Linda Pagli

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.

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.

Network Decontamination in Presence of Local Immunity

We consider the problem of decontaminating a network infected by a mobile virus. The goal is to perform the task using as small a team of antiviral agents as possible, avoiding recontamination of disinfected areas. In all the existing literature, it is assumed that the immunity level of a decontaminated site is nil; that is, a decontaminated node, in absence of an antiviral agent on site, may be re-contaminated by any infected neighbour. The network decontamination problem is studied here under a new model of immunity to recontamination: we consider the case when a decontaminated vertex, after the cleaning agent has gone, will become recontaminated only if a majority of its neighbours are infected. We study the impact that the presence of local immunity has on the number of antiviral agents necessary to decontaminate the entire network. We establish both lower and upper bounds on the number cleaners in the case of (multidimensional) toroidal meshes, graphs of vertex degree at most three (e.g., cubic graphs, binary trees, etc.), and of tree networks. In all cases the established bounds are tight. All upper-bound proofs are constructive; i.e., we exhibit decontamination protocol achieving the claimed bound. We also analyze the total number of moves performed by the agents, and establish tight bounds in some cases.

Routing in times square mode

Abstract We propose a new channel routing model called times square mode (shortly TS , where the grid is composed of horizontal tracks , lines at +60° ( right tracks ), and lines at −60° ( left tracks ). In principle TS makes use of three layers, and can be seen as a variation of the classical Manhattan mode because each layer contains parallel tracks only. Restricting our attention to 2-terminal problems of density d , we present a simple polynomial routing algorithm for TS which produces a standard layout in a channel of width ⌜ 1 2 (d − 1)⌝ ⩽ w ⩽ 2d − 1 , or w ⩽ d + 1 for dense problems. We prove that this layout can be wired in three or four layers, and give a condition, testable in polynomial time, to decide the number of layers needed. Then we outline a wiring algorithm for three or four layers. Finally, we indicate how to extend our approach to multiterminal nets.

Point-of-Failure Shortest-Path Rerouting: Computing the Optimal Swap Edges Distributively

We consider the problem of computing the optimal swap edges of a shortest-path tree. This problem arises in designing systems that offer point-of-failure shortest-path rerouting service in presence of a single link failure: if the shortest path is not affected by the failed link, then the message will be delivered through that path; otherwise, the system will guarantee that, when the message reaches the node where the failure has occurred, the message will then be re-routed through the shortest detour to its destination. There exist highly efficient serial solutions for the problem, but unfortunately because of the structures they use, there is no known (nor foreseeable) efficient distributed implementation for them. A distributed protocol exists only for finding swap edges, not necessarily optimal ones. We present two simple and efficient distributed algorithms for computing the optimal swap edges of a shortest-path tree. One algorithm uses messages containing a constant amount of information, while the other is tailored for systems that allow long messages. The amount of data transferred by the protocols is the same and depends on the structure of the shortest-path spanning-tree; it is no more, and sometimes significantly less, than the cost of constructing the shortest-path tree.

A VLSI tree machine for relational data bases

A VLSI chip for performing relational data base operations is proposed. The chip is a tree of processors (TOP), where each chip has elementary storage and processing capabilities. A relation will be stored in the lowest levels of a TOP. More precisely, every m-tuple will occupy a subtree whose root is s&equil; [log2(m+1)] −1 levels above the leaves. Denoting by h the height of the tree, the upper h-s levels will be used for routing and bookkeeping purposes. A number of basic operations such as allocate and deallocate subtrees, insert and compare m-tuples etc., are defined for the TOPs. Relational operations are effectively performed as simple combinations of basic operations. The architecture of a data base machine based on TOPs is also sketched. Such a machine is feasible with the current VLSI technology and could become attractive in few years if density and performance of VLSI keep improving at the current rate.

Implicit B-trees: New results for the dictionary problem

We reopen the issue of finding an implicit data structure for the dictionary problem. In particular, we examine the problem of maintaining n data values in the first n locations of an array in such a way that we can efficiently perform the operations insert, delete and search. No information other than n and the data is to be retained; and the only operations which we may perform on the data values (other than reads and writes) are comparisons. Our structure supports these operations in O(log/sup 2/ n/log log n) time, marking the first improvement on the problem since the mid 1980s. En route we develop a number of space efficient techniques for handling segments of a large array in a memory hierarchy. We achieve a cost of O(log/sub B/ n) block transfers like in regular B-trees, under the realistic assumption that a block stores B = /spl Omega/(log n) keys, so that reporting r consecutive keys in sorted order has a cost of O(log/sub B/n+r/B) block transfers. Being implicit, our B-tree occupies exactly [n/B] blocks after each update.

Getting close without touching

In this paper we study the Near-Gathering problem for a set of asynchronous, anonymous, oblivious and autonomous mobile robots with limited visibility moving in Look-Compute-Move (LCM) cycles: In this problem, the robots have to get close enough to each other, so that every robot can see all the others, without touching (i.e., colliding) with any other robot. The importance of this problem might not be clear at a first sight: Solving the Near-Gathering problem, it is possible to overcome the limitations of having robots with limited visibility, and it is therefore possible to exploit all the studies (the majority, actually) done on this topic, in the unlimited visibility setting. In fact, after the robots get close enough, they are able to see all the robots in the system, a scenario similar to the one where the robots have unlimited visibility. Here, we present a collision-free algorithm for the Near-Gathering problem, the first to our knowledge, that allows a set of autonomous mobile robots to nearly gather within finite time. The collision-free feature of our solution is crucial in order to combine it with an unlimited visibility protocol. In fact, the majority of the algorithms that can be found on the topic assume that all robots occupy distinct positions at the beginning. Hence, only providing a collision-free Near-Gathering algorithm, as the one presented here, is it possible to successfully combine it with an unlimited visibility protocol, hence overcoming the natural limitations of the limited visibility scenario. In our model, distances are induced by the infinity norm. A discussion on how to extend our algorithm to models with different distance functions, including the usual Euclidean distance, is also presented.

On the Height of Height-Balanced Trees

Height-balanced binary trees with height unbalances up to ¿ are investigated, and the asymptotic value of the height h of such trees is studied for an increasing number of nodes N. It is shown that, in the worst case, the asymptotic value of h is a logarithmic function of N: [h = K log N]<inf>n¿¿</inf>. Specifically, an upper bound for h can be posed as: h ¿ K<inf>1</inf> log (N+2) - K<inf>2</inf> for ¿ ¿ 3; and h ¿ K<inf>1</inf> log (N+K<inf>2</inf>) - K<inf>3</inf> for ¿ = 4. Less strict bounds are posed for ¿ ≫ 4.