Pierluigi Crescenzi

On the Complexity of Protein Folding

We show that the protein folding problem in the two-dimensional H-P model is NP-complete.

A uniform approach to define complexity classes

Abstract Complexity classes are usually defined by referring to computation models and by putting suitable restrictions on them. Following this approach, many proofs of results are tightly bound to the characteristics of the computation model and of its restrictions and, therefore, they sometimes hide the essential properties which insure the obtained results. In order to obtain more general results, a uniform family of computation models which encompasses most of the complexity classes of interest is introduced. As a first initial set of results derivable from the proposed approach, we will give a sufficient and necessary condition for proving separations of relativized complexity classes, a characterization of complexity classes with complete languages and a sufficient condition for proving strong separations of relativized complexity classes. Examples of applications of these results to some specific complexity classes are then given. Additional results related to separations by sparse oracles can be found in Bovet (1991).

A note on optimal area algorithms for upward drawings of binary trees

The goal of this paper is to investigate the area requirements for upward grid drawings of binary trees. First, we show that there is a family of binary trees with n vertices that require ω(n log n) area; this bound is tight to within a constant factor, i.e. any binary tree with n vertices can be drawn in O(n log n) area. Then we present an algorithm for constructing an upward drawing of a complete binary tree with n vertices in O(n) area, and, finally, we extend this result to the drawings of Fibonacci trees.

Approximate solution of NP optimization problems

Abstract This paper presents the main results obtained in the field of approximation algorithms in a unified framework. Most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes, that is, combinatorial properties of problems and approximation preserving reducibilities. In particular, after reviewing the most important combinatorial characterizations of the classes PTAS and FPTAS, we concentrate on the class APX and, as a concluding result, we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial-time approximation preserving reducibility.

Parsimonious flooding in dynamic graphs

An edge-Markovian process with birth-rate <i>p</i> and death-rate <i>q</i> generates sequences of graphs (<i>G</i>₀,<i>G</i>₁,<i>G</i>₂,…) with the same node set [<i>n</i>] such that <i>G_t</i> is obtained from <i>G</i>_t−1 as follows: if <i>e</i> ∉ <i>E</i>(<i>G</i>_<i>t</i>−1) then <i>e</i> ∈ <i>E(G_t)</i> with probability <i>p</i>, and if <i>e</i> ∈ <i>E</i>(<i>G</i>_<i>t</i>−1) then <i>e</i> ∉ <i>E</i>(<i>G_t</i>) with probability <i>q</i>. Clementi et al. (PODC 2008) analyzed thoroughly information dissemination in such dynamic graphs, by establishing bounds on their flooding time--flooding is the basic mechanism in which every node becoming aware of an information at step <i>t</i> forwards this information to all its neighbors at all forthcoming steps <i>t</i>∦ > <i>t</i>. In this paper, we establish tight bounds on the complexity of flooding for all possible birth rates and death rates, completing the previous results by Clementi et al. Moreover, we note that despite its many advantages in term of simplicity and robustness, flooding suffers from its high bandwidth consumption. Hence we also show that flooding in dynamic graphs can be implemented in a more parsimonious manner, so that to save bandwidth, yet preserving efficiency in term of simplicity and completion time. For a positive integer <i>k</i>, we say that the flooding protocol is <i>k</i>-active if each node forwards an information only during the <i>k</i> time steps immediately following the step at which the node receives that information for the first time. We define the <i>reachability threshold</i> for the flooding protocol as the smallest integer <i>k</i> such that, for any source <i>s</i> ∈ [<i>n</i>], the <i>k</i>-active flooding protocol from <i>s</i> completes (i.e., reaches all nodes), and we establish tight bounds for this parameter. We show that, for a large spectrum of parameters <i>p</i> and <i>q</i>, the reachability threshold is by several orders of magnitude smaller than the flooding time. In particular, we show that it is even constant whenever the ratio <i>p/(p + q)</i> exceeds log <i>n/n</i>. Moreover, we also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) allows the flooding protocol to complete in <i>optimal</i> time, i.e., in asymptotically the same number of steps as when being perpetually active. These results demonstrate that flooding can be implemented in a practical and efficient manner in dynamic graphs. The main ingredient in the proofs of our results is a reduction lemma enabling to overcome the time dependencies in edge-Markovian dynamic graphs.

Reversible Execution and Visualization of Programs with LEONARDO

In this paper we present LEONARDO, an integrated environment for software visualization that allows the user to edit, compile, execute, and animate general-purpose C programs. LEONARDO relies on a logic-based approach to visualization: a mapping between concrete and abstract data structures can be declared through a logic visualization language and animations are conceived as reflecting formal properties of algorithms. LEONARDO is able to automatically detect visual events during the execution of programs and simplifies the creation of visualizations according to an incremental approach. Moreover, it guarantees the complete reversibility of computations, bounded only by the potentiality of the working machine, and appears simple to be used. The latest version of LEONARDO is currently available over the Internet at the URLhttp: //www.dis.uniroma1.it/~demetres/Leonardo/.

MeDuSa: a multi-draft based scaffolder

MOTIVATION Completing the genome sequence of an organism is an important task in comparative, functional and structural genomics. However, this remains a challenging issue from both a computational and an experimental viewpoint. Genome scaffolding (i.e. the process of ordering and orientating contigs) of de novo assemblies usually represents the first step in most genome finishing pipelines. RESULTS In this article we present MeDuSa (Multi-Draft based Scaffolder), an algorithm for genome scaffolding. MeDuSa exploits information obtained from a set of (draft or closed) genomes from related organisms to determine the correct order and orientation of the contigs. MeDuSa formalizes the scaffolding problem by means of a combinatorial optimization formulation on graphs and implements an efficient constant factor approximation algorithm to solve it. In contrast to currently used scaffolders, it does not require either prior knowledge on the microrganisms dataset under analysis (e.g. their phylogenetic relationships) or the availability of paired end read libraries. This makes usability and running time two additional important features of our method. Moreover, benchmarks and tests on real bacterial datasets showed that MeDuSa is highly accurate and, in most cases, outperforms traditional scaffolders. The possibility to use MeDuSa on eukaryotic datasets has also been evaluated, leading to interesting results.

On the Complexity of Computing Minimum Energy Consumption Broadcast Subgraphs

We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a sub-logarithmic factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension.

IP Address Lookup Made Fast and Simple

The IP address lookup problem is one of the major bottlenecks in high performance routers. Previous solutions to this problem first describe it in the general terms of longest prefix matching and, then, are experimented on real routing tables T. In this paper, we follow the opposite direction. We start out from the experimental analysis of real data and, based upon our findings, we provide a new and simple solution to the IP address lookup problem. More precisely, our solution for m-bit IP addresses is a reasonable trade-off between performing a binary search on T with O(log |T|) accesses, where |T| is the number of entries in T, and executing a single access on a table of 2m entries obtained by fully expanding T. While the previous results start out from space-efficient data structures and aim at lowering the O(log |T|) access cost, we start out from the expanded table with 2m entries and aim at compressing it without an excessive increase in the number of accesses. Our algorithm takes exactly three memory accesses and occupies O(2m/2 + |T|2) space in the worst case. Experiments on real routing tables for m = 32 show that the space bound is overly pessimistic. Our solution occupies approximately one megabyte for the MaeEast routing table (which has |T| ? 44; 000 and requires approximately 250 KB) and, thus, takes three cache accesses on any processor with 1 MB of L2 cache. According to the measurement obtained by the VTune tool on a Pentium II processor, each lookup requires 3 additional clock cycles besides the ones needed for the memory accesses. Assuming a clock cycle of 3.33 nanoseconds and an L2 cache latency of 15 nanoseconds, search of MaeEast can be estimated in 55 nanoseconds or, equivalently, our method performs 18 millions of lookups per second.

On the complexity of protein folding (extended abstract)

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