6,859 research outputs found
New and Improved Algorithms for Unordered Tree Inclusion
The tree inclusion problem is, given two node-labeled trees P and T (the "pattern tree" and the "text tree"), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is from 1995 and runs in O(poly(m,n) * 2^{2d}) = O^*(2^{2d}) time, where m and n are the sizes of the pattern and text trees, respectively, and d is the maximum outdegree of the pattern tree. Here, we develop a new algorithm that improves the exponent 2d to d by considering a particular type of ancestor-descendant relationships and applying dynamic programming, thus reducing the time complexity to O^*(2^d). We then study restricted variants of the unordered tree inclusion problem where the number of occurrences of different node labels and/or the input trees\u27 heights are bounded. We show that although the problem remains NP-hard in many such cases, it can be solved in polynomial time for c = 2 and in O^*(1.8^d) time for c = 3 if the leaves of P are distinctly labeled and each label occurs at most c times in T. We also present a randomized O^*(1.883^d)-time algorithm for the case that the heights of P and T are one and two, respectively
Deterministic Automata for Unordered Trees
Automata for unordered unranked trees are relevant for defining schemas and
queries for data trees in Json or Xml format. While the existing notions are
well-investigated concerning expressiveness, they all lack a proper notion of
determinism, which makes it difficult to distinguish subclasses of automata for
which problems such as inclusion, equivalence, and minimization can be solved
efficiently. In this paper, we propose and investigate different notions of
"horizontal determinism", starting from automata for unranked trees in which
the horizontal evaluation is performed by finite state automata. We show that a
restriction to confluent horizontal evaluation leads to polynomial-time
emptiness and universality, but still suffers from coNP-completeness of the
emptiness of binary intersections. Finally, efficient algorithms can be
obtained by imposing an order of horizontal evaluation globally for all
automata in the class. Depending on the choice of the order, we obtain
different classes of automata, each of which has the same expressiveness as
CMso.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Matching Tree-Level Matrix Elements with Interleaved Showers
We present an implementation of the so-called CKKW-L merging scheme for
combining multi-jet tree-level matrix elements with parton showers. The
implementation uses the transverse-momentum-ordered shower with interleaved
multiple interactions as implemented in PYTHIA8. We validate our procedure
using e+e--annihilation into jets and vector boson production in hadronic
collisions, with special attention to details in the algorithm which are
formally sub-leading in character, but may have visible effects in some
observables. We find substantial merging scale dependencies induced by the
enforced rapidity ordering in the default PYTHIA8 shower. If this rapidity
ordering is removed the merging scale dependence is almost negligible. We then
also find that the shower does a surprisingly good job of describing the
hardness of multi-jet events, as long as the hardest couple of jets are given
by the matrix elements. The effects of using interleaved multiple interactions
as compared to more simplistic ways of adding underlying-event effects in
vector boson production are shown to be negligible except in a few sensitive
observables. To illustrate the generality of our implementation, we also give
some example results from di-boson production and pure QCD jet production in
hadronic collisions.Comment: 44 pages, 23 figures, as published in JHEP, including all changes
recommended by the refere
Generating loop graphs via Hopf algebra in quantum field theory
We use the Hopf algebra structure of the time-ordered algebra of field
operators to generate all connected weighted Feynman graphs in a recursive and
efficient manner. The algebraic representation of the graphs is such that they
can be evaluated directly as contributions to the connected n-point functions.
The recursion proceeds by loop order and vertex number.Comment: 22 pages, LaTeX + AMS + eepic; new section with alternative recursion
formula added, further minor changes and correction
On the category of Euclidean configuration spaces and associated fibrations
We calculate the Lusternik-Schnirelmann category of the k-th ordered
configuration spaces F(R^n,k) of R^n and give bounds for the category of the
corresponding unordered configuration spaces B(R^n,k) and the sectional
category of the fibrations pi^n_k: F(R^n,k) --> B(R^n,k). We show that
secat(pi^n_k) can be expressed in terms of subspace category. In many cases,
eg, if n is a power of 2, we determine cat(B(R^n,k)) and secat(pi^n_k)
precisely.Comment: This is the version published by Geometry & Topology Monographs on 19
March 200
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