11,082 research outputs found
Time lower bounds for nonadaptive turnstile streaming algorithms
We say a turnstile streaming algorithm is "non-adaptive" if, during updates,
the memory cells written and read depend only on the index being updated and
random coins tossed at the beginning of the stream (and not on the memory
contents of the algorithm). Memory cells read during queries may be decided
upon adaptively. All known turnstile streaming algorithms in the literature are
non-adaptive.
We prove the first non-trivial update time lower bounds for both randomized
and deterministic turnstile streaming algorithms, which hold when the
algorithms are non-adaptive. While there has been abundant success in proving
space lower bounds, there have been no non-trivial update time lower bounds in
the turnstile model. Our lower bounds hold against classically studied problems
such as heavy hitters, point query, entropy estimation, and moment estimation.
In some cases of deterministic algorithms, our lower bounds nearly match known
upper bounds
Cell-Probe Lower Bounds from Online Communication Complexity
In this work, we introduce an online model for communication complexity.
Analogous to how online algorithms receive their input piece-by-piece, our
model presents one of the players, Bob, his input piece-by-piece, and has the
players Alice and Bob cooperate to compute a result each time before the next
piece is revealed to Bob. This model has a closer and more natural
correspondence to dynamic data structures than classic communication models do,
and hence presents a new perspective on data structures.
We first present a tight lower bound for the online set intersection problem
in the online communication model, demonstrating a general approach for proving
online communication lower bounds. The online communication model prevents a
batching trick that classic communication complexity allows, and yields a
stronger lower bound. We then apply the online communication model to prove
data structure lower bounds for two dynamic data structure problems: the Group
Range problem and the Dynamic Connectivity problem for forests. Both of the
problems admit a worst case -time data structure. Using online
communication complexity, we prove a tight cell-probe lower bound for each:
spending (even amortized) time per operation results in at best an
probability of correctly answering a
-fraction of the queries
The Quantum Complexity of Set Membership
We study the quantum complexity of the static set membership problem: given a
subset S (|S| \leq n) of a universe of size m (m \gg n), store it as a table of
bits so that queries of the form `Is x \in S?' can be answered. The goal is to
use a small table and yet answer queries using few bitprobes. This problem was
considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh, where
lower and upper bounds were shown for this problem in the classical
deterministic and randomized models. In this paper, we formulate this problem
in the "quantum bitprobe model" and show tradeoff results between space and
time.In this model, the storage scheme is classical but the query scheme is
quantum.We show, roughly speaking, that similar lower bounds hold in the
quantum model as in the classical model, which imply that the classical upper
bounds are more or less tight even in the quantum case. Our lower bounds are
proved using linear algebraic techniques.Comment: 19 pages, a preliminary version appeared in FOCS 2000. This is the
journal version, which will appear in Algorithmica (Special issue on Quantum
Computation and Quantum Cryptography). This version corrects some bugs in the
parameters of some theorem
Cell-Probe Bounds for Online Edit Distance and Other Pattern Matching Problems
We give cell-probe bounds for the computation of edit distance, Hamming
distance, convolution and longest common subsequence in a stream. In this
model, a fixed string of symbols is given and one -bit symbol
arrives at a time in a stream. After each symbol arrives, the distance between
the fixed string and a suffix of most recent symbols of the stream is reported.
The cell-probe model is perhaps the strongest model of computation for showing
data structure lower bounds, subsuming in particular the popular word-RAM
model.
* We first give an lower bound for
the time to give each output for both online Hamming distance and convolution,
where is the word size. This bound relies on a new encoding scheme and for
the first time holds even when is as small as a single bit.
* We then consider the online edit distance and longest common subsequence
problems in the bit-probe model () with a constant sized input alphabet.
We give a lower bound of which
applies for both problems. This second set of results relies both on our new
encoding scheme as well as a carefully constructed hard distribution.
* Finally, for the online edit distance problem we show that there is an
upper bound in the cell-probe model. This bound gives a
contrast to our new lower bound and also establishes an exponential gap between
the known cell-probe and RAM model complexities.Comment: 32 pages, 4 figure
Compressing Sparse Sequences under Local Decodability Constraints
We consider a variable-length source coding problem subject to local
decodability constraints. In particular, we investigate the blocklength scaling
behavior attainable by encodings of -sparse binary sequences, under the
constraint that any source bit can be correctly decoded upon probing at most
codeword bits. We consider both adaptive and non-adaptive access models,
and derive upper and lower bounds that often coincide up to constant factors.
Notably, such a characterization for the fixed-blocklength analog of our
problem remains unknown, despite considerable research over the last three
decades. Connections to communication complexity are also briefly discussed.Comment: 8 pages, 1 figure. First five pages to appear in 2015 International
Symposium on Information Theory. This version contains supplementary materia
On Data Structures and Asymmetric Communication Complexity
In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the ``asymmetric'' case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds, and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our ``round elimination'' lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the ``round reduction'' techniques used in many previous lower bound proofs
- …