653 research outputs found
Succinct Partial Sums and Fenwick Trees
We consider the well-studied partial sums problem in succint space where one
is to maintain an array of n k-bit integers subject to updates such that
partial sums queries can be efficiently answered. We present two succint
versions of the Fenwick Tree - which is known for its simplicity and
practicality. Our results hold in the encoding model where one is allowed to
reuse the space from the input data. Our main result is the first that only
requires nk + o(n) bits of space while still supporting sum/update in O(log_b
n) / O(b log_b n) time where 2 <= b <= log^O(1) n. The second result shows how
optimal time for sum/update can be achieved while only slightly increasing the
space usage to nk + o(nk) bits. Beyond Fenwick Trees, the results are primarily
based on bit-packing and sampling - making them very practical - and they also
allow for simple optimal parallelization
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
Combining All Pairs Shortest Paths and All Pairs Bottleneck Paths Problems
We introduce a new problem that combines the well known All Pairs Shortest
Paths (APSP) problem and the All Pairs Bottleneck Paths (APBP) problem to
compute the shortest paths for all pairs of vertices for all possible flow
amounts. We call this new problem the All Pairs Shortest Paths for All Flows
(APSP-AF) problem. We firstly solve the APSP-AF problem on directed graphs with
unit edge costs and real edge capacities in
time,
where is the number of vertices, is the number of distinct edge
capacities (flow amounts) and is the time taken
to multiply two -by- matrices over a ring. Secondly we extend the problem
to graphs with positive integer edge costs and present an algorithm with
worst case time complexity, where is
the upper bound on edge costs
Cache-Oblivious Persistence
Partial persistence is a general transformation that takes a data structure
and allows queries to be executed on any past state of the structure. The
cache-oblivious model is the leading model of a modern multi-level memory
hierarchy.We present the first general transformation for making
cache-oblivious model data structures partially persistent
Lower bounds in the quantum cell probe model
We introduce a new model for studying quantum data structure problems --- the "quantum cell probe model". We prove a lower bound for the static predecessor problem in the 'address-only' version of this model where, essentially, we allow quantum parallelism only over the 'address lines' of the queries. This model subsumes the classical cell probe model, and many quantum query algorithms like Grover's algorithm fall into this framework. We prove our lower bound by obtaining a round elimination lemma for quantum communication complexity. A similar lemma was proved by Miltersen, Nisan, Safra and Wigderson for classical communication complexity, but their proof does not generalise to the quantum setting. We also study the static membership problem in the quantum cell probe model. Generalising a result of Yao, we show that if the storage scheme is 'implicit', that is it can only store members of the subset and 'pointers', then any quantum query scheme must make \Omega(\log n) probes. We also consider the one-round quantum communication complexity of set membership and show tight bounds
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
Optimal Color Range Reporting in One Dimension
Color (or categorical) range reporting is a variant of the orthogonal range
reporting problem in which every point in the input is assigned a \emph{color}.
While the answer to an orthogonal point reporting query contains all points in
the query range , the answer to a color reporting query contains only
distinct colors of points in . In this paper we describe an O(N)-space data
structure that answers one-dimensional color reporting queries in optimal
time, where is the number of colors in the answer and is the
number of points in the data structure. Our result can be also dynamized and
extended to the external memory model
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
On dualization in products of forests, in
Abstract. Let P = P1 ×...×Pn be the product of n partially ordered sets, each with an acyclic precedence graph in which either the in-degree or the out-degree of each element is bounded. Given a subset A⊆P,it is shown that the set of maximal independent elements of A in P can be incrementally generated in quasi-polynomial time. We discuss some applications in data mining related to this dualization problem
Separating Hierarchical and General Hub Labelings
In the context of distance oracles, a labeling algorithm computes vertex
labels during preprocessing. An query computes the corresponding distance
from the labels of and only, without looking at the input graph. Hub
labels is a class of labels that has been extensively studied. Performance of
the hub label query depends on the label size. Hierarchical labels are a
natural special kind of hub labels. These labels are related to other problems
and can be computed more efficiently. This brings up a natural question of the
quality of hierarchical labels. We show that there is a gap: optimal
hierarchical labels can be polynomially bigger than the general hub labels. To
prove this result, we give tight upper and lower bounds on the size of
hierarchical and general labels for hypercubes.Comment: 11 pages, minor corrections, MFCS 201
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