An Introduction to Three Algorithms for Sorting in Situ

The purpose of this paper is to introduce three algorithms for sorting in situ. They are dealt with in a single paper so that the introductions can share what they have in common; they are presented in the order of increasing complexity. The rest oi this section is devoted to the terminology that is used throughout this paper and to a central subalgorithm. We assume the reader to be familiar with the notion of a ‘sequence of elements’ and with the corresponding notion of a ‘successor’ or ‘predecessor’ of an element in the sequence. We define a chain to be a finite, non-empty sequence of elements, each having (or owning) an integer value. Each element with a successor in the chain is called the father of that successor; each element with a predecessor in the chain is called the son of that predecessor. The element without father is called the root of the chain, the: element without son is called the leaf of the chain. (Coincidence of root and leaf means the chain being a one-element chain.) The offspring of an element is a recursively defined set of elements: the offspring of a leaf is empty, the offspring of a father is its son together with that son’s offspring. Element e0 dominates element el means that e0 has a value that is at least the value owned by el ; element e dominates a set of elements means that e dominates each member of that set. For an element of a chain, the predicate strong means that the element dominates its offspring; a chain is descending means that each father in the chain dominates its son, and we leave to the reader to convince himself that “chain c is descending” is equivalent to