96 research outputs found
Lower bounds in the quantum cell probe model
We introduce a new model for studying quantum data structure problems --- the "quantum cell probe model". We prove a lower bound for the static predecessor problem in the 'address-only' version of this model where, essentially, we allow quantum parallelism only over the 'address lines' of the queries. This model subsumes the classical cell probe model, and many quantum query algorithms like Grover's algorithm fall into this framework. We prove our lower bound by obtaining a round elimination lemma for quantum communication complexity. A similar lemma was proved by Miltersen, Nisan, Safra and Wigderson for classical communication complexity, but their proof does not generalise to the quantum setting. We also study the static membership problem in the quantum cell probe model. Generalising a result of Yao, we show that if the storage scheme is 'implicit', that is it can only store members of the subset and 'pointers', then any quantum query scheme must make \Omega(\log n) probes. We also consider the one-round quantum communication complexity of set membership and show tight bounds
Optimal Color Range Reporting in One Dimension
Color (or categorical) range reporting is a variant of the orthogonal range
reporting problem in which every point in the input is assigned a \emph{color}.
While the answer to an orthogonal point reporting query contains all points in
the query range , the answer to a color reporting query contains only
distinct colors of points in . In this paper we describe an O(N)-space data
structure that answers one-dimensional color reporting queries in optimal
time, where is the number of colors in the answer and is the
number of points in the data structure. Our result can be also dynamized and
extended to the external memory model
Timeability in Extensive-Form Games
Extensive-form games constitute the standard representation scheme for games
with a temporal component. But do all extensive-form games correspond to
protocols that we can implement in the real world? We often rule out games with
imperfect recall, which prescribe that an agent forget something that she knew
before. In this paper, we show that even some games with perfect recall can be
problematic to implement. Specifically, we show that if the agents have a sense
of time passing (say, access to a clock), then some extensive-form games can no
longer be implemented; no matter how we attempt to time the game, some
information will leak to the agents that they are not supposed to have. We say
such a game is not exactly timeable. We provide easy-to-check necessary and
sufficient conditions for a game to be exactly timeable. Most of the technical
depth of the paper concerns how to approximately time games, which we show can
always be done, though it may require large amounts of time. Specifically, we
show that for some games the time required to approximately implement the game
grows as a power tower of height proportional to the number of players and with
a parameter that measures the precision of the approximation at the top of the
power tower. In practice, that makes the games untimeable. Besides the
conceptual contribution to game theory, we believe our methodology can have
applications to preventing information leakage in security protocols.Comment: 28 pages, 2 figure
Are bitvectors optimal?
We study the it static membership problem: Given a set S of at most n keys drawn from a universe U of size m, store it so that queries of the form "Is u in S?" can be answered by making few accesses to the memory. We study schemes for this problem that use space close to the information theoretic lower bound of bits and yet answer queries by reading a small number of bits of the memory. We show that, for , there is a scheme that stores bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most . We consider schemes that make no error for queries in S but are allowed to err with probability at most for queries not in S. We show that there exist such schemes that store bits and answer queries using just one bitprobe. If multiple probes are allowed, then the number of bits stored can be reduced to for any . The schemes mentioned above are based on probabilistic constructions of set systems with small intersections. We show lower bounds that come close to our upper bounds (for a large range of n and ): Schemes that answer queries with just one bitprobe and error probability must use bits of storage; if the error is restricted to queries not in S, then the scheme must use bits of storage. We also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes
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