34,525 research outputs found
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
Amortized Dynamic Cell-Probe Lower Bounds from Four-Party Communication
This paper develops a new technique for proving amortized, randomized
cell-probe lower bounds on dynamic data structure problems. We introduce a new
randomized nondeterministic four-party communication model that enables
"accelerated", error-preserving simulations of dynamic data structures.
We use this technique to prove an cell-probe
lower bound for the dynamic 2D weighted orthogonal range counting problem
(2D-ORC) with updates and queries, that holds even
for data structures with success probability. This
result not only proves the highest amortized lower bound to date, but is also
tight in the strongest possible sense, as a matching upper bound can be
obtained by a deterministic data structure with worst-case operational time.
This is the first demonstration of a "sharp threshold" phenomenon for dynamic
data structures.
Our broader motivation is that cell-probe lower bounds for exponentially
small success facilitate reductions from dynamic to static data structures. As
a proof-of-concept, we show that a slightly strengthened version of our lower
bound would imply an lower bound for the
static 3D-ORC problem with space. Such result would give a
near quadratic improvement over the highest known static cell-probe lower
bound, and break the long standing barrier for static data
structures
Probabilistic embeddings of the Fr\'echet distance
The Fr\'echet distance is a popular distance measure for curves which
naturally lends itself to fundamental computational tasks, such as clustering,
nearest-neighbor searching, and spherical range searching in the corresponding
metric space. However, its inherent complexity poses considerable computational
challenges in practice. To address this problem we study distortion of the
probabilistic embedding that results from projecting the curves to a randomly
chosen line. Such an embedding could be used in combination with, e.g.
locality-sensitive hashing. We show that in the worst case and under reasonable
assumptions, the discrete Fr\'echet distance between two polygonal curves of
complexity in , where , degrades
by a factor linear in with constant probability. We show upper and lower
bounds on the distortion. We also evaluate our findings empirically on a
benchmark data set. The preliminary experimental results stand in stark
contrast with our lower bounds. They indicate that highly distorted projections
happen very rarely in practice, and only for strongly conditioned input curves.
Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
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Data Structures and Algorithms for Disjoint Set Union Problems
This paper surveys algorithmic techniques and data structures that have been proposed to solve the set union problem and its variants. Their discovery required a new set of algorithmic tools that have proven useful in other areas. Special attention is devoted to recent extensions of the original set union problem, and some effort is made to provide a unifying theoretical framework for this growing body of algorithms
Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model
In this paper, we develop a new communication model to prove a data structure
lower bound for the dynamic interval union problem. The problem is to maintain
a multiset of intervals over with integer coordinates,
supporting the following operations:
- insert(a, b): add an interval to , provided that
and are integers in ;
- delete(a, b): delete a (previously inserted) interval from
;
- query(): return the total length of the union of all intervals in
.
It is related to the two-dimensional case of Klee's measure problem. We prove
that there is a distribution over sequences of operations with
insertions and deletions, and queries, for which any data
structure with any constant error probability requires time
in expectation. Interestingly, we use the sparse set disjointness protocol of
H\aa{}stad and Wigderson [ToC'07] to speed up a reduction from a new kind of
nondeterministic communication games, for which we prove lower bounds.
For applications, we prove lower bounds for several dynamic graph problems by
reducing them from dynamic interval union
A lower bound for the complexity of the union-split-find problem
We prove a Theta(loglog n) (i.e. matching upper and lower) bound on the complexity of the Union-Split-Find problem, a variant of the Union-Find problem. Our lower bound holds for all pointer machine algorithms and does not require the separation assumption used in the lower bound arguments of Tarjan [T79] and Blum [B86]. We complement this with a Theta(log n) bound for the Split-Find problem under the separation assumption. This shows that the separation assumption can imply an exponential loss in efficiency
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