19,109 research outputs found
Arithmetic for Rooted Trees
We propose a new arithmetic for non-empty rooted unordered trees simply
called trees. After discussing tree representation and enumeration, we define
the operations of tree addition, multiplication and stretch, prove their
properties, and show that all trees can be generated from a starting tree of
one vertex. We then show how a given tree can be obtained as the sum or product
of two trees, thus defining prime trees with respect to addition and
multiplication. In both cases we show how primality can be decided in time
polynomial in the number of vertices and we prove that factorization is unique.
We then define negative trees and suggest dealing with tree equations, giving
some preliminary results. Finally we comment on how our arithmetic might be
useful, and discuss preceding studies that have some relations with our. To the
best of our knowledge our approach and results are completely new aside for an
earlier version of this work submitte as an arXiv manuscript.Comment: 18 pages, 8 figure
UPGMA and the normalized equidistant minimum evolution problem
UPGMA (Unweighted Pair Group Method with Arithmetic Mean) is a widely used clustering method. Here we show that UPGMA is a greedy heuristic for the normalized equidistant minimum evolution (NEME) problem, that is, finding a rooted tree that minimizes the minimum evolution score relative to the dissimilarity matrix among all rooted trees with the same leaf-set in which all leaves have the same distance to the root. We prove that the NEME problem is NP-hard. In addition, we present some heuristic and approximation algorithms for solving the NEME problem, including a polynomial time algorithm that yields a binary, rooted tree whose NEME score is within O(log2n) of the optimum
Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations
We consider systems of recursively defined combinatorial structures. We give
algorithms checking that these systems are well founded, computing generating
series and providing numerical values. Our framework is an articulation of the
constructible classes of Flajolet and Sedgewick with Joyal's species theory. We
extend the implicit species theorem to structures of size zero. A quadratic
iterative Newton method is shown to solve well-founded systems combinatorially.
From there, truncations of the corresponding generating series are obtained in
quasi-optimal complexity. This iteration transfers to a numerical scheme that
converges unconditionally to the values of the generating series inside their
disk of convergence. These results provide important subroutines in random
generation. Finally, the approach is extended to combinatorial differential
systems.Comment: 61 page
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