97 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
The MPA graph problem
Given an undirected graph G whose vertices are associated to different subsets of colors, the MPA problem asks for partitioning G into a minimal set of monochromatic connected subgraphs. We prove that MPA is NP-hard as well as its extensions BDMPA where the vertices of G have a bounded maximum degree, PMPA where G is planar, BDPMPA where the maximum vertex degree is bounded and G is planar, and GMPA where G consists of a p Ă— q grid
The MPA Graph Problem
Given an undirected graph G whose vertices are associated to different subsets of colors, the MPA problem asks for partitioning G into a minimal set of monochromatic connected subgraphs. We prove that MPA is NP-hard as well as its extensions BDMPA where the vertices of G have a bounded maximum degree, PMPA where G is planar, BDPMPA where the maximum vertex degree is bounded and G is planar, and GMPA
where G consists of a p by q grid
The MPA Graph Problem
Given an undirected graph G whose vertices are associated to different subsets of colors, the MPA problem asks for partitioning G into a minimal set of monochromatic connected subgraphs. We prove that MPA is NP-hard as well as its extensions BDMPA where the vertices of G have a bounded maximum degree, PMPA where G is planar, BDPMPA where the maximum vertex degree is bounded and G is planar, and GMPA
where G consists of a p by q grid
Recommended from our members
Randomness
"...This famous dialogue was imagined by Giacomo Leopardi1
. In the last sentences he
was expressing hope as mere irony, as chance will remain an enemy. But not
everybody is as pessimistic as Leopardi, nor is convinced that chance can be
characterized in any way, if it exists at all. Let us then look after this fellow, who will
be the main character of the following discussion.
Compact DSOP and partial DSOP Forms
Given a Boolean function f on n variables, a Disjoint Sum-of-Products (DSOP)
of f is a set of products (ANDs) of subsets of literals whose sum (OR) equals
f, such that no two products cover the same minterm of f. DSOP forms are a
special instance of partial DSOPs, i.e. the general case where a subset of
minterms must be covered exactly once and the other minterms (typically
corresponding to don't care conditions of ) can be covered any number of
times. We discuss finding DSOPs and partial DSOP with a minimal number of
products, a problem theoretically connected with various properties of Boolean
functions and practically relevant in the synthesis of digital circuits.
Finding an absolute minimum is hard, in fact we prove that the problem of
absolute minimization of partial DSOPs is NP-hard. Therefore it is crucial to
devise a polynomial time heuristic that compares favorably with the known
minimization tools. To this end we develop a further piece of theory starting
from the definition of the weight of a product p as a functions of the number
of fragments induced on other cubes by the selection of p, and show how product
weights can be exploited for building a class of minimization heuristics for
DSOP and partial DSOP synthesis. A set of experiments conducted on major
benchmark functions show that our method, with a family of variants, always
generates better results than the ones of previous heuristics, including the
method based on a BDD representation of f
La struttura degli algoritmi
Protagonista di questo libro è l'algoritmo, cioè la successione dei passi in cui consiste la risoluzione di un problema. Si cercano le direttrici principali che guidano la fase dell'organizzazione e del progetto di un algoritmo, i metodi per valutarne l'efficienza, i criteri di scelta di un algoritmo fra i tanti che risolvono lo stesso problema
Soluzione di ricorrenze nell'analisi di algoritmi
Si presentano alcuni semplici teoremi per risolvere le relazioni di ricorrenza lineari, bilanciate e di ordine costante, che rappresentano il tempo di funzionamento dei piu' comuni algoritmi ricorsivi
- …