v{C}ern\'y in 1964, that a synchronizing DFA on $n$ states always has a synchronizing word of length at most $(n-1)^2$, and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for $n \leq 4$, and with bounds on the number of symbols for $n \leq 10$. Here we give the full analysis for $n \leq ...

Lower Bounds on Sequence Lengths

In this paper we study the sequence length requirements of distance-based evolutionary tree building algorithms in the Jukes-Cantor model of evolution. By deriving lower bounds on sequence lengths required to recover the evolutionary tree topology correctly , we show that two algorithms, the Short Quartet Method and the Harmonic Greedy Triplets algorithms have optimal sequence length requirements.

Lower Bounds for Generalized Quantum Finite Automata

We obtain several lower bounds on the language recognition power of Nayak’s [12] generalized quantum finite automata (GQFA). Techniques for proving lower bounds on Kondacs and Watrous’ one-way quantum finite automata (KWQFA) were introduced by Ambainis and Freivalds [2], and were expanded in a series of papers. We show that many of these techniques can be adapted to prove lower bounds for GQFAs...

Some Lower Bounds for Estrada Index

Lower Bounds for Dynamic Partial Sums

Let G be a group. The partial sums problem asks to maintain an array A[1 . . n] of group elements, initialized to zeroes (a.k.a. the identity), under the following operations: update(k,∆): modify A[k]← ∆, where ∆ ∈ G. query(k): returns the partial sum ∑k i=1A[i]. For concreteness, let us work on a machine with w-bits words (w ≥ lg n), and take G to be Z/2wZ, i.e. integer arithmetic on machine w...