Coloring inductive graphs on-line

In this paper we consider the problem of on-line graph coloring. In an instance of on-line graph coloring, the nodes are presented one at a time. As each node is presented, its edges to previously presented nodes are also given. Each node must be assigned a color, different from the colors of its neighbors, before the next node is given. LetA(G) be the number of colors used by algorithmA on a graphG and letx(G) be the chromatic number ofG. The performance ratio of an on-line graph coloring algorithm for a class of graphsC is max_{G C(A(G)/(G))}. We consider the class ofd-inductive graphs. A graphG isd-inductive if the nodes ofG can be numbered so that each node has at mostd edges to higher-numbered nodes. In particular, planar graphs are 5-inductive, and chordal graphs arex(G)-inductive. First Fit is the algorithm that assigns each node the lowest-numbered color possible. We show that ifG isd-inductive, then First Fit usesO(d logn) colors onG. This yields an upper bound ofo(logn) on the performance ratio of First Fit on chordal and planar graphs. First Fit does as well as any on-line algorithm ford-inductive graphs: we show that, for anyd and any on-line graph coloring algorithmA, there is ad-inductive graph that forcesA to use (d logn) colors to colorG. We also examine on-line graph coloring with lookahead. An algorithm is on-line with lookaheadl, if it must color nodei after examining only the firstl+i nodes. We show that, forl/logn, the lower bound ofd logn colors still holds.This research was supported by an IBM Graduate Fellowship.     

On the classification of views of piecewise smooth objects

Views of objects composed of smooth surface patches whose intersections form smooth space curves are classified. These views may he described algebraically by mappings and diagrams of mappings from the line to the plane (for the crease) or from the plane to the plane (for the apparent contour). It is possible to derive a finite catalogue of generic views and their transitions, so that every view is either (up to smooth coordinate changes in source and target) equivalent to one of those in the catalogue or is of sufficiently high codimension.     

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.     

Dimensional properties of graphs and digital spaces

Many applications of digital image processing now deal with three- and higher-dimensional images. One way to represent n-dimensional digital images is to use the specialization graphs of subspaces of the Alexandroff topological space ⁿ (where denotes the integers with the Khalimsky line topology). In this paper the dimension of any such graph is defined in three ways, and the equivalence of the three definitions is established. Two of the definitions have a geometric basis and are closely related to the topological definition of inductive dimension; the third extends the Alexandroff dimension to graphs. Diagrams are given of graphs that are dimensionally correct discrete models of Euclidean spaces, n-dimensional spheres, a projective plane and a torus. New characterizations of n-dimensional (digital) surfaces are presented. Finally, the local structure of the space ⁿ is analyzed, and it is shown that ⁿ is an n-dimensional surface for all n1.     

Shortest noncrossing paths in plane graphs

Let G be an undirected plane graph with nonnegative edge length, and letk terminal pairs lie on two specified face boundaries. This paper presents an algorithm for findingk noncrossing paths inG, each connecting a terminal pair, and whose total length is minimum. Noncrossing paths may share common vertices or edges but do not cross each other in the plane. The algorithm runs in timeO(n logn) wheren is the number of vertices inG andk is an arbitrary integer.     

Abductive logics in a belief revision framework

Abduction was first introduced in the epistemological context of scientific discovery. It was more recently analyzed in artificial intelligence, especially with respect to diagnosis analysis or ordinary reasoning. These two fields share a common view of abduction as a general process of hypotheses formation. More precisely, abduction is conceived as a kind of reverse explanation where a hypothesis H can be abduced from events E if H is a good explanation of E. The paper surveys four known schemes for abduction that can be used in both fields. Its first contribution is a taxonomy of these schemes according to a common semantic framework based on belief revision. Its second contribution is to produce, for each non-trivial scheme, a representation theorem linking its semantic framework to a set of postulates. Its third contribution is to present semantic and axiomatic arguments in favor of one of these schemes, ordered abduction, which has never been vindicated in the literature.     

An algebraic technique for context-sensitive parsing

A technique that represents derivations of a context-free grammarG over a semiring and that obtains for a wordw inL(G) the set of all canonical parses forw has previously been described. A state grammar is one of a collection of grammars that place restrictions on the manner of application of context-free-like productions and that generate a noncontext-free language. The context-free properties of a state grammar have been used to extend the algebraic parsing technique for languages generated by state grammars,viz., context-sensitive languages. The extension for state grammars is not unlike that required for other types of grammars in whose collection state grammars are representative.     

The Nonexistence of Ternary [284, 6, 188] Codes

Let [n, k, d] _q codes be linear codes of length n, dimension k, and minimum Hamming distance d over GF(q). Let n _q (k, d) be the smallest value of n for which there exists an [n, k, d] _q code. It is known from [1, 2] that 284 n ₃(6, 188) 285 and 285 n ₃(6, 189) 286. In this paper, the nonexistence of [284, 6, 188]₃ codes is proved, whence we get n ₃(6, 188) = 285 and n ₃(6, 189) = 286.     

View adaptation in the fragment-based approach

View adaptation relies on adapting a set of materialized views in response to schema changes of source relations and/or after view redefinition. Recently, several view selection methods that are based on materializing fragments of the view rather than the whole view have been proposed. We call this approach the fragment-based approach. This paper presents a view adaptation method in the fragment-based approach, which is aimed at exploiting the opportunities to share not only materialized data, but also computation between the different views. In order to do this, the views are modeled using the so-called multiview materialization graph, which represents the views as a bipartite directed acyclic graph whose nodes are operations and fragments of the views. Then, the adaptation is performed regarding all materialized views and not solely the old materialization of the view. However, the data independence is preserved for the views that are not affected by the change. On the contrary, in related work, the adaptation technique is based solely on the old materialization of the same view. We studied the impact of the fragmentation on the adaptation techniques and showed the advantages and drawbacks of this approach.     

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.     

On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups

In this paper, we discuss the minimal number of observables Q ₁, ..., Q , where expectation values at some time instants t ₁, ..., t _r determine the trajectory of a d-level quantum system (qudit) governed by the Gaussian semigroup . We assume that the macroscopic information about the system in question is given by the mean values E _j(Q _i) = tr(Q _i(t _j)) of n selfadjoint operators Q ₁, ..., Q _n at some time instants t ₁ < t ₂ < ... < t _r, where n < d ²– 1 and r deg (, ). Here (, ) stands for the minimal polynomial of the generator of the Gaussian flow (t).     

Doubler and linearizer: an approach toward a unified theory for molecular computing based on DNA complementarity

Two specific mappings called doubler f _d and linearizer are introduced to bridge between two kinds of languages. Specifically, f _d maps string languages into (double-stranded) molecular languages, while performs the opposite mapping. Using these mappings, we obtain new characterizations for the families of sticker languages and of Watson-Crick languages, which lead to not only a unified view of the two families of languages but also provide a helpful view on the computational capability of DNA complementarity. Furthermore, we introduce a special type of a projection f _pr which is composed of f _d and a projection in the usual sense. We show that any recursively enumerable language L can be expressed as f _pr (L _m ) for a minimal linear language L _m . This result can be strengthened to L = f _pr (L _s ), for a specific form of minimal linear language L _s , which provides a simple morphic characterization for the family of recursively enumerable languages. A preliminary version of this article appeared in Onodera and Yokomori (2006).     

The Use of Size Functions for Comparison of Shapes Through Differential Invariants

For comparison of shapes under subgroups of the projective group, we can use a lot of invariants and especially differential invariants coming from multiscale analysis. But such invariants, as we have to compute curvature, are very sensitive to the noise induced by the dicretization grid. In order to resolve this problem we use size functions which can recognize the qualitative similarity between graphs of functions that should be theorically coinciding but, unfortunately, change their values due to the presence of noise. Moreover, we focus this study on a projective differential invariant which allows to decide if one shape can be considered as the deformation of another one by a rotation of the camera.     

Multilayer grid embeddings for VLSI

In this paper we propose two new multilayer grid models for VLSI layout, both of which take into account the number of contact cuts used. For the first model in which nodes exist only on one layer, we prove a tight area × (number of contact cuts) = (n ²) tradeoff for embeddingn-node planar graphs of bounded degree in two layers. For the second model in which nodes exist simultaneously on all layers, we give a number of upper bounds on the area needed to embed groups using no contact cuts. We show that anyn-node graph of thickness 2 can be embedded on two layers inO(n ²) area. This bound is tight even if more layers and any number of contact cuts are allowed. We also show that planar graphs of bounded degree can be embedded on two layers inO(n ^3/2(logn)²) area.Some of our embedding algorithms have the additional property that they can respect prespecified grid placements of the nodes of the graph to be embedded. We give an algorithm for embeddingn-node graphs of thicknessk ink layers usingO(n ³) area, using no contact cuts, and respecting prespecified node placements. This area is asymptotically optimal for placement-respecting algorithms, even if more layers are allowed, as long as a fixed fraction of the edges do not use contact cuts. Our results use a new result on embedding graphs in a single-layer grid, namely an embedding ofn-node planar graphs such that each edge makes at most four turns, and all nodes are embedded on the same line.The first author's research was partially supported by NSF Grant No. MCS 820-5167.     

Graph expressions and graph rewritings

We define an algebraic structure for the set of finite graphs, a notion of graph expression for defining them, and a complete set of equational rules for manipulating graph expressions. (By agraph we mean an oriented hypergraph, the hyperedges of which are labeled with symbols from a fixed finite ranked alphabet and that is equipped with a finite sequence of distinguished vertices). The notion of a context-free graph grammar is introduced (based on the substitution of a graph for a hyperedge in a graph). The notion of an equational set of graphs follows in a standard way from the algebraic structure. As in the case of context-free languages, a set of graphs is contextfree iff it is equational. By working at the level of expressions, we derive from the algebraic formalism a notion of graph rewriting which is as powerful as the usual one (based on a categorical approach) introduced by Ehrig, Pfender, and Schneider.This work has been supported by the PRC Mathématiques et Informatique. Reprints can be requested from B. Courcelle by electronic mail at the following address (UUCP network): mcvax!inria!geocub!courcell.     

Strongly adaptive token distribution

The token distribution (TD) problem, an abstract static variant of load balancing, is defined as follows: letM be a (parallel processor) network with setP of processors. Initially, each processorP P has a certain amountl(P) of tokens. The goal of a TD algorithm, run onM, is to distribute the tokens evenly among the processors. In this paper we introduce the notion of strongly adaptive TD algorithms, i.e., algorithms whose running times come close to the best possible runtime, the off-line complexity of the TD problem, for each individual initial token distributionl. Until now, only weakly adaptive algorithms have been considered, where the running time is measured in terms of the maximum initial load max{l(P)P P}.We design an almost optimal, strongly adaptive algorithm on mesh-connected networks of arbitrary dimension extended by a single 1-bit bus. This result shows that an on-line TD algorithm can come close to the optimal (off-line) bound for each individual initial load. Furthermore, we exactly characterize the off-line complexity of arbitrary initial token distributions on arbitrary networks. As an intermediate result, we design almost optimal weakly adaptive algorithms for TD on mesh-connected networks of arbitrary dimension.This research was partially supported by DFG-Forschergruppe Effiziente Nutzung massiv paralleler Systeme, Teilprojekt 4, by the ESPRIT Basic Research Action No. 7141 (ALCOM II), and by the Volkswagen-stiftung. A preliminary version was presented at the 20th ICALP, 1993, see [9].     

Tight bounds for oblivious routing in the hypercube

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires (N/d) parallel communication steps in the worst case. This is an improvement upon the (N/d ^3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in (N/logN) steps. The best previously known upper bound was (N). Our algorithm may be practical for smallN (up to about 2¹⁴ nodes).C. Kaklamanis was supported in part by NSF Grant NSF-CCR-87-04513. T. Tsantilas was supported in part by NSF Grants NSF-DCR-86-00379 and NSF-CCR-89-02500.     

Sparse QR factorization on a massively parallel computer

This paper shows that QR factorization of large, sparse matrices can be performed efficiently on massively parallel SIMD (single instruction stream/multiple data stream) computers such as the Connection Machine CM-2. The problem is cast as a dataflow graph, whose nodes are mapped to a virtual dataflow machine in such a way that only nearest-neighbor communication is required. This virtual machine is implemented by programming the CM-2 processors to support a restricted dataflow protocol. Execution results for several test matrices show that good performance can be obtained without relying on nested dissection techniques.     

Context-free graph languages of bounded degree are generated by apex graph grammars

The apex graph grammars generate precisely the context-free graph languages of bounded degree, independently of whether one considers hyperedge replacement systems or (boundary or confluent) NLC or edNCE graph grammars. The main feature of apex graph grammars is that nodes cannot be passed from nonterminal to nonterminal. The proof is based on a normal form result for arbitrary hyperedge replacement systems that forbids passing chains. This generalizes Greibach Normal Form.