684 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]
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
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
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
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs
A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ?}) update time and O(m^{1 -?}) query time, for any small ? > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching
Data Structures Lower Bounds and Popular Conjectures
In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li [Li and Li, 2004] and several data structure problems such as non-adaptive function inversion of Hellman [M. Hellman, 1980] and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation
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