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 × 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

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 $f$) 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