27,649 research outputs found
Instance complexity of Boolean functions
In the area of query complexity of Boolean functions, the most widely studied
cost measure of an algorithm is the worst-case number of queries made by it on
an input. Motivated by the most natural cost measure studied in online
algorithms, the competitive ratio, we consider a different cost measure for
query algorithms for Boolean functions that captures the ratio of the cost of
the algorithm and the cost of an optimal algorithm that knows the input in
advance. The cost of an algorithm is its largest cost over all inputs.
Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boolean
functions, and dubbed it instance complexity. Grossman et al. showed, among
other results, that monotone Boolean functions with instance complexity 1 are
precisely those that depend on one or two variables.
We complement the above-mentioned result of Grossman et al. by completely
characterizing the instance complexity of symmetric Boolean functions. As a
corollary we conclude that the only symmetric Boolean functions with instance
complexity 1 are the Parity function and its complement. We also study the
instance complexity of some graph properties like Connectivity and k-clique
containment.
In all the Boolean functions we study above, and those studied by Grossman et
al., the instance complexity turns out to be the ratio of query complexity to
minimum certificate complexity. It is a natural question to ask if this is the
correct bound for all Boolean functions. We show a negative answer in a very
strong sense, by analyzing the instance complexity of the Greater-Than and
Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the
input size for both of these functions, while we exhibit algorithms for which
the instance complexity is a constant
Weak Parity
We study the query complexity of Weak Parity: the problem of computing the
parity of an n-bit input string, where one only has to succeed on a 1/2+eps
fraction of input strings, but must do so with high probability on those inputs
where one does succeed. It is well-known that n randomized queries and n/2
quantum queries are needed to compute parity on all inputs. But surprisingly,
we give a randomized algorithm for Weak Parity that makes only
O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only
O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of
Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove
lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in
the quantum case for any eps>0. We show that improving our lower bounds is
intimately related to two longstanding open problems about Boolean functions:
the Sensitivity Conjecture, and the relationships between query complexity and
polynomial degree.Comment: 18 page
Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs
In this paper, we present a quantum algorithm for dynamic programming
approach for problems on directed acyclic graphs (DAGs). The running time of
the algorithm is , and the running time of the
best known deterministic algorithm is , where is the number of
vertices, is the number of vertices with at least one outgoing edge;
is the number of edges. We show that we can solve problems that use OR,
AND, NAND, MAX and MIN functions as the main transition steps. The approach is
useful for a couple of problems. One of them is computing a Boolean formula
that is represented by Zhegalkin polynomial, a Boolean circuit with shared
input and non-constant depth evaluating. Another two are the single source
longest paths search for weighted DAGs and the diameter search problem for
unweighted DAGs.Comment: UCNC2019 Conference pape
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
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