[pt] Neste trabalho, apresentamos algoritmos aproximativos para dois problemas de otimização em árvores. Na primeira parte, consideramos o Problema de Atribuição de k-Hotlinks. Seja G= (V,E) um grafo direcionado acíclico representando um web site, onde nós correspondem a páginas e arcos correspondem a hyperlinks. Nesse contexto, hotlink são definidos como atalhos (novos arcos) adicionados às páginas de G de modo a reduzir o tempo gasto pelos usuários para alcançarem as informações desejadas. Neste trabalho consideramos o problema onde G é uma árvore enraizada e o objetivo é minimizar o tempo médio gasto pelos usuários atribuindo no máximo k hotlinks a cada nó. Para a versão mais estudada desse problema onde no máximo um hotlink pode ser atribuído a cada nó, provamos a existência de um FPTAS. Isso representa uma significante melhora em relação ao algoritmo com aproximação constante obtido recentemente em [Jacobs, WADS 2007]. Além disso, desenvolvemos o primeiro algoritmo com aproximação constante para a versão mais geral onde k hotlinks podem ser atribuídos a cada nó. Na segunda parte deste trabalho, consideramos o problema de computar estratégias eficientes para realizar buscas em árvores. Como uma generalização da tradicional busca binária em listas ordenadas, suponha que se deseja encontrar um nó específico (porém desconhecido) de uma árvore realizando consultas em seus arcos, onde cada consulta indica a extremidade do arco mais próxima ao nó desejado. Dada a probabilidade de cada nó ser aquele procurado, o objetivo é computar uma estratégia de busca que minimize o número esperado de consultas. Aplicações práticas desse problema incluem sincronização de file systems e testes de software. Apresentamos um algoritmo linear que obtém a primeira aproximação constante para esse problema. Isso representa uma melhora significativa em relação à O(log n)-aproximação anterior. / [en] Here we present a study on two optimization problems in trees: the k- Hotlink Assignment Problem and the problem of Binary Searching in Trees. As a result, we obtain improved approximation algorithms for both problem. The k-Hotlink Assignment Problem can be defined as follows. Let G = (V,E) be a directed acyclic graph representing a web site, where nodes correspond to pages and arcs to hyperlinks. In this context, hotlinks are defined as shortcuts (new arcs) added to web pages of G in order to reduce the time spent by users to reach their desired information. Here we consider the problem where G is a rooted directed tree and the goal is minimizing the expected time spent by users by assigning at most k hotlinks to each node. For the most studied version of this problem where at most one hotlink can be assigned from each node, we prove the existence of an FPTAS, improving upon the constant factor algorithm recently obtained in [Jacobs, WADS 2007]. In addition, we develop the first constant factor approximation algorithm for the most general version where k hotlinks can be assigned from each node. In the second part of this work, we consider the problem of computing efficient strategies for searching in trees. As a generalization of the classical binary search for ordered lists, suppose one wishes to find a (unknown) specific node of a tree by asking queries to its arcs, where each query indicates the endpoint closer to the desired node. Given the likelihood of each node being the one searched, the objective is to compute a search strategy that minimizes the expected number of queries. Practical applications of this problem include file system synchronization and software testing. Here we present a linear time algorithm which is the first constant factor approximation for this problem. This represents a significant improvement over previous O(log n)-approximation.

Ο παγκόσμιος ιστός έχει εδραιωθεί πλέον ως το δημοφιλέστερο μέσο ανάκτησης πληροφοριών. Όπως είναι λογικό, όσο παλαιώνει τόσο μεγαλύτερη πληροφορία εμπεριέχει. Πληθαίνουν έτσι εκείνοι οι ιστότοποι που γιγαντώνονται άναρχα και ενώ σαν στόχο έχουν να προσφέρουν την πληροφορία στον χρήστη που τους επισκέπτεται, λόγω του τεράστιου όγκου της, κάνουν συχνά δύσκολη την πρόσβαση σε συγκεκριμένα κομμάτια αυτής. Με στόχο την αντιμετώπιση αυτής της κατάστασης, αναπτύσσονται τα τελευταία χρόνια αλγόριθμοι ανάθεσης υπερσυνδέσμων σε ιστοτόπους. Η λογική τους είναι ο εντοπισμός της πιο δημοφιλούς ή πιθανής πληροφορίας και η εξασφάλιση καλύτερης πρόσβασης σε αυτήν, αναθέτοντας υπερσυνδέσμους (hotlinks) προς τις ιστοσελίδες που την περιέχουν. Οι αλγόριθμοι αυτοί εφαρμόζονται όχι σε πραγματικές αναπαραστάσεις ιστοτόπων, αλλά κατά κανόνα στα αντίστοιχα κατευθυνόμενα άκυκλα γραφήματα (DAG) αυτών. Όπως είναι γνωστό κανένας ιστότοπος δεν έχει μορφή DAG, με συνέπεια να υπάρχει μία απόσταση από τη θεωρητική ανεύρεση υπερσυνδέσμων και την πιθανή εφαρμογή τους στην πραγματικότητα. Σε αυτήν την εργασία ασχολούμαστε αρχικά με την μεθοδική καταγραφή της πραγματικής συνδεσμολογίας ενός ιστότοπου, που αποτελεί ένα πρώτο βήμα στην ανάθεση υπερσυνδέσμων σε πραγματικούς ιστοτόπους. Αυτό επιτυγχάνεται με την κατάλληλη προδιαγραφή και υλοποίηση μιας δικτυακής μηχανής αναζήτησης, ώστε να ανταποκρίνεται στις ανάγκες μας. Προτείνουμε στη συνέχεια το εργαλείο ‘HotLink Visualizer’, το οποίο αρχικά μετατρέπει την πληροφορία της συνδεσμολογίας ενός ιστοτόπου σε απλά δεδομένα μορφής πίνακα και στη συνέχεια οπτικοποιεί το αποτέλεσμα. Τέλος, υλοποιεί την απευθείας ανάθεση υπερσυνδέσμων προσθέτοντας αυτόματα μέσα στις σελίδες του ιστοτόπου τους υπερσυνδέσμους και οπτικοποιεί εκ νέου το αποτέλεσμα. Παρέχει έτσι τη δυνατότητα διατήρησης διαφορετικών εκδόσεων της μορφής ενός ιστοτόπου, ανάλογα με το σύνολο από υπερσυνδέσμους που έχουν ανατεθεί σε αυτό. / The World Wide Web has become established as the most popular source of information retrieval. As expected, the older it gets the more information it contains and thus the number of the web sites with gigantic growth and bad information access rates is constantly increased within it. During the last years the matter is being addressed with the development of several hotlink assignment algorithms for web sites. The main idea behind those algorithms is to spot the most popular or more likely to be accessed piece of information and provide better access to it by assigning links (hotlinks) to the web pages containing it. These algorithms are not applied to the actual representations of these web sites but usually to their corresponding direct acyclic graphs (DAGs). However, it is widely known that a web site in its true form is not a DAG, since there can be found hundreds of links pointing to just one page. Hence, there is a gap between the theoretical determination of a set of hotlinks and the possible application of this set to a real web site. In this paper we first address the issue of recording and persisting the exact map of a web site with its full connectivity, which can be considered as a first step towards the assignment of hotlinks in real web sites. We succeed in that, with the appropriate specification and implementation of a web crawler, with functionality suited to our specific needs. We then propose an administrative tool, the ‘Hotlink Visualizer’, which, after persisting in tabular data all the necessary information to capture a web site’s real map, visualizes the outcome and implements hotlink additions by adding with an automated procedure the generated hotlinks in the web pages of the site. Thus we have the ability to maintain in row data different forms and versions of the originally parsed web site, as it can be formed from the assignment of different hotlink sets to it.