Fabrizio d'Amore

On the optimal binary plane partition for sets of isothetic rectangles

Abstract We present a binary space partition algorithm for a set of disjoint isothetic rectangles. It recursively splits the set by means of isothetic cutting lines, until no two rectangles belong to the same portion of the plane. Rectangles intersected by a cutting line are split. We provide an algorithm that given n rectangles builds a linear sized Binary Space Partition with no empty regions by means of at most 4 n -1 cuts. The algorithm runs in O( n log n ) worst-case space and time. We generalize and improve a previous result achieved on isothetic line segments by Paterson and Yao.

On optimal cuts of hyperrectangles

We are given a set ofn d-dimensional (possibly intersecting) isothetic hyperrectangles. The topic of this paper is the separation of these rectangles by means of a cutting isothetic hyperplane. Thereby we assume that a rectangle which is intersected by the cutting plane iscut into two non-overlapping hyperrectangles. We investigate the behavior of several kinds of balancing functions, as well as their linear combination and present optimal and practical algorithms for computing the corresponding balanced cuts. In addition, we give tight worst-case bounds for the quality of the balanced cuts.ZusammenfassungGegeben sei eine Menge vonn (ggf. überlappenden) isothetischen Hyperrechtecken imd-dimensionalen Raum. Diese Arbeit beschäftigt sich mit Zerlegungen dieser Hyperrechteckmenge durch Schnitthyperebenen, wobei wir annehmen, daß jedes von einer Hyperebene geschnittene Hyperrechteck in zwei nicht-überlappende Hyperrechtecke zerschnitten wird. Wir untersuchen das Verhalten einiger Balancierungskriterien für Schnitte und präsentieren optimale and praktikable Algorithmen zur Berechnung der entsprechenden balancierten Schnitte. Schließlich geben wir auch scharfe Worst-case-Schranken für die bestmöglich erreichbare Qualität der balancierten Schnitte an.

The weighted list update problem and the lazy adversary

Abstract The list update problem consists in maintaining a dictionary as an unsorted linear list. Any request specifies an item to be found by sequential scanning through the list. After an item has been found, the list may be rearranged in order to reduce the cost of processing a sequence of requests. Several kinds of adversaries can be considered to analyze the behavior of heuristics for this problem. The move-to-front (MTF) heuristic is 2-competitive against a strong adversary, matching the deterministic lower bound for this problem [Sleator and Tarjan (1985)]. But, for this problem, moving elements does not help the adversary. A lazy adversary has the limitation that he can use only a static arrangement of the list to process (off-line) the sequence of requests: still, no algorithm can be better than 2-competitive against the lazy adversary [Bentley and McGeogh (1985)]. In this paper we consider the weighted list update problem (WLUP), where the cost of accessing an item depends on the item itself. It is shown that MTF is not competitive by any constant factor for this problem against a lazy adversary. Two heuristics, based on the MTF strategy, are presented for WLUP: random move-to-front is randomized and uses biased coins; counting move-to-front is deterministic, and replaces coins by counters. Both are shown to be 2-competitive against a lazy adversary. This is optimal for the deterministic case. We apply this approach for searching items in a tree, proving that any c -competitive heuristic for the weighted list update problem provides a c -competitive heuristic for the tree update problem .

Robust Region Approach to the Computation of Geometric Graphs (Extended Abstract)

We present robust algorithms for computing the minimum spanning tree, the nearest neighbor graph and the relative neighborhood graph of a set of points in the plane, under the L2 metric. Our algorithms are asymptotically optimal, and use only double precision arithmetic. As a side effect of our results, we solve a question left open by Katajainen [11] about the computation of relative neighborhood graphs.

Maintaining Maxima under Boundary Updates

Given a set of point P ∈ ℝ2, we consider the well-known maxima problem, consisting of reporting the maxima (not dominated) points of P, in the dynamic setting of boundary updates. In this setting we allow insertions and deletions of points at the extremities of P: this permits to move a resizable vertical window on the point set.

Incremental Hive Graph

The hive graph is a rectangular graph satisfying some additional condition widely used in computational geometry for solving several kinds of fundamental queries. It has been introduced by Chazelle

Competitive algorithms for the weighted list update problem

In this paper we present some deterministic and randomized algorithms for the Weight List Update Problem. In this framework a cost (weight) is associated to each item. The algorithms consist in modifying the well known Move-To-Front heuristic by adding randomness or counters in order to decide whether moving the accessed item. We prove that Random Move-To-Front and Counting Move-To-Front are 2-competitive against any static adversary, and that deterministic Move-To-Front does not share this property. We apply this approach to the management of non-modifiable trees by means of lists of successors proving that 2-competitivity property still holds.

On the Embedding of Refinements of 2-dimensional Grids

We consider the problem of constructing embeddings of 2-dimensional FEM graphs into grids. Our goal is to minimize the edge-congestion and dilation and optimize the load. We introduce some heuristics, analyze their performance, and present experimental results comparing the heuristics with the methods based on the usage of standard graph partitioning libraries.

The List Update Problem and the Retrieval of Sets

We consider the list update problem under a sequence of requests of sets of items, and for this problem we investigate the competitiveness features of two algorithms. We prove that algorithm Move-Set-to-Front (MSF) is (1+βs)-competitive, where β is the size of the largest requested set, and that a lower bound is roughly 2. We provide an upper bound to the MSF competitive ratio by relating it to the size n of the list, obtaining that the algorithm is (1+n/4)-competitive in general, and O(√n)-competitive if we add a not too restrictive constraint to the sizes of the requested sets.

On-line algorithms for networks of temporal constraints

We consider a semi-dynamic setting for the Temporal Constraint Satisfaction Problem (TCSP), where we are requested to maintain the path-consistency of a network under a sequence of insertions of new (further) constraints between pairs of variables. We show how to maintain the path-consistency in O(nR3) amortized time on a sequence of Θ(n2) insertions, where n is the number of vertices of the network and R is its range, defined as the maximum size of the minimum interval containing all the intervals of a single constraint.Furthermore we extend our algorithms to deal with more general temporal networks where variables can be points and/or intervals and constraints can also be defined on pairs of different kinds of variables. For such cases our algorithms maintain their performance. Finally, we adapt our algorithms to also maintain the arc-consistency of such generalized networks in O(R) amortized time for Θ(n2) insertions.