449 research outputs found
Approximate Hamming distance in a stream
We consider the problem of computing a -approximation of the
Hamming distance between a pattern of length and successive substrings of a
stream. We first look at the one-way randomised communication complexity of
this problem, giving Alice the first half of the stream and Bob the second
half. We show the following: (1) If Alice and Bob both share the pattern then
there is an bit randomised one-way communication
protocol. (2) If only Alice has the pattern then there is an
bit randomised one-way communication protocol.
We then go on to develop small space streaming algorithms for
-approximate Hamming distance which give worst case running time
guarantees per arriving symbol. (1) For binary input alphabets there is an
space and
time streaming -approximate Hamming distance algorithm. (2) For
general input alphabets there is an
space and time streaming
-approximate Hamming distance algorithm.Comment: Submitted to ICALP' 201
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
Element Distinctness, Frequency Moments, and Sliding Windows
We derive new time-space tradeoff lower bounds and algorithms for exactly
computing statistics of input data, including frequency moments, element
distinctness, and order statistics, that are simple to calculate for sorted
data. We develop a randomized algorithm for the element distinctness problem
whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than
previous lower bounds for comparison-based algorithms, showing that element
distinctness is strictly easier than sorting for randomized branching programs.
This algorithm is based on a new time and space efficient algorithm for finding
all collisions of a function f from a finite set to itself that are reachable
by iterating f from a given set of starting points. We further show that our
element distinctness algorithm can be extended at only a polylogarithmic factor
cost to solve the element distinctness problem over sliding windows, where the
task is to take an input of length 2n-1 and produce an output for each window
of length n, giving n outputs in total. In contrast, we show a time-space
tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to
compute the number of distinct elements over sliding windows. The same lower
bound holds for computing the low-order bit of F_0 and computing any frequency
moment F_k, k neq 1. This shows that those frequency moments and the decision
problem F_0 mod 2 are strictly harder than element distinctness. We complement
this lower bound with a T in O(n^2/S) comparison-based deterministic RAM
algorithm for exactly computing F_k over sliding windows, nearly matching both
our lower bound for the sliding-window version and the comparison-based lower
bounds for the single-window version. We further exhibit a quantum algorithm
for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider
the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437
New Unconditional Hardness Results for Dynamic and Online Problems
There has been a resurgence of interest in lower bounds whose truth rests on
the conjectured hardness of well known computational problems. These
conditional lower bounds have become important and popular due to the painfully
slow progress on proving strong unconditional lower bounds. Nevertheless, the
long term goal is to replace these conditional bounds with unconditional ones.
In this paper we make progress in this direction by studying the cell probe
complexity of two conjectured to be hard problems of particular importance:
matrix-vector multiplication and a version of dynamic set disjointness known as
Patrascu's Multiphase Problem. We give improved unconditional lower bounds for
these problems as well as introducing new proof techniques of independent
interest. These include a technique capable of proving strong threshold lower
bounds of the following form: If we insist on having a very fast query time,
then the update time has to be slow enough to compute a lookup table with the
answer to every possible query. This is the first time a lower bound of this
type has been proven
Upper and lower bounds for dynamic data structures on strings
We consider a range of simply stated dynamic data structure problems on
strings. An update changes one symbol in the input and a query asks us to
compute some function of the pattern of length and a substring of a longer
text. We give both conditional and unconditional lower bounds for variants of
exact matching with wildcards, inner product, and Hamming distance computation
via a sequence of reductions. As an example, we show that there does not exist
an time algorithm for a large range of these problems
unless the online Boolean matrix-vector multiplication conjecture is false. We
also provide nearly matching upper bounds for most of the problems we consider.Comment: Accepted at STACS'1
Silicon Burning II: Quasi-Equilibrium and Explosive Burning
Having examined the application of quasi-equilibrium to hydrostatic silicon
burning in Paper I of this series, Hix & Thielemann (1996), we now turn our
attention to explosive silicon burning. Previous authors have shown that for
material which is heated to high temperature by a passing shock and then cooled
by adiabatic expansion, the results can be divided into three broad categories;
\emph{incomplete burning}, \emph{normal freezeout} and \emph{-rich
freezeout}, with the outcome depending on the temperature, density and cooling
timescale. In all three cases, we find that the important abundances obey
quasi-equilibrium for temperatures greater than approximately 3 GK, with
relatively little nucleosynthesis occurring following the breakdown of
quasi-equilibrium. We will show that quasi-equilibrium provides better
abundance estimates than global nuclear statistical equilibrium, even for
normal freezeout and particularly for -rich freezeout. We will also
examine the accuracy with which the final nuclear abundances can be estimated
from quasi-equilibrium.Comment: 27 pages, including 15 inline figures. LaTeX 2e with aaspp4 and
graphicx packages. Accepted to Ap
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