Embedding multidimensional grids into optimal hypercubes

Let $G$ and $H$ be graphs, with $|V(H)|\geq |V(G)|$, and $f:V(G)\rightarrow V(H)$ a one to one map of their vertices. Let $dilation(f) = max\{ dist_{H}(f(x),f(y)): xy\in E(G) \}$, where $dist_{H}(v,w)$ is the distance between vertices $v$ and $w$ of $H$. Now let $B(G,H)$ = $min_{f}\{ dilation(f) \}$, over all such maps $f$. The parameter $B(G,H)$ is a generalization of the classic and well studied "bandwidth" of $G$, defined as $B(G,P(n))$, where $P(n)$ is the path on $n$ points and $n = |V(G)|$. Let $[a_{1}\times a_{2}\times \cdots \times a_{k} ]$ be the $k$-dimensional grid graph with integer values $1$ through $a_{i}$ in the $i$'th coordinate. In this paper, we study $B(G,H)$ in the case when $G = [a_{1}\times a_{2}\times \cdots \times a_{k} ]$ and $H$ is the hypercube $Q_{n}$ of dimension $n = \lceil log_{2}(|V(G)|) \rceil$, the hypercube of smallest dimension having at least as many points as $G$. Our main result is that $$B( [a_{1}\times a_{2}\times \cdots \times a_{k} ],Q_{n}) \le 3k,$$ provided $a_{i} \geq 2^{22}$ for each $1\le i\le k$. For such $G$, the bound $3k$ improves on the previous best upper bound $4k+O(1)$. Our methods include an application of Knuth's result on two-way rounding and of the existence of spanning regular cyclic caterpillars in the hypercube.Comment: 47 pages, 8 figure

Branching Programs for Tree Evaluation

Abstract. The problem FT h d (k) consists in computing the value in [k] = {1,..., k} taken by the root of a balanced d-ary tree of height h whose internal nodes are labelled with d-ary functions on [k] and whose leaves are labelled with elements of [k]. We propose FT h d (k) as a good candidate for witnessing L � LogDCFL. We observe that the latter would follow from a proof that k-way branching programs solving FT h d (k) require Ω(k unbounded function(h) ) size. We introduce a “state sequence ” method that can match the size lower bounds on FT h d (k) obtained by the Ne˘ciporuk method and can yield slightly better (yet still subquadratic) bounds for some nonboolean functions. Both methods yield the tight bounds Θ(k 3) and Θ(k 5/2) for deterministic and nondeterministic branching programs solving FT 3 2 (k) respectively. We propose as a challenge to break the quadratic barrier inherent in the Ne˘ciporuk method by adapting the state sequence method to handle FT 4 d (k).

The Vertex Separation And Search Number Of A Graph

We relate two concepts in graph theory and algorithmic complexity, namely the search number and the vertex separation of a graph. Let s (G ) denote the search number and vs (G ) denote the vertex separation of a connected, undirected graph G . We show that vs (G ) s (G ) vs (G ) + 2 and we give a simple transformation from G to G such that vs (G ) = s (G ). We characterize those trees having a given vertex separation and describe the smallest such trees. We also note that there exist trees for which the difference between search number and vertex separation is indeed 2. We give algorithms that, for any tree T , compute vs (T ) in linear time and compute an optimal layout with respect to vertex separation in time O (n log n ). Vertex separation has previously been related to progressive black/white pebble demand and has been shown to be identical to a variant of search number, node search number, and to path width, which has been related directly to gate matrix layout cost. All these..

Bounding the Size of K-tuple Covers

Suppose there are n applications and n processors. A pair cover is a set S of one-to-one mappings (assignments) of the applications to the processors such that, for every pair (Ai,Aj) of applications and every pair (p,q) of processors, there is an assignment f in S that maps (Ai,Aj) to (p,q). More generally, we consider, for all k/spl ges/1, minimum size k-tuple covers. We improve bounds given earlier in by Latifi, where the application for k-tuple covers was fault tolerance of the multidimensional star network. Let F(n,k) denote the minimum cardinality k-tuple cover for n applications and processors. We give bounds for F(n,k) that are within a small multiplicative factor of optimum

Towards Regular Languages Over Infinite Alphabets

Motivated by formal models recently proposed in the context of XML, we study automata and logics on strings over infinite alphabets. These are conservative extensions of classical automata and logics defining the regular languages on finite alphabets. Specically, we consider register and pebble automata, and extensions of first-order logic and monadic second-order logic. For each type of automaton we consider one-way and two-way variants, as well as deterministic, non-deterministic, and alternating control. We investigate the expressiveness and complexity of the automata, their connection to the logics, as well as standard decision problems. Some of our results answer open questions of Kaminski and Francez on register automata