The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed f-vectors and h-vectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their Stanley-Reisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the k-equal partition lattice (the intersection lattice of the k-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the k-equal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.

Documents can be represented as ordered labelled trees. Finding the editing distance between documents is a particular case of the general problem for trees. We give a detailed survey of previous results, presenting them in a single notation to elucidate their commonalities. We then discuss two ways of extending these results---first, by changing the set of primitive editing operations used by existing algorithms and, second, by post-processing the output of the algorithms to recognize patterns of change significant to documents. Finally, we provide extensions of the first type. Our algorithm allows subtree operations but is otherwise similar to that of Zhang and Shasha. This is a corrected and expanded version of Technical Report 91-315. y This report was completed during a sabbatical at INRIA (Institute National de Recherche en Informatique et en Automatique) in Rocquencourt, France. Contents 1 Introduction 3 2 Background 5 2.1 String-to-String Correction: Wagner and Fischer ...

Let T(n) denote the set of all bitstrings with n 1's and n 0's that satisfy the property that in every prefix the number of 0's does not exceed the number of 1's. This is a well known representation of binary trees. We consider algorithms for generating the elements of T(n) that satisfy one of the following constraints: (a) successive bitstrings differ by the transposition of two bits, or (b) successive bitstrings differ by the transposition of two adjacent bits. In case (a) a constant average time generation algorithm is presented. In case (b) we show that such generation is possible if and only if n is even or n ! 5. A constant average time algorithm is presented in this case as well. 1 Introduction Binary Trees are of fundamental importance in computer science. In recent years there has been some interest in algorithms that generate all binary trees with a fixed number of nodes (for example, Ruskey and Hu [17], Proskurowski [10], Zaks [20], Pallo [9], Zerling [21]) or restricted cl...

In contrast to traditional integer sequences for the representation of binary trees, a kind of character sequence (words) is presented for binary trees based on a grammar GBT and for full binary trees based on a grammar GFBT. The properties of words derived from GBT (GFBT) are discussed in depth, including necessary and suf"cient conditions for a word, pre"x and suf"x of &(GBT) (&(GFBT)) and algorithms are given and analysed for the enumeration of words of &(GBT) (&(GFBT)) lexicographically and in other ways. By modifying an algorithm for the enumeration of words in &(GBT), an algorithm is obtained to enumerate binary trees with a computer representation in an average time of O(1) per tree. The problem with non-isomorphic

We show that the space of all binary Huffman codes for a finite alphabet defines a lattice, ordered by the imbalance of the code trees. Representing code trees as path-length sequences, we show that the imbalance ordering is closely related to a majorization ordering on realvalued sequences that correspond to discrete probability density functions. Furthermore, this tree imbalance is a partial ordering that is consistent with the total orderings given by either the external path length (sum of tree path lengths), or the entropy determined by the tree structure. On the imbalance lattice, we show the weighted path-length of a tree (the usual objective function for Huffman coding) is a submodular function, as is the corresponding function on the majorization lattice. Submodular functions are discrete analogues of convex functions. These results give perspective on Huffman coding, and suggest new approaches to coding as optimization over a lattice. 1 Introduction Traditionally, the Huffm...

We construct a one-to-one mapping between binary vectors of length n and preorder codewords of regular, ordered, oriented, rooted, binary trees having N n + 2 log n nodes. The mappings in both directions can be organized in such a way that complexities of all transformations are measured by linear functions of n: The approach is then completely extended to non-regular binary trees and partially extended to D-ary trees with D > 2:

In this paper we present a polynomial time algorithm for finding the shortest sequence of rotations that converts one binary tree into another when both binary trees are of a restricted form. The initial tree must be a degenerate tree, where every node has exactly one child, and the destination binary tree must also be degenerate, of a more restricted nature. Previous work on rotation distance has focused on approximation algorithms. Our algorithm is the only known non-trivial polynomial time algorithm for exact rotation distance between special cases of binary trees. 1.

We classify a type of language called a reflectable language. We then develop a generic algorithm that can be used to list all strings of length n for any reflectable language in Gray code order. The algorithm generalizes Gray code algorithms developed independently for k-ary strings, restricted growth strings, and k-ary trees, as each of these objects can be represented by a reflectable language. Finally, we apply the algorithm to open meanderic systems which can also be represented by a reflectable language. 1

. This is a direct continuation of Shellable Nonpure Complexes and Posets, I ; which appeared in Transactions of the American Mathematical Society 348(1996), 1299-1327. 8. Interval-generated lattices and dominance order In this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets. Let F = fI 1 ; I 2 ; : : : ; I ng be a family of intervals of integers, by which is meant sets of the form [a; b] = fa; a + 1; : : : ; bg, a b. We assume that there are no containments among these intervals, and that they are ordered so that their left and right endpoints are increasing. Let L(F) be the lattice of all sets that are unions of subfamilies of F , ordered by inclusion. Such interval-generated lattices L(F) were introduced and studied by Greene [G]. Define an edge-labeling of L(F) as follows. If A ! B is a covering and a = max(B n A), then (8.1) (A ! B) = ae \Gammaa; if (a + 1) 2 A and a is the left endpoint of some ...

problem under consideration, when N is the smallest odd integer satisfying (1), has been addressed by many authors ([3, 4, 5], and other papers). However, implementation of the known coding algorithms seems to be difficult if n is big enough. We develop the approach, proposed for solving the data transmission problem in asynchronous mode [6], and include a small redundancy in the representation, which allows us to avoid the difficulties. The inequality (3/2) log n which follows from (1) and Stirling's approximation formula for the factorial, can be helpful in the evaluation of the number of additional redundant bits. TABLE 1. Some parameters of the code for regular binary trees. n#Nn/N 7 9 23 0.3043 15 9 31 0.4839 31 11 51 0.6078 63 11 83 0.7590 127 13 151 0.8411 255 15 283 0.9011 511 15 539 0.9480 1023 17 1055 0.9697 2095 17 2127 0.9850 4095 19 4131 0.9913 8191 19 8227 0.9956 16,383 21 16,423 0.9976 32,767 23 32,811 0.9987 THEOREM 1. Let n be a given odd integer and let # be