694 research outputs found
Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs
Let be a graph where each vertex is associated with a label. A
Vertex-Labeled Approximate Distance Oracle is a data structure that, given a
vertex and a label , returns a -approximation of
the distance from to the closest vertex with label in . Such
an oracle is dynamic if it also supports label changes. In this paper we
present three different dynamic approximate vertex-labeled distance oracles for
planar graphs, all with polylogarithmic query and update times, and nearly
linear space requirements
Speeding up shortest path algorithms
Given an arbitrary, non-negatively weighted, directed graph we
present an algorithm that computes all pairs shortest paths in time
, where is the number of
different edges contained in shortest paths and is a running
time of an algorithm to solve a single-source shortest path problem (SSSP).
This is a substantial improvement over a trivial times application of
that runs in . In our algorithm we use
as a black box and hence any improvement on results also in improvement
of our algorithm.
Furthermore, a combination of our method, Johnson's reweighting technique and
topological sorting results in an all-pairs
shortest path algorithm for arbitrarily-weighted directed acyclic graphs.
In addition, we also point out a connection between the complexity of a
certain sorting problem defined on shortest paths and SSSP.Comment: 10 page
A simpler and more efficient algorithm for the next-to-shortest path problem
Given an undirected graph with positive edge lengths and two
vertices and , the next-to-shortest path problem is to find an -path
which length is minimum amongst all -paths strictly longer than the
shortest path length. In this paper we show that the problem can be solved in
linear time if the distances from and to all other vertices are given.
Particularly our new algorithm runs in time for general
graphs, which improves the previous result of time for sparse
graphs, and takes only linear time for unweighted graphs, planar graphs, and
graphs with positive integer edge lengths.Comment: Partial result appeared in COCOA201
Dynamic Range Majority Data Structures
Given a set of coloured points on the real line, we study the problem of
answering range -majority (or "heavy hitter") queries on . More
specifically, for a query range , we want to return each colour that is
assigned to more than an -fraction of the points contained in . We
present a new data structure for answering range -majority queries on a
dynamic set of points, where . Our data structure uses O(n)
space, supports queries in time, and updates in amortized time. If the coordinates of the points are integers,
then the query time can be improved to . For constant values of , this improved query
time matches an existing lower bound, for any data structure with
polylogarithmic update time. We also generalize our data structure to handle
sets of points in d-dimensions, for , as well as dynamic arrays, in
which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201
Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search
The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a
general technique for constructing a data structure to answer approximate near
neighbor queries by using a distribution over locality-sensitive
hash functions that partition space. For a collection of points, after
preprocessing, the query time is dominated by evaluations
of hash functions from and hash table lookups and
distance computations where is determined by the
locality-sensitivity properties of . It follows from a recent
result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive
hash functions can be reduced to , leaving the query time to be
dominated by distance computations and
additional word-RAM operations. We state this result as a general framework and
provide a simpler analysis showing that the number of lookups and distance
computations closely match the Indyk-Motwani framework, making it a viable
replacement in practice. Using ideas from another locality-sensitive hashing
framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of
additional word-RAM operations to .Comment: 15 pages, 3 figure
Cross-Document Pattern Matching
We study a new variant of the string matching problem called cross-document
string matching, which is the problem of indexing a collection of documents to
support an efficient search for a pattern in a selected document, where the
pattern itself is a substring of another document. Several variants of this
problem are considered, and efficient linear-space solutions are proposed with
query time bounds that either do not depend at all on the pattern size or
depend on it in a very limited way (doubly logarithmic). As a side result, we
propose an improved solution to the weighted level ancestor problem
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
Widest Paths and Global Propagation in Bounded Value Iteration for Stochastic Games
Solving stochastic games with the reachability objective is a fundamental
problem, especially in quantitative verification and synthesis. For this
purpose, bounded value iteration (BVI) attracts attention as an efficient
iterative method. However, BVI's performance is often impeded by costly end
component (EC) computation that is needed to ensure convergence. Our
contribution is a novel BVI algorithm that conducts, in addition to local
propagation by the Bellman update that is typical of BVI, global propagation of
upper bounds that is not hindered by ECs. To conduct global propagation in a
computationally tractable manner, we construct a weighted graph and solve the
widest path problem in it. Our experiments show the algorithm's performance
advantage over the previous BVI algorithms that rely on EC computation.Comment: v2: an URL to the implementation is adde
âEqually unequal or unequally equalâ: Adopting a substantive equality approach to gender discrimination in Nigeria
The purpose of this article is to critically assess the approach of Nigerian courts to interpreting section 42 of the Constitution. This article argues that Nigerian courts are yet to develop a substantive equality approach to interpreting section 42 of the Constitution. Rather, the courts have tended to adopt the formal equality approach to interpreting the section. Analysing some decisions of the Court of Appeal and the Supreme Court, the article argues that in order to safeguard womenâs rights and address gender inequality in the country, Nigerian courts should lean towards substantive equality approach to the interpretation of section 42 of the Constitution. This is not only consistent with Nigeriaâs obligations under international law but also crucial to addressing historical imbalances between men and women in the country
Longest Increasing Subsequence under Persistent Comparison Errors
We study the problem of computing a longest increasing subsequence in a
sequence of distinct elements in the presence of persistent comparison
errors. In this model, every comparison between two elements can return the
wrong result with some fixed (small) probability , and comparisons cannot
be repeated. Computing the longest increasing subsequence exactly is impossible
in this model, therefore, the objective is to identify a subsequence that (i)
is indeed increasing and (ii) has a length that approximates the length of the
longest increasing subsequence.
We present asymptotically tight upper and lower bounds on both the
approximation factor and the running time. In particular, we present an
algorithm that computes an -approximation in time , with
high probability. This approximation relies on the fact that that we can
approximately sort elements in time such that the maximum
dislocation of an element is at most . For the lower bounds, we
prove that (i) there is a set of sequences, such that on a sequence picked
randomly from this set every algorithm must return an -approximation with high probability, and (ii) any -approximation
algorithm for longest increasing subsequence requires
comparisons, even in the absence of errors
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