Ami Litman

On covering problems of codes

LetC be a binary code of lengthn (i.e., a subset of {0, 1}n). TheCovering Radius of C is the smallest integerr such that each vector in {0, 1}n is at a distance at mostr from some code word. Our main result is that the decision problem associated with the Covering Radius of arbitrary binary codes is NP-complete.This result is established as follows. TheRadius of a binary codeC is the smallest integerr such thatC is contained in a radius-r ball of the Hamming metric space 〈{0, 1}n,d〉. It is known [K] that the problems of computing the Radius and the Covering Radius are equivalent. We show that the 3SAT problem is polynomially reducible to the Radius decision problem.A central tool in our reduction is a metrical characterization of the set ofdoubled vectors of length 2n: {v=(v1v2 …v2n) | ∀i:v2i=v2i−1}. We show that there is a setY ⊂ {0, 1}2n such that for everyv ε {0, 1}2n:v is doubled iffY is contained in the radius-n ball centered atv; moreover,Y can be constructed in time polynomial inn.

Combinatorial variability of Vapnik-Chervonenkis classes with applications to sample compression schemes

Abstract The classical binary classification problem is investigated when it is known in advance that the posterior probability function (or regression function) belongs to some class of functions. We introduce and analyze methods which effectively exploit this knowledge. These methods are based on minimizing the empirical risk over a carefully selected “skeleton” of the class of regression functions. The skeletons are coverings of the class based on metrics, especially fitted for classification. A new scale-sensitive dimension is introduced which is more suitable for the studied classification problem than other, previously defined, dimension measures. This fact is demonstrated by performance bounds for the skeleton estimates in terms of the new dimension. 2

Computing with Snakes in Directed Networks of Automata

We consider unidirectional, strongly connected networks of identical finite-state automata, of bounded in- and out-degree but unknown topology and unbounded sizen. Protocols which are quadratic or linear innare provided which accomplish the following tasks: wake up and report when done; construct spanning trees out from the root and in to the root; conduct breadth-first and depth-first searches; send a message from the end-point of an (unidirectional) arc to its start-point; run a slow clock; and solve the firing squad synchronization problem. Our protocols are highly parallel and use a new technique?a special kind of moving data structure we call asnake.

On the capabilities of systolic systems

This paper studies transformations of systems into systolic systems with related functionality. It distinguishes two antithetical transformation methods: one syntactic, the other semantic.The syntactic method considers the topology of the system, but ignores its behavior and the behavior of its combinational units. We use retiming and introduce two new basic syntactic techniques: tiling and bypassing. Using these, we present syntactic transformations that perform the following: conversion of a semisystolic system to a systolic one; elimination of either broadcast or instant-accumulation from a system that is otherwise systolic; and speeding up a systolic system by any constant factor. Leiserson and Saxe [10] have developed transformations to accomplish the first two tasks, but failed to preserve the behavior of the system. Our transformations leave the behavior of the system intact.The semantic method considers the functionality of the system as a whole, but ignores its internal structure. A system is called Ψ-homogeneous if all its combinational units are identical and equal to the given unit Ψ. We show that every semisystolic system can be transformed into a Ψ-homogeneous systolic system, where Ψ depends only on the alphabet used by the system to communicate with the external world. As a special case, any regular language L ⊂ ∑* is defined by some Ψ-homogeneous systolic system, where Ψ depends only on ∑.For binary systems, this technique produces a systolic system with a feasible clock period ofO(i + log(o)), wherei ando are the numbers of input and output ports of the system. This clock period is independent of the size and complexity of the given system.

Layered cross product—a technique to construct interconnection networks

We introduce a Layered Cross Product — A Technique to Construct Interconnection Networks Shimon Even* and Ami Litmant Computer Science Department Technion, Israel Institute of Technology Haifa, Israel 32000 Layered Cross Product, LCP, of layered graphs and show that several well known networks are LCP-S of simple layered graphs, such as trees. Some important properties of these networks are shown to be trivial consequences, once a network is presented as an LCP of simpler graphs. We believe that this new tool will make the construction of new networks easier, and it will simplify the study of the properties of known and new networks.

A Systematic Design and Explanation of the Atrubin Multiplier

The Atrubin systolic array, for multiplying two serially supplied integers in real-time, was invented in 1962, but to date, no simple explanation of its operation, or proof of its validity, has been published.

On distributed smooth scheduling

This paper studies evenly distributed sets of natural numbersand their applications to scheduling in a distributed environment.Such sets, called smooth sets, have the property that theirquantity within each interval is proportional to the size of theinterval, up to a bounded additive deviation; namely, forπ,Δ ∈ R a set A of natural numbers is(π, Δ)-smooth if abs(&vbar;I&vbar; ·π-- &vbar;I∩A&vbar;) 9 Δ for anyinterval I ⊂ N. The current paper studies scheduling <i>persistentclients</i> on a single slot-oriented<i>resource</i> in a flexible, predictable anddistributed manner. Each client γ has a given<i>rate</i> ρ_γ thatdefines the share of the resource he is entitled to receive and thegoal is a smooth schedule in which, for some predefined Δ,each client γ is served in aρ_γ,Δ)-smooth set of slots(natural numbers). The paper focuses on a <i>distributedenvironment</i> where each client by itself (without anyinter-client communication) <i>resolves</i> (computes),slot after slot, whether or not it owns this slot. The paperpresents extremely efficient schedules under which a clientresolves each slot in a constant time. The paper considers two scheduling frameworks. The first one,the <i>Flat Scheduling Framework</i>, is the commonproblem where the rates of the clients are given a priori. In thesecond and novel framework, the <i>Open-Market SchedulingFramework</i>, fractions of the resource are bought and soldby <i>dealers</i>. Each dealer, upon receiving his setof slots, may choose either to become a client and use his share,or to remain a dealer and sell fractions of his share to otherdealers. In this framework, the allocation process is highlydistributed; moreover, fractions of several resources can becombined into a single virtual resource of new capabilities. The paper presents two scheduling techniques. Both techniques,in both frameworks, produce smooth schedules with highly efficientdistributed resolutions --- a client resolves each slot in<i>O</i>(1) time on a RAM with a moderate number ofmemory words, all of a small size. Each technique has its pros andcons. For example, one technique utilizes 100% of the resource butits resolution algorithm requires a number of words which is linearin the number of clients; the other technique utilizes only 99% ofthe resource but its resolution algorithm requires just<i>O</i>(1) words. One of these techniques yields a solution to TijdemansHierarchial Chairman Assignment Problem which outperforms priorsolutions. The other technique naturally extends to the problem ofscheduling multiple resources, under the restriction that a clientmay be served concurrently by at most one resource. The extensionyields the first solution to this problem having efficientdistributed resolution. Prior solutions produce a special type ofsmooth scheduling called <i>P-fair scheduling</i>, arecentralized, and are less efficient than ours.

FAST, MINIMAL, AND OBLIVIOUS ROUTING ALGORITHMS ON THE MESH WITH BOUNDED QUEUES

This paper studies fast, deterministic permutation routing algorithms with bounded queues on the n×n mesh. Our main result is an O(n)-step, strongly-dimensional (and thus also source-oblivious and minimal) permutation routing algorithm. This algorithm works under a relaxed model in which nodes can freely send data to their neighbors. In a more prevalent model, the standard model, data may be sent only when accompanied by a packet. Under this model we present the following two algorithms: an O(n log n)-step strongly-dimensional algorithm and an O(n)-step oblivious and weakly-dimensional (and thus also minimal) algorithm. As said, all these algorithms store only O(1) packets in a node. Moreover, they use only O(log n) state bits in a node and transfer only O(log n) data bits on an edge in a step. All our routing algorithms are based on the following new technique of open-loop flow control. An algorithm is composed of two stages: setup and transportation. The setup stage computes certain values and stores th...

A tight layout of the butterfly network

We establish upper and lower bounds on the layout area of the butterfly network, which differ only in low-order terms. Specifically, the N-input, N-output butterfly network can be laid out in area (1 + o(1)) N^2, while no layout of the network can have area smaller than (1 - o(1)) N^2. These results improve both the known upper bound and the known lower bound on the area of butterfly network layouts.

On Centralized Smooth Scheduling

This paper studies evenly distributed sets of natural numbers and their applications to scheduling in a centralized environment. Such sets, called smooth sets, have the property that their quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation; namely, for ρ,Δ∈ℝ a set A of natural numbers is (ρ,Δ)-smooth if abs(|I|⋅ρ−|I∩A|)<Δ for any interval I⊂ℕ.The paper studies scheduling persistent clients on a single slot-oriented resource in a flexible and predictable manner. Each client γ has a given rateργ that defines the share of the resource he is entitled to receive and the goal is a smooth schedule in which, for some predefined Δ, each client γ is served in a (ργ,Δ)-smooth set of slots (natural numbers).The paper considers a centralized environment where a single algorithm computes the user of the current slot. It constructs a smooth schedule with a very efficient algorithm that computes the user of each slot in O(log log q) time and O(n) space, where n is the number of clients and q≜max {ργ/ργ′ | γ,γ′∈Γ}; in many practical applications this O(log log q) value is actually a small constant.Our scheduling technique is based on a reduction from allocation of slots to allocation of sub-intervals of the unit interval. This technique naturally extends to the problem of scheduling multiple resources, even under the restriction that a client can be served concurrently by at most one resource. This paper constructs such a schedule in which the users of each slot are computed very fast—in O(mlog log q) time per slot and O(n) space where m is the number of resources; this result is a significant improvement on the prior fastest algorithm that produces such a schedule (actually of a special type—a P-fair schedule) in O(mlog n) time per slot and O(n) space.Moreover, the paper introduces a novel approach to multi-resource scheduling in which each resource independently computes, slot after slot, what client to serve in this slot. Under this approach the paper constructs a smooth schedule which is computed in O(n) space and O(log log q) time per slot.