Traversal of an Unknown Directed Graph by a Finite Robot

A covering path in a directed graph is a path passing through all vertices and arcs of the graph, with each arc being traversed only in the direction of its orientation. A covering path exists for any initial vertex only if the graph is strongly connected, i.e., any of its vertices can be reached from any other vertex by some path. The strong connectivity is the only restriction on the considered class of graphs. As is known, on the class of such graphs, the covering path length is (nm), where n is the number of vertices and m is the number of arcs. For any graph, there exists a covering path of length O(nm), and there exist graphs with covering paths of the minimum length (nm). The traversal of an unknown graph implies that the topology of the graph is not a priori known, and we learn it only in the course of traversing the graph. At each vertex, one can see which arcs originate from the vertex, but one can learn to which vertex a given arc leads only after traversing this arc. This is similar to the problem of traversing a maze by a robot in the case where the plan of the maze is not available. If the robot is a general-purpose computer without any limitations on the number of its states, then traversal algorithms with the same estimate O(nm) are known. If the number of states is bounded, then this robot is a finite automaton. Such a robot is an analogue of the Turing machine, where the tape is replaced by a graph and the cells are assigned to the graph vertices and arcs. Currently, the lower estimate of the length of the traversal by a finite robot is not known. In 1971, the author of this paper suggested a robot with the traversal length O(nm + n ²logn). The algorithm of the robot is based on the construction of the output directed spanning tree of the graph and on the breadth-first search (BFS) on this tree. In 1993, Afek and Gafni [1] suggested an algorithm with the same estimate of the covering path length, which was also based on constructing a spanning tree but used the depth-first search (DFS) method. In this paper, an algorithm is suggested that combines the breadth-first search with the backtracking (suggested by Afek and Gafni), which made it possible to reach the estimate O(nm + n ²loglogn). The robot uses a constant number of memory bits for each vertex and arc of the graph.     

n-Dimensional Processor Arrays with Optical dBuses

dBus-array(k,n) is an n-dimensional processor array of kⁿ nodes connected via k^n–1 dBuses. A dBus is a unidirectional bus which receives signals from a set of k nodes (input set), and transmits signals to a different set of k nodes (output set). Two optical implementations of the dBus-array(k,n) are discussed. One implementation uses the wavelength division multiplexing as in the wavelength division multiple access channel hypercube WMCH [7]. WMCH(k,n) and dBus-array(k,n) have the same diameter and about the same average internode distance, while the dBus-array requires only one tunable transmitter/receiver per node, compared to n tunable transmitters/receivers per node for the WMCH. The other implementation uses one fixed-wavelength transmitter/receiver per node and the dilated slipped banyan switching network (DSB) [17] to combine time division and wavelength division multiplexing.     

The input/output complexity of transitive closure

Suppose a directed graph has its arcs stored in secondary memory, and we wish to compute its transitive closure, also storing the result in secondary memory. We assume that an amount of main memory capable of holdings values is available, and thats lies betweenn, the number of nodes of the graph, ande, the number of arcs. The cost measure we use for algorithms is theI/O complexity of Kung and Hong, where we count 1 every time a value is moved into main memory from secondary memory, or vice versa.In the dense case, wheree is close ton ², we show that I/O equal toO(n ³/s) is sufficient to compute the transitive closure of ann-node graph, using main memory of sizes. Moreover, it is necessary for any algorithm that is standard, in a sense to be defined precisely in the paper. Roughly, standard means that paths are constructed only by concatenating arcs and previously discovered paths. For the sparse case, we show that I/O equal toO(n ²e/s) is sufficient, although the algorithm we propose meets our definition of standard only if the underlying graph is acyclic. We also show that(n ²e/s) is necessary for any standard algorithm in the sparse case. That settles the I/O complexity of the sparse/acyclic case, for standard algorithms. It is unknown whether this complexity can be achieved in the sparse, cyclic case, by a standard algorithm, and it is unknown whether the bound can be beaten by nonstandard algorithms.We then consider a special kind of standard algorithm, in which paths are constructed only by concatenating arcs and old paths, never by concatenating two old paths. This restriction seems essential if we are to take advantage of sparseness. Unfortunately, we show that almost another factor ofn I/O is necessary. That is, there is an algorithm in this class using I/OO(n ³e/s) for arbitrary sparse graphs, including cyclic ones. Moreover, every algorithm in the restricted class must use(n ³e/s/log³ n) I/O, on some cyclic graphs.The work of this author was partially supported by NSF grant IRI-87-22886, IBM contract 476816, Air Force grant AFOSR-88-0266 and a Guggenheim fellowship.     

A New Class of Depth-Size Optimal Parallel Prefix Circuits

Given n values x₁, x₂, ... ,x_n and an associative binary operation o, the prefix problem is to compute x₁ox₂o··· ox_i, 1in. Many combinational circuits for solving the prefix problem, called prefix circuits, have been designed. It has been proved that the size s(D(n)) and the depth d(D(n)) of an n-input prefix circuit D(n) satisfy the inequality d(D(n))+s(D(n))2n–2; thus, a prefix circuit is depth-size optimal if d(D(n))+s(D(n))=2n–2. In this paper, we construct a new depth-size optimal prefix circuit SL(n). In addition, we can build depth-size optimal prefix circuits whose depth can be any integer between d(SL(n)) and n–1. SL(n) has the same maximum fan-out lgn+1 as Snir's SN(n), but the depth of SL(n) is smaller; thus, SL(n) is faster. Compared with another optimal prefix circuit LYD(n), d(LYD(n))+2d(SL(n))d(LYD(n)). However, LYD(n) may have a fan-out of at most 2 lgn–2, and the fan-out of LYD(n) is greater than that of SL(n) for almost all n12. Because an operation node with greater fan-out occupies more chip area and is slower in VLSI implementation, in most cases, SL(n) needs less area and may be faster than LYD(n). Moreover, it is much easier to design SL(n) than LYD(n).     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Nonrobustness Property of the Individual Ergodic Theorem

Main laws of probability theory, when applied to individual sequences, have a robustness property under small violations of randomness. For example, the law of large numbers for the symmetric Bernoulli scheme holds for a sequence where the randomness deficiency of its initial fragment of length n grows as o(n). The law of iterated logarithm holds if the randomness deficiency grows as o(loglogn). We prove that Birkhoff's individual ergodic theorem is nonrobust in this sense. If the randomness deficiency grows arbitrarily slowly on initial fragments of an infinite sequence, this theorem can be violated. An analogous nonrobustness property holds for the Shannon–McMillan–Breiman theorem.     

Shallow circuits and concise formulae for multiple addition and multiplication

A theory is developed for the construction of carry-save networks with minimal delay, using a given collection of carry-save adders each of which may receive inputs and produce outputs using several different representation standards.The construction of some new carry-save adders is described. Using these carry-save adders optimally, as prescribed by the above theory, we get {, , }-circuits of depth 3.48 log₂ n and {, , }-circuits of depth 4.95 log₂ n for the carry-save addition ofn numbers of arbitrary length. As a consequence we get multiplication circuits of the same depth. These circuits put out two numbers whose sum is the result of the multiplication. If a single output number is required then the depth of the multiplication circuits increases respectively to 4.48 log₂ n and 5.95 log₂ n.We also get {, , }-formulae of sizeO (n ^3.13) and {, }-formulae of sizeO (n ^4.57) for all the output bits of a carry-save addition ofn numbers. As a consequence we get formulae of the same size for the majority function and many other symmetric Boolean functions.     

MAAN: A Multi-Attribute Addressable Network for Grid Information Services

Recent structured Peer-to-Peer (P2P) systems such as Distributed Hash Tables (DHTs) offer scalable key-based lookup for distributed resources. However, they cannot be simply applied to grid information services because grid resources need to be registered and searched using multiple attributes. This paper proposes a Multi-Attribute Addressable Network (MAAN) that extends Chord to support multi-attribute and range queries. MAAN addresses range queries by mapping attribute values to the Chord identifier space via uniform locality preserving hashing. It uses an iterative or single attribute dominated query routing algorithm to resolve multi-attribute based queries. Each node in MAAN only has O(logN) neighbors for N nodes. The number of routing hops to resolve a multi-attribute range query is O(logN+N×s_min), where s_min is the minimum range selectivity on all attributes. When s_min=, it is logarithmic to the number of nodes, which is scalable to a large number of nodes and attributes. We also measured the performance of our MAAN implementation and the experimental results are consistent with our theoretical analysis.     

Hierarchy theorems for two-way finite state transducers

Summary For a family of languages , CAL() is defined as the family of images of under nondeterministic two-way finite state transducers, while FINITE · VISIT() is the closure of under deterministic two-way finite state transducers; CAL⁰()= and for n0, CAL ⁿ⁺¹()=CAL ⁿ (CAL()). For any semiAFL , if FINITE · VISIT() CAL(), then CAL ⁿ () forms a proper hierarchy and for every n0, FINITE · VISIT(CALⁿ()) CAL ⁿ⁺¹() FINITE · VISIT(CAL ⁿ⁺¹()). If is a SLIP semiAFL or a weakly k-iterative full semiAFL or a semiAFL contained in any full bounded AFL, then FINITE · VISIT() CAL() and in the last two cases, FINITE · VISIT(). If is a substitution closed full principal semiAFL and FINITE · VISIT(), then FINITE · VISIT() CAL(). If is a substitution closed full principal semiAFL generated by a language without an infinite regular set and ₁ is a full semiAFL, then is contained in CAL^m(₁) if and only if it is contained in ₁. Among the applications of these results are the following. For the following families , CAL ⁿ () forms a proper hierarchy: =INDEXED, =ETOL, and any semiAFL contained in CF. The family CF is incomparable with CAL ^m (NESA) where NESA is the family of one-way nonerasing stack languages and INDEXED is incomparable with CAL ^m (STACK) where STACK is the family of one-way stack languages.This work was supported in part by the National Science Foundation under Grants No. DCR74-15091 and MCS-78-04725     

Long unavoidable patterns

Summary We examine long unavoidable patterns, unavoidable in the sense of Bean, Ehrenfeucht, McNulty. Zimin and independently Schmidt have shown that there is only one unavoidable pattern of length 2 ⁿ -1 on an alphabet with n letters; this pattern is a quasi-power in the sense of Schützenberger. We characterize the unavoidable words of length 2 ⁿ -2 and 2 ⁿ -3. Finally we show that every sufficiently long unavoidable word has a certain quasi-power as a subword.This work was done while the author stayed at LITP, Université Paris 6, France     

Drawings of graphs on surfaces with few crossings

We give drawings of a complete graphK _n withO(n ⁴ log² g/g) many crossings on an orientable or nonorientable surface of genusg 2. We use these drawings ofK _n and give a polynomial-time algorithm for drawing any graph withn vertices andm edges withO(m ² log² g/g) many crossings on an orientable or nonorientable surface of genusg 2. Moreover, we derive lower bounds on the crossing number of any graph on a surface of genusg 0. The number of crossings in the drawings produced by our algorithm are within a multiplicative factor ofO(log² g) from the lower bound (and hence from the optimal) for any graph withm 8n andn ²/m g m/64.The research of the third and the fourth authors was partially supported by Grant No. 2/1138/94 of the Slovak Academy of Sciences and by EC Cooperative action IC1000 Algorithms for Future Technologies (Project ALTEC). A preliminary version of this paper was presented at WG93 and published in Lecture Notes in Computer Science, Vol. 790, 1993, pp. 388–396.     

Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered: in a set of admissible feedback controls v, with range inE ⁿ , find one minimizing the expectationE _sx { _s ^T L(t, (t), (t, (t)))dt + W _T ((T))} for all (s, x) [0,T) E ⁿ , whereL(t, x, ) = (/12)m ² – U(t, x) is the classical action integrand and is an-dimensional diffusion process in the weak sense, (see Bensoussan, 1982) with drift and diffusion coefficientD constant > 0.W _T andU are given real functions. Sufficiency conditions for the existence of such an optimal feedback control are given. Dedicated to George Leitmann Recommended by G.J. Olsder Presented at the Third Workshop on Control Mechanics in honor of George Leitmann, January 22–24, 1990, University of Southern California, Los Angeles, California (USA).     

Beating the Logarithmic Lower Bound: Randomized Preemptive Disjoint Paths and Call Control Algorithms

We consider the maximum disjoint paths problem and its generalization, the call control problem, in the on-line setting. In the maximum disjoint paths problem, we are given a sequence of connection requests for some communication network. Each request consists of a pair of nodes, that wish to communicate over a path in the network. The request has to be immediately connected or rejected, and the goal is to maximize the number of connected pairs, such that no two paths share an edge. In the call control problem, each request has an additional bandwidth specification, and the goal is to maximize the total bandwidth of the connected pairs (throughput), while satisfying the bandwidth constraints (assuming each edge has unit capacity). These classical problems are central in routing and admission control in high speed networks and in optical networks.We present the first known constant-competitive algorithms for both problems on the line. This settles an open problem of Garay et al. and of Leonardi. Moreover, to the best of our knowledge, all previous algorithms for any of these problems, are (logn)-competitive, where n is the number of vertices in the network (and obviously noncompetitive for the continuous line). Our algorithms are randomized and preemptive. Our results should be contrasted with the (logn) lower bounds for deterministic preemptive algorithms of Garay et al. and the (logn) lower bounds for randomized non-preemptive algorithms of Lipton and Tomkins and Awerbuch et al. Interestingly, nonconstant lower bounds were proved by Canetti and Irani for randomized preemptive algorithms for related problems but not for these exact problems.     

On Bounding Solutions of Underdetermined Systems

Sufficient conditions for the existence and uniqueness of a solution x* D (R ⁿ ) of Y(x) = 0 where : R ⁿ R ^m (m n) with C ²(D) where D R ⁿ is an open convex set and Y = (x)⁺ are given, and are compared with similar results due to Zhang, Li and Shen (Reliable Computing 5(1) (1999)). An algorithm for bounding zeros of f (·) is described, and numerical results for several examples are given.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

Dynamic half-space range reporting and its applications

We consider the half-space range-reporting problem: Given a setS ofn points in ^d, preprocess it into a data structure, so that, given a query half-space , allk points ofS can be reported efficiently. We extend previously known static solutions to dynamic ones, supporting insertions and deletions of points ofS. For a given parameterm,n m n ^d/2 and an arbitrarily small positive constant , we achieveO(m ¹⁺) space and preprocessing time, O((n/m ^d/2 logn+k) query time, and O(m¹⁺n) amortized update time (d 3). We present, among others, the following applications: an O(n¹⁺)-time algorithm for computing convex layers in ³, and an output sensitive algorithm for computing a level in an arrangements of planes in ³, whose time complexity is O((b+n) n, whereb is the size of the level.Work by the first author has been supported by National Science Foundation Grant CCR-91-06514. A preliminary version of this paper appeared in Agarwalet al. [2], which also contains the results of [20] on dynamic bichromatic closest pair and minimum spanning trees.     

Proof Reflection in Coq

We formalize natural deduction for first-order logic in the proof assistant Coq, using de Bruijn indices for variable binding. The main judgment we model is of the form d[:], stating that d is a proof term of formula under hypotheses it can be viewed as a typing relation by the Curry–Howard isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, we define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction.     

Representation of Algebraic Functions As Power and Fractional Power Series of Special Type

A Maple procedure is described by means of which an algebraic function given by an equation f(x y) = 0 can be expanded into a fractional power series (Puiseux series)

To the Metrical Rigidity of Binary Codes

A code C in the n-dimensional metric space E ⁿ over GF(2) is called metrically rigid if each isometry I : C E ⁿ can be extended to an isometry of the whole space E ⁿ . For n large enough, metrical rigidity of any length-n binary code that contains a 2-(n, k, )-design is proved. The class of such codes includes, for instance, all families of uniformly packed codes of large enough lengths that satisfy the condition d – 2, where d is the code distance and is the covering radius.     

Gradually intractable problems and nondeterministic log-space lower bounds

The paper places five different problems (thek-pebble game problem, two problems aboutk finite automata, the reachability problem for Petri nets withk tokens, and the teachability problem for graphs whose k-dimensional edge sets are described by Cartesian products ofk factors) into the hierarchyNL _k of problems solvable by nondeterministic Turing machines ink-log₂ n space (and binary tape alphabet, to avoid tape speed-up). The results, when combined with the conjecture thatNL _i contains problems that requireO(n ^k ) deterministic time, show that these problems, while inP for every fixed value ofk, have polynomial deterministic time complexities with the degree of the polynomial growing linearly with the parameterk, and hence are, in this sense, gradually intractable.