The two dimensional mesh is widely considered to be a promising parallel architecture in its scalability. In this architecture, processors are naturally placed at intersections of horizontal and vertical grids, while there can be three different types of communication links: (i) The first type is the most popular model, called a mesh-connected computer: Each processor is connected to its four neighbours by local connections. (ii) Each processor of the second type is connected to a couple of (row and column) buses. The system is then called a mesh of buses. (iii) The third model is equipped with both buses and local connections, which is called a mesh-connected computer with buses. Mesh routing has received considerable attention for the last two decades, and a variety of algorithms have been proposed. This paper provides an overview of lower and upper bounds for algorithms, with pointers to the literature, and suggests further research directions for mesh routing.

Our basic idea of generating random benchmark circuits, i.e., not generating them directly but applying random transformations to initial circuits was presented at DAC'94. In this paper we make the two major improvements towards the goal of random benchmarking: i.e., increasing the generality, the naturality, the security of random circuits: One is controlling fan-ins of logic gates in the random circuits, and the other is producing the initial circuit also at random but under some control of its on-set size and complexity. Experimental data claiming merits of those improvements are also given.

It is known that the original Grover Search (GS) can be modified to use a general value for the phase θ of the diffusion transform. Then, if the number of answers is relatively large, this modified GS can find one of the answers with probability one in a single iteration. However, such a quick and error-free GS can only be possible if we can initially adjust the value of θ correctly against the number of answers, and this seems very hard in usual occasions. A natural question now arises: Can we enjoy a merit even if GS is used without such an adjustment? In this paper, we give a positive answer using the balls-and-bins game in which the random sampling of bins is replaced by the quantum sampling, i.e., a single round of modified GS. It is shown that by using the quantum sampling: (i) The maximum load can be improved quadratically for the static model of the game and this improvement is optimal. (ii) That is also improved to O(1) for the continuous model if we have a certain knowledge about the total number of balls in the bins after the system becomes stable.

Cowen gave a universal compact routing algorithm with a stretch factor of three and table-size of O(n2/3log4/3n) based on a simple and practical model. (The table-size is later improved to O(n1/2log3/2n).) This paper considers, using the same model, how the necessary table-size differs if the stretch factor must be less than three. It is shown that: (i) There is a routing algorithm with a stretch factor of two whose table-size is (n -+ 2)log n. (ii) There is a network for which any routing algorithm that follows the model and with a stretch factor of less than three needs a table-size of (n - 2)log n in at least one node. Thus, we can only reduce roughly an additive log n (i.e., table-entries) from the trivial table-size of n log n which obviously enables shortest-path routing. Furthermore it turns out that we can reduce only an additive log n (i.e., only one table-entry) from the trivial n log n if we have to achieve a stretch factor of less than two. Thus the algorithm (i) is (roughly) tight both in its stretch factor and in its table-size.

In this paper, we consider two types of promises for (k-)CNF formulas which can help to find a satisfying assignment of a given formula. The first promise is the Hamming distance between truth assignments. Namely, we know in advance that a k-CNF formula with n variables, if satisfiable, has a satisfying assignment with at most pn variables set to 1. Then we ask whether or not the formula is satisfiable. It is shown that for k 3 and (i) when p=nc (-1 < c 0), the problem is NP-hard; and (ii) when p=log n/n, there exists a polynomial-time deterministic algorithm. The algorithm is based on the exponential-time algorithm recently presented by Schoning. It is also applied for coloring k-uniform hypergraphs. The other promise is the number of satisfying assignments. For a CNF formula having 2n/2nε (0 ε < 1) satisfying assignments, we present a subexponential-time deterministic algorithm based on the inclusion-exclusion formula.

The main purpose of this paper is to show that we can exploit the difference (l1-norm and l2-norm) in the probability calculation between quantum and probabilistic computations to claim the difference in their space efficiencies. It is shown that there is a finite language L which contains sentences of length up to O(nc+1) such that: (i) There is a one-way quantum finite automaton (qfa) of O(nc+4) states which recognizes L. (ii) However, if we try to simulate this qfa by a probabilistic finite automaton (pfa) using the same algorithm, then it needs Ω(n2c+4) states. It should be noted that we do not prove real lower bounds for pfa's but show that if pfa's and qfa's use exactly the same algorithm, then qfa's need much less states.

In the CNN problem, a "scene" appears on the two-dimensional plane, at different positions sequentially, and a "camera crew" has to shoot the scene whenever it appears. If a scene appears at some position, the camera crew does not have to move to the position exactly, but has only to move to a point that lies in the same horizontal or vertical line with the scene. Namely it is enough to move either to the same row or to the same column. The goal is to minimize the total moving distance of the camera crew. This problem has been quite popular in the last decade but it is still open whether or not there is a competitive algorithm, i.e., an algorithm with competitive ratio bounded by a constant. In this paper we study this problem under a natural restriction that the server can move only along the X-axis and the Y-axis. It is shown that there exists a competitive algorithm for this restricted version, namely there is an online algorithm for this "axis-bound CNN" with competitive ratio 9.0.

The planar Hajos calculus (PHC) is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. The degree-d planar Hajos calculus (PHC(dd)) is PHC with the restriction that all the graphs that appear in the construction (including a final graph) must have maximum degree at most d. We prove the followings: (1) If PHC is polynomially bounded, then for any d ≥ 4, PHC(dd+2) can generate any non-3-colorable planar graphs of maximum degree at most d in polynomial steps. (2) If PHC can generate any non-3-colorable planar graphs of maximum degree 4 in polynomial steps, then PHC is polynomially bounded.

An instance of the classical stable marriage problem requires all participants to submit a strictly ordered preference list containing all members of the opposite sex. However, considering applications in real-world, we can think of two natural relaxations, namely, incomplete preference lists and ties in the lists. Either variation leaves the problem polynomially solvable, but it is known that finding a maximum cardinality stable matching is NP-hard when both variations are allowed. It is easy to see that the size of any two stable matchings differ by at most a factor of two, and so, an approximation algorithm with a factor two is trivial. A few approximation algorithms have been proposed with approximation ratio better than two, but they are only for restricted instances, such as restricting occurrence of ties and/or lengths of ties. Up to the present, there is no known approximation algorithm with ratio better than two for general inputs. In this paper, we give the first nontrivial result for approximation of factor less than two for general instances. Our algorithm achieves the ratio for an arbitrarily positive constant c, where N denotes the number of men in an input.

Chen and Kanj considered the VERTEX COVER problem for graphs with perfect matchings (VC-PM). They showed that: (i) There is a reduction from general VERTEX COVER to VC-PM, which guarantees that if one can achieve an approximation factor of less than two for VC-PM, then one can do so for general VERTEX COVER as well. (ii) There is an algorithm for VC-PM whose approximation factor is given as 1.069+0.069 where is the average degree of the given graph. In this paper we improve (ii). Namely we give a new VC-PM algorithm which greatly outperforms the above one and its approximation factor is roughly . Our algorithm also works for graphs with "large" matchings, although its approximation factor is degenerated.

In STS-based mapping, it is necessary to obtain the correct order of probes in a DNA sequence from a given set of fragments or an equivalently a hybridization matrix A. It is well-known that the problem is formulated as the combinatorial problem of obtaining a permutation of A's columns so that the resulting matrix has a consecutive-one property. If the data (the hybridization matrix) is error free and includes enough information, then the above column order uniquely determines the correct order of the probes. Unfortunately this does not hold if the data include errors, and this has been a popular research target in computational biology. Even if there is no error, ambiguities in the probe order may still remain. This in fact happens because of the lack of some information regarding the data, but almost no further investigation has previously been made. In this paper, we define a measure of such imperfectness of the data as the minimum amount of the additional fragments that are needed to uniquely fix the probe order. Polynomial-time algorithms to compute such additional fragments of the minimum cost are presented. A computer simulation using genes of human chromosome 20 is also noted.

The Hajos calculus is a nondeterministic procedure which generates the class of non-3-colorable graphs. If all non-3-colorable graphs can be constructed in polynomial steps by the calculus, then NP = co-NP holds. Up to date, however, it remains open whether there exists a family of graphs that cannot be generated in polynomial steps. To attack this problem, we propose two graph calculi PHC and PHC* that generate non-3-colorable planar graphs, where intermediate graphs in the calculi are also restricted to be planar. Then we prove that PHC and PHC* are sound and complete. We also show that PHC* can polynomially simulate PHC.