Salvador Roura

Randomized binary search trees

In this paper, we present randomized algorithms over binary search trees such that: (a) the insertion of a set of keys, in any fixed order, into an initially empty tree always produces a random binary search tree; (b) the deletion of any key from a random binary search tree results in a random binary search tree; (c) the random choices made by the algorithms are based upon the sizes of the subtrees of the tree; this implies that we can support accesses by rank without additional storage requirements or modification of the data structures; and (d) the cost of any elementary operation, measured as the number of visited nodes, is the same as the expected cost of its standard deterministic counterpart; hence, all search and update operations have guaranteed expected cost O(log n), but now irrespective of any assumption on the input distribution.

Optimal Sampling Strategies in Quicksort and Quickselect

It is well known that the performance of quicksort can be improved by selecting the median of a sample of elements as the pivot of each partitioning stage. For large samples the partitions are better, but the amount of additional comparisons and exchanges to find the median of the sample also increases. We show in this paper that the optimal sample size to minimize the average total cost of quicksort, as a function of the size n of the current subarray size, is

Improved master theorems for divide-and-conquer recurrences

a\cdot \sqrt{n} + o(\sqrt{n}\,)

Jutge.org: an educational programming judge

. We give a closed expression for a, which depends on the selection algorithm and the costs of elementary comparisons and exchanges. Moreover, we show that selecting the medians of the samples as pivots is not the best strategy when exchanges are much more expensive than comparisons. We also apply the same ideas and techniques to the analysis of quickselect and get similar results.

On the competitiveness of the move-to-front rule

This paper presents new theorems to analyze divide-and-conquer recurrences, which improve other similar ones in several aspects. In particular, these theorems provide more information, free us almost completely from technicalities like floors and ceilings, and cover a wider set of toll functions and weight distributions, stochastic recurrences included.

An Improved Master Theorem for Divide-and-Conquer Recurrences

Jutge.org is an open access educational online programming judge where students can try to solve more than 800 problems using 22 programming languages. The verdict of their solutions is computed using exhaustive test sets run under time, memory and security restrictions. By contrast to many popular online judges, Jutge.org is designed for students and instructors: On one hand, the problem repository is mainly aimed to beginners, with a clear organization and gradding. On the other hand, the system is designed as a virtual learning environment where instructors can administer their own courses, manage their roster of students and tutors, add problems, attach documents, create lists of problems, assignments, contests and exams. This paper presents Jutge.org and offers some case studies of courses using it.

Complete voting systems with two classes of voters: weightedness and counting

We consider the list access problem and show that one questionable assumption in the original cost model presented by Sleator and Tarjan (1985) and subsequent literature allowed for several competitiveness results of the move-to-front rule (MTF). We present an off-line algorithm for the list access problem and prove that, under a more realistic cost model, no on-line algorithm can be c-competitive for any constant c, MTF included.

Digital access to comparison-based tree data structures and algorithms

We present a new master theorem for the study of divide-and-conquer recursive definitions, which improves the old one in several aspects. In particular, it provides more information, frees us completely from technicalities like floors and ceilings, and covers a wider set of toll functions and weight distributions, stochastic recurrences included.

A New Method for Balancing Binary Search Trees

We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games.

Quad-kd trees

This paper presents a simple method of building tree data structures, which only requires visiting O(log N) nodes and comparing O(D) digits per search or update, where N is the number of keys and D is the length of the keys. These bounds hold independently of the order of the updates and of the digits of the keys. The additional space required by the method is asymptotically dismissable when compared to the space used by the keys and pointers. The proposed method applies either to fixed-length base-2 keys or to variable-length string keys and permits saving space for common prefixes. The same ideas can be applied to achieve algorithms that have the best bounds of quicksort or of mergesort and radixsort together.