Expansion of layouts of complete binary trees into grids

AbstractLet Th be the complete binary tree of height h. Let M be the infinite grid graph with vertex set Z2, where two vertices (x1,y1) and (x2,y2) of M are adjacent if and only if |x1−x2|+|y1−y2|=1. Suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of M. Motivated by issues in optimal VLSI design, we show that the point expansion ratio n(T)/n(Th)=n(T)/(2h+1−1) is bounded below by 1.122 for h sufficiently large. That is, we give bounds on how many vertices of degree 2 must be inserted along the edges of Th in order that the resulting tree can be laid out in the grid. Concerning the constructive end of VLSI design, suppose that T is a tree which is a subdivision of Th and is also isomorphic to a subgraph of the n×n grid graph. Define the expansion ratio of such a layout to be n2/n(Th)=n2/(2h+1−1). We show constructively that the minimum possible expansion ratio over all layouts of Th is bounded above by 1.4656 for sufficiently large h. That is, we give efficient layouts of complete binary trees into square grids, making improvements upon the previous work of others. We also give bounds for the point expansion and expansion problems for layouts of Th into extended grids, i.e. grids with added diagonals

Parity Games of Bounded Tree- and Clique-Width

Abstract. In this paper it is shown that deciding the winner of a parity game is in LogCFL, if the underlying graph has bounded tree-width, and in LogDCFL, if the tree-width is at most 2. It is also proven that parity games of bounded clique-width can be solved in LogCFL via a log-space reduction to the bounded tree-width case, assuming that a k-expression for the parity game is part of the input.

Complexity and decidability for chain code picture languages

AbstractA picture description is a word over the alphabet &{u, d, r, l}, where u means “go one unit line up from the current point”, and d, r, and l are interpreted analogously with down, right, and left instead of up. By this, a picture description describes a walk in the plane—its trace is the picture it describes. A set of picture descriptions describes a (chain code) picture language.This paper investigates complexity and decidability questions for these picture languages. Thus it is shown that the membership problem is NP-complete for regular picture languages (i.e., picture languages described by regular languages of picture descriptions), and that it is undecidable whether two regular picture description languages describe a picture in common. After this we investigate so-called stripe picture languages (all pictures are within a stripe defined by two parallel lines), providing ‘better’ complexity and decidability results: Membership is decidable in linear time for regular stripe picture languages. Emptiness of intersection and equivalence is decidable for regular stripe picture languages

Min cut is NP-complete for edge weighted trees

AbstractWe show that the Min Cut Linear Arrangement Problem (Min Cut) is NP-complete for trees with polynomial size edge weights and derive from this the NP-completeness of Min Cut for planar graphs with maximum vertex degree 3. This is used to show the NP-completeness of Search Number, Vertex Separation, Progressive Black/White Pebble Demand, and Topological Bandwidth for planar graphs with maximum vertex degree 3

Computing LOGCFL certificates

AbstractThe complexity class LOGCFL consists of all languages (or decision problems) which are logspace reducible to a context-free language. Since LOGCFL is included in AC1, the problems in LOGCFL are highly parallelizable.By results of Ruzzo (JCSS 21 (1980) 218), the complexity class LOGCFL can be characterized as the class of languages accepted by alternating Turing machines (ATMs) which use logarithmic space and have polynomially sized accepting computation trees. We show that for each such ATM M recognizing a language A in LOGCFL, it is possible to construct an LLOGCFL transducer TM such that TM on input w∈A outputs an accepting tree for M on w. It follows that computing single LOGCFL certificates is feasible in functional AC1 and is thus highly parallelizable.Wanke (J. Algorithms 16 (1994) 470) has recently shown that for any fixed k, deciding whether the treewidth of a graph is at most k is in the complexity-class LOGCFL. As an application of our general result, we show that the task of computing a tree-decomposition for a graph of constant treewidth is in functional LOGCFL, and thus in AC1.We also show that the following tasks are all highly parallelizable: Computing a solution to an acyclic constraint satisfaction problem; computing an m-coloring for a graph of bounded treewidth; computing the chromatic number and minimal colorings for graphs of bounded tree- width