354 research outputs found
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Crosslinking in parallel
A crosslink is a double link established between the two entries of an edge in an adjacency list representation of a graph. Crosslinks play important roles in several parallel algorithms as they provide constant time access between the two entries of an edge; the existence of crosslinks is usually assumed. We consider the problem of establishing crosslinks in a crosslink-less adjacency list for graphs that belong to a class of graphs called the linearly contractible graphs, and show that cross-links can be established optimally in O(log n log*n) time using a CREW PRAM and optimally in O(log n) time using a CRCW PRAM for such graphs
Dynamic Ordered Sets with Approximate Queries, Approximate Heaps and Soft Heaps
We consider word RAM data structures for maintaining ordered sets of integers whose select and rank operations are allowed to return approximate results, i.e., ranks, or items whose rank, differ by less than Delta from the exact answer, where Delta=Delta(n) is an error parameter. Related to approximate select and rank is approximate (one-dimensional) nearest-neighbor. A special case of approximate select queries are approximate min queries. Data structures that support approximate min operations are known as approximate heaps (priority queues). Related to approximate heaps are soft heaps, which are approximate heaps with a different notion of approximation.
We prove the optimality of all the data structures presented, either through matching cell-probe lower bounds, or through equivalences to well studied static problems. For approximate select, rank, and nearest-neighbor operations we get matching cell-probe lower bounds. We prove an equivalence between approximate min operations, i.e., approximate heaps, and the static partitioning problem. Finally, we prove an equivalence between soft heaps and the classical sorting problem, on a smaller number of items.
Our results have many interesting and unexpected consequences. It turns out that approximation greatly speeds up some of these operations, while others are almost unaffected. In particular, while select and rank have identical operation times, both in comparison-based and word RAM implementations, an interesting separation emerges between the approximate versions of these operations in the word RAM model. Approximate select is much faster than approximate rank. It also turns out that approximate min is exponentially faster than the more general approximate select. Next, we show that implementing soft heaps is harder than implementing approximate heaps. The relation between them corresponds to the relation between sorting and partitioning.
Finally, as an interesting byproduct, we observe that a combination of known techniques yields a deterministic word RAM algorithm for (exactly) sorting n items in O(n log log_w n) time, where w is the word length. Even for the easier problem of finding duplicates, the best previous deterministic bound was O(min{n log log n,n log_w n}). Our new unifying bound is an improvement when w is sufficiently large compared with n
Wavelet Trees Meet Suffix Trees
We present an improved wavelet tree construction algorithm and discuss its
applications to a number of rank/select problems for integer keys and strings.
Given a string of length n over an alphabet of size , our
method builds the wavelet tree in time,
improving upon the state-of-the-art algorithm by a factor of .
As a consequence, given an array of n integers we can construct in time a data structure consisting of machine words and
capable of answering rank/select queries for the subranges of the array in
time. This is a -factor improvement in
query time compared to Chan and P\u{a}tra\c{s}cu and a -factor
improvement in construction time compared to Brodal et al.
Next, we switch to stringological context and propose a novel notion of
wavelet suffix trees. For a string w of length n, this data structure occupies
words, takes time to construct, and simultaneously
captures the combinatorial structure of substrings of w while enabling
efficient top-down traversal and binary search. In particular, with a wavelet
suffix tree we are able to answer in time the following two
natural analogues of rank/select queries for suffixes of substrings: for
substrings x and y of w count the number of suffixes of x that are
lexicographically smaller than y, and for a substring x of w and an integer k,
find the k-th lexicographically smallest suffix of x.
We further show that wavelet suffix trees allow to compute a
run-length-encoded Burrows-Wheeler transform of a substring x of w in time, where s denotes the length of the resulting run-length encoding.
This answers a question by Cormode and Muthukrishnan, who considered an
analogous problem for Lempel-Ziv compression.Comment: 33 pages, 5 figures; preliminary version published at SODA 201
Deterministic sub-linear space LCE data structures with efficient construction
Given a string of symbols, a longest common extension query
asks for the length of the longest common prefix of the
th and th suffixes of . LCE queries have several important
applications in string processing, perhaps most notably to suffix sorting.
Recently, Bille et al. (J. Discrete Algorithms 25:42-50, 2014, Proc. CPM 2015:
65-76) described several data structures for answering LCE queries that offers
a space-time trade-off between data structure size and query time. In
particular, for a parameter , their best deterministic
solution is a data structure of size which allows LCE queries to be
answered in time. However, the construction time for all
deterministic versions of their data structure is quadratic in . In this
paper, we propose a deterministic solution that achieves a similar space-time
trade-off of query time using
space, but significantly improve the construction time to
.Comment: updated titl
Parallel Wavelet Tree Construction
We present parallel algorithms for wavelet tree construction with
polylogarithmic depth, improving upon the linear depth of the recent parallel
algorithms by Fuentes-Sepulveda et al. We experimentally show on a 40-core
machine with two-way hyper-threading that we outperform the existing parallel
algorithms by 1.3--5.6x and achieve up to 27x speedup over the sequential
algorithm on a variety of real-world and artificial inputs. Our algorithms show
good scalability with increasing thread count, input size and alphabet size. We
also discuss extensions to variants of the standard wavelet tree.Comment: This is a longer version of the paper that appears in the Proceedings
of the IEEE Data Compression Conference, 201
Fast deterministic processor allocation
Interval allocation has been suggested as a possible formalization for the PRAM of the (vaguely defined) processor allocation problem, which is of fundamental importance in parallel computing. The interval allocation problem is, given nonnegative integers , to allocate nonoverlapping subarrays of sizes from within a base array of cells. We show that interval allocation problems of size can be solved in time with optimal speedup on a deterministic CRCW PRAM. In addition to a general solution to the processor allocation problem, this implies an improved deterministic algorithm for the problem of approximate summation. For both interval allocation and approximate summation, the fastest previous deterministic algorithms have running times of . We also describe an application to the problem of computing the connected components of an undirected graph
Perfectly Oblivious (Parallel) RAM Revisited, and Improved Constructions
Oblivious RAM (ORAM)
is a technique for compiling any RAM program to an oblivious counterpart, i.e.,
one whose access patterns do not leak information about the secret inputs.
Similarly, Oblivious Parallel RAM (OPRAM) compiles a
{\it parallel} RAM program to an oblivious counterpart.
In this paper, we care about ORAM/OPRAM with {\it perfect security}, i.e.,
the access patterns must be {\it identically distributed}
no matter what the program\u27s memory request sequence is.
In the past, two types of perfect ORAMs/OPRAMs
have been considered:
constructions whose performance bounds hold {\it in expectation} (but may occasionally
run more slowly);
and constructions whose performance bounds hold {\it deterministically} (even though
the algorithms themselves are randomized).
In this paper, we revisit the performance metrics for perfect
ORAM/OPRAM, and
show novel constructions that achieve asymptotical improvements
for all performance metrics.
Our first result
is a new perfectly secure OPRAM
scheme with {\it expected} overhead.
In comparison, prior literature
has been stuck at for more than a decade.
Next, we show how to construct a perfect ORAM
with
{\it deterministic} simulation overhead. We further show how
to make the scheme parallel, resulting in an perfect OPRAM
with
{\it deterministic} simulation overhead.
For perfect ORAMs/OPRAMs
with deterministic performance bounds, our results achieve
{\it subexponential} improvement over the state-of-the-art.
Specifically, the best known prior scheme
incurs more than deterministic simulation overhead
(Raskin and Simkin, Asiacrypt\u2719); moreover, their scheme works
only for the sequential setting and is {\it not} amenable to parallelization.
Finally, we additionally consider perfect ORAMs/OPRAMs
whose performance bounds hold with high probability.
For this new performance metric, we show new constructions
whose simulation overhead is upper bounded by
except with negligible in probability, i.e., we prove
high-probability performance bounds that match the expected
bounds mentioned earlier
An information theoretic necessary condition for perfect reconstruction
This article proposes a new information theoretic necessary condition for
reconstructing a discrete random variable based on the knowledge of a set
of discrete functions of . The reconstruction condition is derived from the
Shannon's Lattice of Information (LoI) \cite{Shannon53} and two entropic
metrics proposed respectively by Shannon and Rajski.
This theoretical material being relatively unknown and/or dispersed in
different references, we provide a complete and synthetic description of the
LoI concepts like the total, common and complementary informations with
complete proofs. The two entropic metrics definitions and properties are also
fully detailled and showed compatible with the LoI structure. A new geometric
interpretation of the Lattice structure is then investigated that leads to a
new necessary condition for reconstructing the discrete random variable
given a set ,..., of elements of the lattice generated by
.
Finally, this condition is derived in five specific examples of
reconstruction of from a set of deterministic functions of : the
reconstruction of a symmetric random variable from the knowledge of its sign
and of its absolute value, the reconstruction of a binary word from a set of
binary linear combinations, the reconstruction of an integer from its prime
signature (Fundamental theorem of arithmetics) and from its reminders modulo a
set of coprime integers (Chinese reminder theorem), and the reconstruction of
the sorting permutation of a list from a set of 2-by-2 comparisons. In each
case, the necessary condition is shown compatible with the corresponding
well-known results.Comment: 17 pages, 9 figure
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