75 research outputs found
Threesomes, Degenerates, and Love Triangles
The 3SUM problem is to decide, given a set of real numbers, whether any
three sum to zero. It is widely conjectured that a trivial -time
algorithm is optimal and over the years the consequences of this conjecture
have been revealed. This 3SUM conjecture implies lower bounds on
numerous problems in computational geometry and a variant of the conjecture
implies strong lower bounds on triangle enumeration, dynamic graph algorithms,
and string matching data structures.
In this paper we refute the 3SUM conjecture. We prove that the decision tree
complexity of 3SUM is and give two subquadratic 3SUM
algorithms, a deterministic one running in
time and a randomized one running in time with
high probability. Our results lead directly to improved bounds for -variate
linear degeneracy testing for all odd . The problem is to decide, given
a linear function and a set , whether . We show the
decision tree complexity of this problem is .
Finally, we give a subcubic algorithm for a generalization of the
-product over real-valued matrices and apply it to the problem of
finding zero-weight triangles in weighted graphs. We give a
depth- decision tree for this problem, as well as an
algorithm running in time
Encoding 3SUM
We consider the following problem: given three sets of real numbers, output a
word-RAM data structure from which we can efficiently recover the sign of the
sum of any triple of numbers, one in each set. This is similar to a previous
work by some of the authors to encode the order type of a finite set of points.
While this previous work showed that it was possible to achieve slightly
subquadratic space and logarithmic query time, we show here that for the
simpler 3SUM problem, one can achieve an encoding that takes
space for inputs sets of size and allows constant
time queries in the word-RAM
Dynamic Set Intersection
Consider the problem of maintaining a family of dynamic sets subject to
insertions, deletions, and set-intersection reporting queries: given , report every member of in any order. We show that in the word
RAM model, where is the word size, given a cap on the maximum size of
any set, we can support set intersection queries in
expected time, and updates in expected time. Using this algorithm
we can list all triangles of a graph in
expected time, where and
is the arboricity of . This improves a 30-year old triangle enumeration
algorithm of Chiba and Nishizeki running in time.
We provide an incremental data structure on that supports intersection
{\em witness} queries, where we only need to find {\em one} .
Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected
time, where . Finally, we provide time/space tradeoffs for
the fully dynamic set intersection reporting problem. Using words of space,
each update costs expected time, each reporting query
costs expected time where
is the size of the output, and each witness query costs expected time.Comment: Accepted to WADS 201
Geometric Pattern Matching Reduces to k-SUM
We prove that some exact geometric pattern matching problems reduce in linear time to o k-SUM when the pattern has a fixed size k. This holds in the real RAM model for searching for a similar copy of a set of k ? 3 points within a set of n points in the plane, and for searching for an affine image of a set of k ? d+2 points within a set of n points in d-space.
As corollaries, we obtain improved real RAM algorithms and decision trees for the two problems. In particular, they can be solved by algebraic decision trees of near-linear height
Deterministic Time-Space Tradeoffs for k-SUM
Given a set of numbers, the -SUM problem asks for a subset of numbers
that sums to zero. When the numbers are integers, the time and space complexity
of -SUM is generally studied in the word-RAM model; when the numbers are
reals, the complexity is studied in the real-RAM model, and space is measured
by the number of reals held in memory at any point.
We present a time and space efficient deterministic self-reduction for the
-SUM problem which holds for both models, and has many interesting
consequences. To illustrate:
* -SUM is in deterministic time and space
. In general, any
polylogarithmic-time improvement over quadratic time for -SUM can be
converted into an algorithm with an identical time improvement but low space
complexity as well. * -SUM is in deterministic time and space
, derandomizing an algorithm of Wang.
* A popular conjecture states that 3-SUM requires time on the
word-RAM. We show that the 3-SUM Conjecture is in fact equivalent to the
(seemingly weaker) conjecture that every -space algorithm for
-SUM requires at least time on the word-RAM.
* For , -SUM is in deterministic time and
space
- …