Short Proofs for Cut-and-Paste Sorting of Permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most \flr{2n/3} such moves and that some permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure

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

Bounds for Permutation Arrays under Kendall Tau Metric

Permutation arrays under the Kendall-$\tau$ metric have been considered for error-correcting codes. Given $n$ and $d\in [1..\binom{n}{2}]$, the task is to find a large permutation array of permutations on $n$ symbols with pairwise Kendall-$\tau$ distance at least $d$. Let $P(n,d)$ denote the maximum size of any permutation array of permutations on $n$ symbols with pairwise Kendall-$\tau$ distance $d$. New algorithms and several theorems are presented, giving improved lower bounds for $P(n,d)$. Also, $(n,m,d)$-arrays are defined, which are permutation arrays on n symbols with Kendall-$\tau$ distance d, with the restriction that symbols {1...(n-m)} appear in increasing order. Let $P(n,m,d)$ denote the maximum size of any $(n,m,d)$-array. For example, (n,m,d)-arrays are useful for recursively computing lower bounds for $P(n,d)$. Lower and upper bounds are given for $P(n.m,d)$

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).

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.