10 research outputs found
Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds
This paper proves the first super-logarithmic lower bounds on the cell probe
complexity of dynamic boolean (a.k.a. decision) data structure problems, a
long-standing milestone in data structure lower bounds.
We introduce a new method for proving dynamic cell probe lower bounds and use
it to prove a lower bound on the operational
time of a wide range of boolean data structure problems, most notably, on the
query time of dynamic range counting over ([Pat07]). Proving an
lower bound for this problem was explicitly posed as one of
five important open problems in the late Mihai P\v{a}tra\c{s}cu's obituary
[Tho13]. This result also implies the first lower bound for the
classical 2D range counting problem, one of the most fundamental data structure
problems in computational geometry and spatial databases. We derive similar
lower bounds for boolean versions of dynamic polynomial evaluation and 2D
rectangle stabbing, and for the (non-boolean) problems of range selection and
range median.
Our technical centerpiece is a new way of "weakly" simulating dynamic data
structures using efficient one-way communication protocols with small advantage
over random guessing. This simulation involves a surprising excursion to
low-degree (Chebychev) polynomials which may be of independent interest, and
offers an entirely new algorithmic angle on the "cell sampling" method of
Panigrahy et al. [PTW10]
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
Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication
The conjectured hardness of Boolean matrix-vector multiplication has been
used with great success to prove conditional lower bounds for numerous
important data structure problems, see Henzinger et al. [STOC'15]. In recent
work, Larsen and Williams [SODA'17] attacked the problem from the upper bound
side and gave a surprising cell probe data structure (that is, we only charge
for memory accesses, while computation is free). Their cell probe data
structure answers queries in time and is succinct in the
sense that it stores the input matrix in read-only memory, plus an additional
bits on the side. In this paper, we essentially settle the
cell probe complexity of succinct Boolean matrix-vector multiplication. We
present a new cell probe data structure with query time
storing just bits on the side. We then complement our data
structure with a lower bound showing that any data structure storing bits
on the side, with must have query time satisfying . For , any data structure must have . Since lower bounds in the cell probe model also apply to
classic word-RAM data structures, the lower bounds naturally carry over. We
also prove similar lower bounds for matrix-vector multiplication over
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