605 research outputs found
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
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
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
Lower Bounds for Oblivious Near-Neighbor Search
We prove an lower bound on the dynamic
cell-probe complexity of statistically
approximate-near-neighbor search () over the -dimensional
Hamming cube. For the natural setting of , our result
implies an lower bound, which is a quadratic
improvement over the highest (non-oblivious) cell-probe lower bound for
. This is the first super-logarithmic
lower bound for against general (non black-box) data structures.
We also show that any oblivious data structure for
decomposable search problems (like ) can be obliviously dynamized
with overhead in update and query time, strengthening a classic
result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page
Faster Worst Case Deterministic Dynamic Connectivity
We present a deterministic dynamic connectivity data structure for undirected
graphs with worst case update time and constant query time. This improves on the previous best
deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein
Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time
. All other algorithms for dynamic connectivity are either
randomized (Monte Carlo) or have only amortized performance guarantees
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