Trees are a useful data type, but they are not routinely included in parallel programming systems because their irregular structure makes them seem hard to compute with efficiently. We present a method for constructing implementations of skeletons, high-level homomorphic operations on trees, that execute in parallel. In particular, we consider the case where the size of the tree is much larger than the the number of processors available, so that tree data must be partitioned. The approach uses the theory of categorical data types to derive implementation templates based on tree contraction. Many useful tree operations can be computed in time logarithmic in the size of their argument, on a wide range of parallel systems. 1 Contribution One common approach to general-purpose parallel computation is based on packaging complex operations as templates, or skeletons [3, 12]. Skeletons encapsulate the control and data flow necessary to compute useful operations. This permits software to be...

. A downwards accumulation is a higher-order operation that distributes information downwards through a data structure, from the root towards the leaves. The concept was originally introduced in an ad hoc way for just a couple of kinds of tree. We generalize the concept to an arbitrary regular datatype; the resulting denition is co-inductive. 1 Introduction The notion of scans or accumulations on lists is well known, and has proved very fruitful for expressing and calculating with programs involving lists [4]. Gibbons [7, 8] generalizes the notion of accumulation to various kinds of tree; that generalization too has proved fruitful, underlying the derivations of a number of tree algorithms, such as the parallel prex algorithm for prex sums [15, 8], Reingold and Tilford's algorithm for drawing trees tidily [21, 9], and algorithms for query evaluation in structured text [16, 23]. There are two varieties of accumulation on lists: leftwards and rightwards. Leftwards accumulation ...

While tree contraction algorithms play an important role in e#cient tree computation in parallel, it is di#cult to develop such algorithms due to the strict conditions imposed on contracting operators. In this paper, we propose a systematic method of deriving e#cient tree contraction algorithms from recursive functions on trees in any shape. We identify a general recursive form that can be parallelized to obtain e#cient tree contraction algorithms, and present a derivation strategy for transforming general recursive functions to parallelizable form. We illustrate our approach by deriving a novel parallel algorithm for the maximum connected-set sum problem on arbitrary trees, the tree-version of the famous maximum segment sum problem.

Parallelism is useful in the storage and access of structured documents. Fast parallel algorithms for search in structured text are already known, but they will not supplant the use of indexes to speed up searching until massively parallel architectures become routinely available. However, parallel algorithms suggest new kinds of indexes that provide powerful search capability and performance even on modestly-parallel computers. We present a generalisation of indexes based on regular languages, called indexing languages, that are chosen to be homomorphic images of languages generated by typical search patterns. Precomputing properties of text strings relative to indexing languages makes it fast to exclude large parts of the text from consideration before executing a direct search. 1 Background Search times in document archives are improved by building indexes giving the location of (usually) words as offsets in the structure. This kind of index has three major drawbacks: it is expens...