179 research outputs found
Conditional Lower Bounds for Space/Time Tradeoffs
In recent years much effort has been concentrated towards achieving
polynomial time lower bounds on algorithms for solving various well-known
problems. A useful technique for showing such lower bounds is to prove them
conditionally based on well-studied hardness assumptions such as 3SUM, APSP,
SETH, etc. This line of research helps to obtain a better understanding of the
complexity inside P.
A related question asks to prove conditional space lower bounds on data
structures that are constructed to solve certain algorithmic tasks after an
initial preprocessing stage. This question received little attention in
previous research even though it has potential strong impact.
In this paper we address this question and show that surprisingly many of the
well-studied hard problems that are known to have conditional polynomial time
lower bounds are also hard when concerning space. This hardness is shown as a
tradeoff between the space consumed by the data structure and the time needed
to answer queries. The tradeoff may be either smooth or admit one or more
singularity points.
We reveal interesting connections between different space hardness
conjectures and present matching upper bounds. We also apply these hardness
conjectures to both static and dynamic problems and prove their conditional
space hardness.
We believe that this novel framework of polynomial space conjectures can play
an important role in expressing polynomial space lower bounds of many important
algorithmic problems. Moreover, it seems that it can also help in achieving a
better understanding of the hardness of their corresponding problems in terms
of time
Improved Bounds for 3SUM, -SUM, and Linear Degeneracy
Given a set of real numbers, the 3SUM problem is to decide whether there
are three of them that sum to zero. Until a recent breakthrough by Gr{\o}nlund
and Pettie [FOCS'14], a simple -time deterministic algorithm for
this problem was conjectured to be optimal. Over the years many algorithmic
problems have been shown to be reducible from the 3SUM problem or its variants,
including the more generalized forms of the problem, such as -SUM and
-variate linear degeneracy testing (-LDT). The conjectured hardness of
these problems have become extremely popular for basing conditional lower
bounds for numerous algorithmic problems in P.
In this paper, we show that the randomized -linear decision tree
complexity of 3SUM is , and that the randomized -linear
decision tree complexity of -SUM and -LDT is , for any odd
. These bounds improve (albeit randomized) the corresponding
and decision tree bounds
obtained by Gr{\o}nlund and Pettie. Our technique includes a specialized
randomized variant of fractional cascading data structure. Additionally, we
give another deterministic algorithm for 3SUM that runs in time. The latter bound matches a recent independent bound by Freund
[Algorithmica 2017], but our algorithm is somewhat simpler, due to a better use
of word-RAM model
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
Solving -SUM using few linear queries
The -SUM problem is given input real numbers to determine whether any
of them sum to zero. The problem is of tremendous importance in the
emerging field of complexity theory within , and it is in particular open
whether it admits an algorithm of complexity with . Inspired by an algorithm due to Meiser (1993), we show
that there exist linear decision trees and algebraic computation trees of depth
solving -SUM. Furthermore, we show that there exists a
randomized algorithm that runs in
time, and performs linear queries on the input. Thus, we show
that it is possible to have an algorithm with a runtime almost identical (up to
the ) to the best known algorithm but for the first time also with the
number of queries on the input a polynomial that is independent of . The
bound on the number of linear queries is also a tighter bound
than any known algorithm solving -SUM, even allowing unlimited total time
outside of the queries. By simultaneously achieving few queries to the input
without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we
deepen the understanding of this canonical problem which is a cornerstone of
complexity-within-.
We also consider a range of tradeoffs between the number of terms involved in
the queries and the depth of the decision tree. In particular, we prove that
there exist -linear decision trees of depth
Data Structure Lower Bounds for Document Indexing Problems
We study data structure problems related to document indexing and pattern
matching queries and our main contribution is to show that the pointer machine
model of computation can be extremely useful in proving high and unconditional
lower bounds that cannot be obtained in any other known model of computation
with the current techniques. Often our lower bounds match the known space-query
time trade-off curve and in fact for all the problems considered, there is a
very good and reasonable match between the our lower bounds and the known upper
bounds, at least for some choice of input parameters. The problems that we
consider are set intersection queries (both the reporting variant and the
semi-group counting variant), indexing a set of documents for two-pattern
queries, or forbidden- pattern queries, or queries with wild-cards, and
indexing an input set of gapped-patterns (or two-patterns) to find those
matching a document given at the query time.Comment: Full version of the conference version that appeared at ICALP 2016,
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