72 research outputs found
Incremental and Decremental Maintenance of Planar Width
We present an algorithm for maintaining the width of a planar point set
dynamically, as points are inserted or deleted. Our algorithm takes time
O(kn^epsilon) per update, where k is the amount of change the update causes in
the convex hull, n is the number of points in the set, and epsilon is any
arbitrarily small constant. For incremental or decremental update sequences,
the amortized time per update is O(n^epsilon).Comment: 7 pages; 2 figures. A preliminary version of this paper was presented
at the 10th ACM/SIAM Symp. Discrete Algorithms (SODA '99); this is the
journal version, and will appear in J. Algorithm
Low-Cost Piezoelectric Sensors for Time Domain Load Monitoring of Metallic Structures During Operational and Maintenance Processes
The versatility of piezoelectric sensors in measurement techniques and their performance in applications has given rise to an increased interest in their use for structural and manufacturing component monitoring. They enable wireless and sensor network solutions to be developed in order to directly integrate the sensors into machines, fixtures and tools. Piezoelectric sensors increasingly compete with strain-gauges due to their wide operational temperature range, load and strain sensing accuracy, low power consumption and low cost. This research sets out the use of piezoelectric sensors for real-time monitoring of mechanical strength in metallic structures in the ongoing operational control of machinery components. The behaviour of aluminium and steel structures under flexural strength was studied using piezoelectric sensors. Variations in structural behaviour and geometry were measured, and the load and μstrains during operational conditions were quantified in the time domain at a specific frequency. The lead zirconium titanate (PZT) sensors were able to distinguish between material types and thicknesses. Moreover, this work covers frequency selection and optimisation from 20 Hz to 300 kHz. Significant differences in terms of optimal operating frequencies and sensitivity were found in both structures. The influence of the PZT voltage applied was assessed to reduce power consumption without signal loss, and calibration to μstrains and loads was performed.This research was funded by Basque Government, grant number KK-2019/00051-SMARTRESNAK and
by the European Commission, grant number 869884- RECLAIM
Dynamic Connectivity in Disk Graphs
Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated
radius rs > 0. Let D(S) be the disk intersection graph defined by S, i.e., the graph
with vertex set S and an edge between two distinct sites s, t ∈ S if and only if the
disks with centers s, t and radii rs , rt intersect. Our goal is to design data structures
that maintain the connectivity structure of D(S) as sites are inserted and/or deleted
in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S.
For this case, we describe a data structure that has O(log2 n) amortized update time
and O(log n/ log log n) query time. Second, we look at disk graphs with bounded
radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a parameter Ψ that is
known in advance. Here, we not only investigate the fully dynamic case, but also the
incremental and the decremental scenario, where only insertions or only deletions of
sites are allowed. In the fully dynamic case, we achieve amortized expected update
time O(Ψ log4 n) and query time O(log n/ log log n). This improves the currently
best update time by a factor of Ψ. In the incremental case, we achieve logarithmic
dependency on Ψ, with a data structure that has O(α(n)) amortized query time and
O(log Ψ log4 n) amortized expected update time, where α(n) denotes the inverse Ackermann
function. For the decremental setting, we first develop an efficient decremental
disk revealing data structure: given two sets R and B of disks in the plane, we can delete
disks from B, and upon each deletion, we receive a list of all disks in R that no longer
intersect the union of B. Using this data structure, we get decremental data structures
with a query time of O(log n/ log log n) that supports deletions in O(n log Ψ log4 n)
overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 n)
overall expected time for disk graphs with arbitrary radii, assuming that the deletion
sequence is oblivious of the internal random choices of the data structures
Deterministic Decremental Reachability, SCC, and Shortest Paths via Directed Expanders and Congestion Balancing
Let be a weighted, digraph subject to a sequence of adversarial
edge deletions. In the decremental single-source reachability problem (SSR), we
are given a fixed source and the goal is to maintain a data structure that
can answer path-queries for any . In the more
general single-source shortest paths (SSSP) problem the goal is to return an
approximate shortest path to , and in the SCC problem the goal is to
maintain strongly connected components of and to answer path queries within
each component. All of these problems have been very actively studied over the
past two decades, but all the fast algorithms are randomized and, more
significantly, they can only answer path queries if they assume a weaker model:
they assume an oblivious adversary which is not adaptive and must fix the
update sequence in advance. This assumption significantly limits the use of
these data structures, most notably preventing them from being used as
subroutines in static algorithms. All the above problems are notoriously
difficult in the adaptive setting. In fact, the state-of-the-art is still the
Even and Shiloach tree, which dates back all the way to 1981 and achieves total
update time . We present the first algorithms to break through this
barrier:
1) deterministic decremental SSR/SCC with total update time
2) deterministic decremental SSSP with total update time .
To achieve these results, we develop two general techniques of broader
interest for working with dynamic graphs: 1) a generalization of expander-based
tools to dynamic directed graphs, and 2) a technique that we call congestion
balancing and which provides a new method for maintaining flow under
adversarial deletions. Using the second technique, we provide the first
near-optimal algorithm for decremental bipartite matching.Comment: Reuploaded with some generalizations of previous theorem
Two Approaches to Building Time-Windowed Geometric Data Structures
Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.\u27s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.\u27s O(n log^2 n) and O(n log n loglog n) solutions respectively.
Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems
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Dynamic Algorithms for Shortest Paths and Matching
There is a long history of research in theoretical computer science devoted to designing efficient algorithms for graph problems. In many modern applications the graph in question is changing over time, and we would like to avoid rerunning our algorithm on the entire graph every time a small change occurs. The evolving nature of graphs motivates the dynamic graph model, in which the goal is to minimize the amount of work needed to reoptimize the solution when the graph changes. There is a large body of literature on dynamic algorithms for basic problems that arise in graphs. This thesis presents several improved dynamic algorithms for two fundamental graph problems: shortest paths, and matching
A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow
We give a deterministic time algorithm that computes exact
maximum flows and minimum-cost flows on directed graphs with edges and
polynomially bounded integral demands, costs, and capacities. As a consequence,
we obtain the first running time improvement for deterministic algorithms that
compute maximum-flow in graphs with polynomial bounded capacities since the
work of Goldberg-Rao [J.ACM '98].
Our algorithm builds on the framework of
Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow
by computing a sequence of -approximate undirected minimum-ratio
cycles. We develop a deterministic dynamic graph data-structure to compute such
a sequence of minimum-ratio cycles in an amortized time per edge
update. Our key technical contributions are deterministic analogues of the
vertex sparsification and edge sparsification components of the data-structure
from Chen et al. For the vertex sparsification component, we give a method to
avoid the randomness in Chen et al. which involved sampling random trees to
recurse on. For the edge sparsification component, we design a deterministic
algorithm that maintains an embedding of a dynamic graph into a sparse spanner.
We also show how our dynamic spanner can be applied to give a deterministic
data structure that maintains a fully dynamic low-stretch spanning tree on
graphs with polynomially bounded edge lengths, with subpolynomial average
stretch and subpolynomial amortized time per edge update.Comment: Accepted to FOCS 202
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