4,230 research outputs found
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
Let f be a univariate polynomial with real coefficients, f in R[X].
Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes
methods) are widely used for isolating the real roots of f in a given interval.
In this paper, we consider a simple subdivision algorithm whose primitives are
purely numerical (e.g., function evaluation). The complexity of this algorithm
is adaptive because the algorithm makes decisions based on local data. The
complexity analysis of adaptive algorithms (and this algorithm in particular)
is a new challenge for computer science. In this paper, we compute the size of
the subdivision tree for the SqFreeEVAL algorithm.
The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is
well-known in several communities. The algorithm itself is simple, but prior
attempts to compute its complexity have proven to be quite technical and have
yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on
the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark
problem of isolating all real roots of an integer polynomial f of degree d and
whose coefficients can be written with at most L bits.
Our proof uses two amortization-based techniques: First, we use the algebraic
amortization technique of the standard Mahler-Davenport root bounds to
interpret the integral in terms of d and L. Second, we use a continuous
amortization technique based on an integral to bound the size of the
subdivision tree. This paper is the first to use the novel analysis technique
of continuous amortization to derive state of the art complexity bounds
Near Optimal Subdivision Algorithms for Real Root Isolation
We describe a subroutine that improves the running time of any subdivision
algorithm for real root isolation. The subroutine first detects clusters of
roots using a result of Ostrowski, and then uses Newton iteration to converge
to them. Near a cluster, we switch to subdivision, and proceed recursively. The
subroutine has the advantage that it is independent of the predicates used to
terminate the subdivision. This gives us an alternative and simpler approach to
recent developments of Sagraloff (2012) and Sagraloff-Mehlhorn (2013), assuming
exact arithmetic.
The subdivision tree size of our algorithm using predicates based on
Descartes's rule of signs is bounded by , which is better by
compared to known results. Our analysis differs in two key
aspects. First, we use the general technique of continuous amortization from
Burr-Krahmer-Yap (2009), and second, we use the geometry of clusters of roots
instead of the Davenport-Mahler bound. The analysis naturally extends to other
predicates.Comment: 19 pages, 3 figure
An Improved Algorithm for Incremental DFS Tree in Undirected Graphs
Depth first search (DFS) tree is one of the most well-known data structures
for designing efficient graph algorithms. Given an undirected graph
with vertices and edges, the textbook algorithm takes time to
construct a DFS tree. In this paper, we study the problem of maintaining a DFS
tree when the graph is undergoing incremental updates. Formally, we show: Given
an arbitrary online sequence of edge or vertex insertions, there is an
algorithm that reports a DFS tree in worst case time per operation, and
requires preprocessing time.
Our result improves the previous worst case update time
algorithm by Baswana et al. and the time by Nakamura and
Sadakane, and matches the trivial lower bound when it is required
to explicitly output a DFS tree.
Our result builds on the framework introduced in the breakthrough work by
Baswana et al., together with a novel use of a tree-partition lemma by Duan and
Zhan, and the celebrated fractional cascading technique by Chazelle and Guibas
Case study on economic return on investments for safety and emergency lighting in road tunnels
While planning a double-hole road tunnel with a length higher than one km, it is important to pay attention to the safety factor if an accident occurs. If there is a power outage, in order to avoid critical situations that could jeopardize the safety of the people present (facilitating the stream coming out from the tunnel and the arrival of the emergency personnel), it is really important to guarantee uninterrupted lighting of roadways, mandatory emergency lay-bys, and ways of escape. Uninterrupted service of the lighting systems supply must be guaranteed, in accordance with the current regulations, through the exertion of UPS (Uninterruptible Power Supply) and power units. During tunnel construction, such devices represent a cost that must be amortized. In this case study, which takes into consideration a section of a road tunnel characterized by emergency lay-bys and ways of escape, emergency and security lighting were planned and installation and management costs were evaluated. The goal of this research was the creation of a cash flow thanks to the energy generated by photovoltaic panels, in a way that the service life of the system (25 years) coincided with the amortization of the costs of the backup electrical equipment installation (complying with the regulations). The possibility of over-dimensioning the UPS and providing it with a proper photovoltaic panel surface (235 kWp) to generate and exchange electric energy with the grid was taken into consideration
Space-Efficient Re-Pair Compression
Re-Pair is an effective grammar-based compression scheme achieving strong
compression rates in practice. Let , , and be the text length,
alphabet size, and dictionary size of the final grammar, respectively. In their
original paper, the authors show how to compute the Re-Pair grammar in expected
linear time and words of working space on top
of the text. In this work, we propose two algorithms improving on the space of
their original solution. Our model assumes a memory word of bits and a re-writable input text composed by such words. Our
first algorithm runs in expected time and uses
words of space on top of the text for any parameter
chosen in advance. Our second algorithm runs in expected
time and improves the space to words
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