0000000001257227
AUTHOR
Krišjānis Prūsis
A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity
Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy, is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity and 0-sensitivity and 0-block sensitivity. Previously the best known lower bound was $C_1(f)\geq \frac{bs_0(f)}{2 s_0(f)}$, achieved by Kenyon and Kutin. We improve this to $C_1(f)\geq \frac{3 bs_0(f)}{2 s_0(f)}$. While this improvement is only by a constant factor, this is quite important, as it precludes achi…
Error-Free Affine, Unitary, and Probabilistic OBDDs
We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las-Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata counterparts of these models.
Error-Free Affine, Unitary, and Probabilistic OBDDs
We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata versions of these models.
Būla funkciju bloku jūtīgums
Darbā tiek pētīta neatrisināta problēma par divu Būla funkciju sarežģītības mēru - jūtīguma un bloku jūtīguma - sakarību. Darba ietvaros tiek aplūkota arī šo mēru saistība ar vēl vienu sarežģītības mēru - sertifikātu sarežģītību, jo funkcijās, kas maksimizē meklēto atšķirību, sertifikātu sarežģītība ir vienāda ar jūtīgumu. Darbā tiek pierādīta apakšējā robeža sertifikātu sarežģītībai attiecībā pret bloku jūtīguma un jūtīguma attiecību. Tā izslēdz iespēju uzlabot meklēto atšķirību, izmantojot konstrukciju, ar ko iegūti līdzšinējie rezultāti. Šī robeža arī ir asimptotiski cieša, jo labākā zināmā konstrukcija to sasniedz.
Doubling the success of quantum walk search using internal-state measurements
In typical discrete-time quantum walk algorithms, one measures the position of the walker while ignoring its internal spin/coin state. Rather than neglecting the information in this internal state, we show that additionally measuring it doubles the success probability of many quantum spatial search algorithms. For example, this allows Grover's unstructured search problem to be solved with certainty, rather than with probability 1/2 if only the walker's position is measured, so the additional measurement yields a search algorithm that is twice as fast as without it, on average. Thus the internal state of discrete-time quantum walks holds valuable information that can be utilized to improve a…
Stationary states in quantum walk search
When classically searching a database, having additional correct answers makes the search easier. For a discrete-time quantum walk searching a graph for a marked vertex, however, additional marked vertices can make the search harder by causing the system to approximately begin in a stationary state, so the system fails to evolve. In this paper, we completely characterize the stationary states, or 1-eigenvectors, of the quantum walk search operator for general graphs and configurations of marked vertices by decomposing their amplitudes into uniform and flip states. This infinitely expands the number of known stationary states and gives an optimization procedure to find the stationary state c…
On Block Sensitivity and Fractional Block Sensitivity
We investigate the relation between the block sensitivity bs(f) and fractional block sensitivity fbs(f) complexity measures of Boolean functions. While it is known that fbs(f) = O(bs(f)2), the best known separation achieves $${\rm{fbs}}\left( f \right) = \left( {{{\left( {3\sqrt 2 } \right)}^{ - 1}} + o\left( 1 \right)} \right){\rm{bs}}{\left( f \right)^{3/2}}$$ . We improve the constant factor and show a family of functions that give fbs(f) = (6−1/2 − o(1)) bs(f)3/2.
All Classical Adversary Methods Are Equivalent for Total Functions
We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions and are equal to the fractional block sensitivity fbs( f ). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between fbs( f ) and other adversary bounds, as well as between the adversary bounds themselves. We also show that, for partial functions, fractional block sensitivity canno…
Exact affine counter automata
We introduce an affine generalization of counter automata, and analyze their ability as well as affine finite automata. Our contributions are as follows. We show that there is a language that can be recognized by exact realtime affine counter automata but by neither 1-way deterministic pushdown automata nor realtime deterministic k-counter automata. We also show that a certain promise problem, which is conjectured not to be solved by two-way quantum finite automata in polynomial time, can be solved by Las Vegas affine finite automata. Lastly, we show that how a counter helps for affine finite automata by showing that the language MANYTWINS, which is conjectured not to be recognized by affin…
A Potential Field Function for Overlapping Point Set and Graph Cluster Visualization
In this paper we address the problem of visualizing overlapping sets of points with a fixed positioning in a comprehensible way. A standard visualization technique is to enclose the point sets in isocontours generated by bounding a potential field function. The most commonly used functions are various approximations of the Gaussian distribution. Such an approach produces smooth and appealing shapes, however it may produce an incorrect point nesting in generated regions, e.g. some point is contained inside a foreign set region. We introduce a different potential field function that keeps the desired properties of Gaussian distribution, and in addition guarantees that every point belongs to a…
A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity
Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy [7], is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity and 0-sensitivity and 0-block sensitivity.
Starpībkopu atbalsta bibliotēka
Darbā “Starpībkopu atbalsta bibliotēka” aprakstīta C++ bibliotēkas darbam ar starpībkopām izstrāde. Bibliotēka ļauj pārbaudīt starpībkopu ar konkrētiem parametriem esamību,(Lai gan to nevar pateikt visiem parametru komplektiem.) konstruēt sešu tipu starpībkopas, un ir veidota tā, lai būtu viegli to papildināt ar vēl citiem starpībkopu veidiem. Bibliotēka tiek pasniegta pirmkoda viedā.
Oscillatory Localization of Quantum Walks Analyzed by Classical Electric Circuits
We examine an unexplored quantum phenomenon we call oscillatory localization, where a discrete-time quantum walk with Grover's diffusion coin jumps back and forth between two vertices. We then connect it to the power dissipation of a related electric network. Namely, we show that there are only two kinds of oscillating states, called uniform states and flip states, and that the projection of an arbitrary state onto a flip state is bounded by the power dissipation of an electric circuit. By applying this framework to states along a single edge of a graph, we show that low effective resistance implies oscillatory localization of the quantum walk. This reveals that oscillatory localization occ…
Full Characterization of Oscillatory Localization of Quantum Walks
Discrete-time quantum walks are well-known for exhibiting localization, a quantum phenomenon where the walker remains at its initial location with high probability. In companion with a joint Letter, we introduce oscillatory localization, where the walker alternates between two states. The walk is given by the flip-flop shift, which is easily defined on non-lattice graphs, and the Grover coin. Extremely simple examples of the localization exist, such as a walker jumping back and forth between two vertices of the complete graph. We show that only two kinds of states, called flip states and uniform states, exhibit exact oscillatory localization. So the projection of an arbitrary state onto the…