1,024 research outputs found
Faster deterministic sorting and priority queues in linear space
The RAM complexity of deterministic linear space sorting of integers in words is improved from to . No better bounds are known for polynomial space. In fact, the techniques give a deterministic linear space priority queue supporting insert and delete in amortized time and find-min in constant time. The priority queue can be implemented using addition, shift, and bit-wise boolean operations
Four Soviets Walk the Dog-Improved Bounds for Computing the Fr\'echet Distance
Given two polygonal curves in the plane, there are many ways to define a
notion of similarity between them. One popular measure is the Fr\'echet
distance. Since it was proposed by Alt and Godau in 1992, many variants and
extensions have been studied. Nonetheless, even more than 20 years later, the
original algorithm by Alt and Godau for computing the Fr\'echet
distance remains the state of the art (here, denotes the number of edges on
each curve). This has led Helmut Alt to conjecture that the associated decision
problem is 3SUM-hard.
In recent work, Agarwal et al. show how to break the quadratic barrier for
the discrete version of the Fr\'echet distance, where one considers sequences
of points instead of polygonal curves. Building on their work, we give a
randomized algorithm to compute the Fr\'echet distance between two polygonal
curves in time on a pointer machine
and in time on a word RAM. Furthermore, we show that
there exists an algebraic decision tree for the decision problem of depth
, for some . We believe that this
reveals an intriguing new aspect of this well-studied problem. Finally, we show
how to obtain the first subquadratic algorithm for computing the weak Fr\'echet
distance on a word RAM.Comment: 34 pages, 15 figures. A preliminary version appeared in SODA 201
Can We Overcome the Barrier for Oblivious Sorting?
It is well-known that non-comparison-based techniques
can allow us to sort elements in time
on a Random-Access Machine (RAM). On the other hand, it is a long-standing open
question whether
(non-comparison-based) circuits can sort
elements from the domain
with boolean gates.
We consider weakened forms of this question: first, we consider
a restricted class of sorting where the number of distinct keys
is much smaller than the input length; and second, we
explore Oblivious RAMs and probabilistic circuit families, i.e.,
computational models that are
somewhat more powerful than circuits but much weaker than RAM.
We show that Oblivious RAMs and probabilistic circuit families
can sort -bit keys in time or circuit
complexity where is the input length.
Our algorithms work in the balls-and-bins model, i.e., not only can they
sort an array of numerical keys --- if each key additionally
carries an opaque ball, our algorithms
can also move the balls into the correct order.
We further show that
in such a balls-and-bins model, it is impossible
to sort -bit keys
in time, and thus
the -bit-key assumption
is necessary for overcoming the barrier.
Finally, we optimize the IO efficiency of our oblivious algorithms
for RAMs --- we show that even the -bit special
case of our algorithm can solve open questions
regarding whether there exist oblivious
algorithms for tight compaction and selection in linear IO
A subquadratic algorithm for 3XOR
Given a set of binary words of equal length , the 3XOR problem
asks for three elements such that , where denotes the bitwise XOR operation. The problem can be easily solved on
a word RAM with word length in time . Using Han's fast
integer sorting algorithm (2002/2004) this can be reduced to . With randomization or a sophisticated deterministic dictionary
construction, creating a hash table for with constant lookup time leads to
an algorithm with (expected) running time . At present, seemingly no
faster algorithms are known. We present a surprisingly simple deterministic,
quadratic time algorithm for 3XOR. Its core is a version of the Patricia trie
for , which makes it possible to traverse the set in ascending
order for arbitrary in linear time.
Furthermore, we describe a randomized algorithm for 3XOR with expected
running time . The
algorithm transfers techniques to our setting that were used by Baran, Demaine,
and P\u{a}tra\c{s}cu (2005/2008) for solving the related int3SUM problem (the
same problem with integer addition in place of binary XOR) in expected time
. As suggested by Jafargholi and Viola (2016), linear hash functions
are employed. The latter authors also showed that assuming 3XOR needs expected
running time one can prove conditional lower bounds for triangle
enumeration just as with 3SUM. We demonstrate that 3XOR can be reduced to other
problems as well, treating the examples offline SetDisjointness and offline
SetIntersection, which were studied for 3SUM by Kopelowitz, Pettie, and Porat
(2016)
Compressed Subsequence Matching and Packed Tree Coloring
We present a new algorithm for subsequence matching in grammar compressed
strings. Given a grammar of size compressing a string of size and a
pattern string of size over an alphabet of size , our algorithm
uses space and or time. Here
is the word size and is the number of occurrences of the pattern. Our
algorithm uses less space than previous algorithms and is also faster for
occurrences. The algorithm uses a new data structure
that allows us to efficiently find the next occurrence of a given character
after a given position in a compressed string. This data structure in turn is
based on a new data structure for the tree color problem, where the node colors
are packed in bit strings.Comment: To appear at CPM '1
- …