Fabrizio Luccio

Structuring labeled trees for optimal succinctness, and beyond

Consider an ordered, static tree /spl Tscr/ on t nodes where each node has a label from alphabet set /spl Sigma/. Tree /spl Tscr/ may be of arbitrary degree and of arbitrary shape. Say, we wish to support basic navigational operations such as find the parent of node u, the ith child of u, and any child of it with label /spl alpha/. In a seminal work over fifteen years ago, Jacobson (1989) observed that pointer-based tree representations are wasteful in space and introduced the notion of succinct data structures. He studied the special case of unlabeled trees and presented a succinct data structure of 2t + o(t) bits supporting navigational operations in O(1) time. The space used is asymptotically optimal with the information-theoretic lower bound averaged over all trees. This led to a slew of results on succinct data structures for arrays, trees, strings and multisets. Still, for the fundamental problem of structuring labeled trees succinctly, few results, if any, exist even though labeled trees arise frequently in practice, e.g. in the data as in markup text (XML) or in augmented data structures. We present a novel approach to the problem of succinct manipulation of labeled trees by designing what we call the xbw transform of the tree, in the spirit of the well-known Burrows-Wheeler transform for strings. The xbw transform uses path-sorting and grouping to linearize the labeled tree /spl Tscr/ into two coordinated arrays, one capturing the structure and the other the labels. Using the properties of the xbw transform, we (i) derive the first-known (near-)optimal results for succinct representation of labeled trees with O(1) time for navigation operations, (ii) optimally support the powerful subpath search operation for the first time, and (iii) introduce a notion of tree entropy and present linear time algorithms for compressing a given labeled tree up to its entropy beyond the information-theoretic lower bound averaged over all tree inputs. Our xbw transform is simple and likely to spur new results in the theory of tree compression and indexing, and may have some practical impact in XML data processing.

Compressing and indexing labeled trees, with applications

Consider an ordered, static tree T where each node has a label from alphabet Σ. Tree T may be of arbitrary degree and shape. Our goal is designing a compressed storage scheme of T that supports basic navigational operations among the immediate neighbors of a node (i.e. parent, ith child, or any child with some label,…) as well as more sophisticated path-based search operations over its labeled structure. We present a novel approach to this problem by designing what we call the XBW-transform of the tree in the spirit of the well-known Burrows-Wheeler transform for strings [1994]. The XBW-transform uses path-sorting to linearize the labeled tree T into two coordinated arrays, one capturing the structure and the other the labels. For the first time, by using the properties of the XBW-transform, our compressed indexes go beyond the information-theoretic lower bound, and support navigational and path-search operations over labeled trees within (near-)optimal time bounds and entropy-bounded space. Our XBW-transform is simple and likely to spur new results in the theory of tree compression and indexing, as well as interesting application contexts. As an example, we use the XBW-transform to design and implement a compressed index for XML documents whose compression ratio is significantly better than the one achievable by state-of-the-art tools, and its query time performance is order of magnitudes faster.

A preliminary study of a diagonal channel-routing model

The layout of two-terminal nets in a VLSI channel is realized in a new diagonal channel-routing model (DCRM), where the tracks are segments respectively displayed at +45 ° and −45 ° on the two layers of the channel. A new definition of channel density is introduced, and a lower bound to the channel width is derived by the application of an algorithm, whose complexity is evaluated as a function of the channel density, and other parameters of the problem.A simple linear-time algorithm is proposed, which produces an optimal layout (i.e., it requires a channel of minimum width) if the length of the longest net equals the lower bound for the channel width. In any case, the number of vias is at most one for each net. Some particular solutions are proposed for problems with long nets.Specific problems are much easier in DCRM than in the classical Manhattan model. For example, any shift-by-i can be realized in DCRM in a channel of widthi.

Compressing and searching XML data via two zips

XML is fast becoming the standard format to store, exchange and publish over the web, and is getting embedded in applications. Two challenges in handling XML are its size (the XML representation of a document is significantly larger than its native state) and the complexity of its search (XML search involves path and content searches on labeled tree structures). We address the basic problems of compression, navigation and searching of XML documents. In particular, we adopt recently proposed theoretical algorithms [11] for succinct tree representations to design and implement a compressed index for XML, called XBZIPiNDEX, in which the XML document is maintained in a highly compressed format, and both navigation and searching can be done uncompressing only a tiny fraction of the data. This solution relies on compressing and indexing two arrays derived from the XML data. With detailed experiments we compare this with other compressed XML indexing and searching engines to show that XBZIPiNDEX has compression ratio up to 35% better than the ones achievable by those other tools, and its time performance on some path and content search operations is order of magnitudes faster: few milliseconds over hundreds of MBs of XML files versus tens of seconds, on standard XML data sources.

A 2 d channel router for the diagonal model

Abstract In this paper we consider two-terminal channel routing problems in a Diagonal Model (DM), where the grid consists of right and left tracks displayed at +45° and −45° on two layers. As in the Manhattan Model, the tracks are displaced in two layers with the left tracks on one layer and the right tracks on the other. InDM we prove a new lower bound to the channel width equal to d , and present an algorithm that obtains 2 d + 3 as an upper bound, where d is the channel density.

On a new Boolean function with applications

Consider a hypercube of 2/sup n/ points described by n Boolean variables and a subcube of 2/sup m/ points, m/spl les/n. As is well-known, the Boolean function with value 1 in the points of the subcube can be expressed as the product (AND) of n-m variables. The standard synthesis of arbitrary functions exploits this property. We extend the concept of subcube to the more powerful pseudocube. The basic set is still composed of 2/sup m/ points, but has a more general form. The function with value 1 in a pseudocube, called pseudoproduct, is expressed as the AND of n-m EXOR-factors, each containing at most m+1 variables. Subcubes are special cases of pseudocubes and their corresponding pseudoproducts reduce to standard products. An arbitrary Boolean function can be expressed as a sum of pseudoproducts (SPP). This expression is in general much shorter than the standard sum of products, as demonstrated on some known benchmarks. The logical network of an n-bit adder is designed in SPP, as a relevant example of application of this new technique. A class of symmetric functions is also defined, particularly suitable for SPP representation.

Dynamic monopolies in tori

Let G be a simple connected graph where every node is colored either black or white. Consider now the following repetitive process on G: each node recolors itself, at each local time step, with the color held by the majority of its neighbors. Depending on the initial assignment of colors to the nodes and on the definition of majority, different dynamics can occur. We are interested in dynamos; i.e., initial assignments of colors which lead the system to a monochromatic configuration in a finite number of steps. In the context of distributed computing and communication networks, this repetitive process is particularly important in that it describes the impact that a set of initial faults can have in majority-based systems (where black nodes correspond to faulty elements and white to non-faulty ones). In this paper, we study two particular forms of dynamos (irreversible and monotone) in tori, focusing on the minimum number of initial black elements needed to reach the fixed point. We derive lower and upper bounds on the size of dynamos for three types of tori, under different assumptions on the majority rule (simple and strong). These bounds are tight within an additive constant. The upper bounds are constructive: for each topology and each majority rule, we exhibit a dynamo of the claimed size.

Simple and efficient string matching with k mismatches

Given two strings P and T, m=|P|, n=|T|, m 0. We follow a new approach to SMk based on the determination of the permutations of P in T, and propose two algorithms for its solution. The first algorithm is very simple. It runs in time O(n log |AP|+rm), where AP is the alphabet of P, and r<n is the number of occurrences in T of the permutations of P with up to k mismatches. The second algorithm, which makes use of a suffix tree, runs in time O(n log |AP|+dk), where d⩽r is upper bounded by the number of distinct permutations of P in T with up to k mismatches. An extensive set of runs shows that rm<nk and d⪡n, and that the running times are strongly reduced, thus making our algorithms important in practice.

On the upper bound on the rotation distance of binary trees

Abstract The rotation distance d ( T 1 , T 2 ) between two binary search trees T 1 and T 2 on n nodes is the minimum number of rotations needed to transform T 1 into T 2 . Sleator, Tarjan and Thurston have proved an upper bound 2 n −6 to d ( T 1 , T 2 ), using triangulation of polyhedra. Makinen has subsequently shown that a weaker upper bound of 2 n −5 can be simply proved with elementary tree concepts. In this note we present a new tree transformation algorithm, leading to an elementary proof of the bound 2 n −6.

A new scheme for the deterministic simulation of PRAMs in VLSI

A deterministic scheme for the simulation of (n, m)-PRAM computation is devised. Each PRAM step is simulated on a bounded degree network consisting of a mesh-of-trees (MT) of siden. The memory is subdivided inn modules, each local to a PRAM processor. The roots of the MT contain these processors and the memory modules, while the otherO(n2) nodes have the mere capabilities of packet switchers and one-bit comparators. The simulation algorithm makes a crucial use of pipelining on the MT, and attains a time complexity ofO(log2n/log logn). The best previous time bound wasO(log2n) on a different interconnection network withn processors. While the previous simulation schemes use an intermediate MPC model, which is in turn simulated on a bounded degree network, our method performs the simulation directly with a simple algorithm.