We introduce a novel approach to the classical problem of in-situ, stable merging, where \in-situ" means the use of no more than O(log 2 n) bits of extra memory for lists of size n. Shu emerge reduces the merging problem to the problem of realising the \perfect shu e" permutation, that is, the exact interleaving of two, equal length lists. The algorithm is recursive, using a logarithmic number of variables, and so does not use absolutely minimum storage, i.e., a xed number of variables. A simple method of realising the perfect shu e uses one extra bit per element, and so is not in-situ. We show that the perfect shu e can be attained using absolutely minimum storage and in linear time, at the expense of doubling the number of moves, relative to the simple method. We note that there is a worst case for Shu emerge requiring time (n log n), where n is the sum of the lengths of the input lists. We also present an analysis of a variant of Shu emerge which uses a generalised shu e and which has a provable average case time complexity of O(n log log m), where m is the length of the shortest input list. It is unlikely that the generalised shu e can be achieved in-situ. Linear time, in-situ, stable merging has previously been demonstrated. We present experimental evidence indicating that Shu emerge, although almost certainly not asymptotically linear, might be of value in practice. The relative simplicity of the basic method, particularly with respect to stability, also recommends it. This work was supported by the Natural Sciences and Engineering Research Council of Canada We describe a novel merging algorithm with certain properties of theoretical interest and possibly of practical value. The merging problem is to produce one sorted list from an input of two sorted lists. The properties of interest are execution time, memory usage and \stability". An \in-situ" merge, sort or other permutation, rearranges the subject elements within the space that they occupy, in contrast to the standard merge algorithm which duplicates the input space. We use the precise but tight de nition of \in-situ" Knu73] (Chapter 5, Section 5, Exercise 3) which allows the use of no more than O(log 2 n) bits over and above that space occupied by the elements themselves, where n is the number of elements. This de nition permits recursion, so long as the stack depth is restricted to O(log n), but does not allow the use of extra space proportional to n. It also implies that the lists be represented by arrays, or parts of arrays. The extra space used by the pointers in a linked list representation would be at least (pointer size n), and each pointer would require at least log n bits. Originally, what we are here referring to as \in-situ", was called \minimum storage", e.g., in the citation just given Knu73]. An even tighter restriction, sometimes referred to as \absolutely minimum storage", permits only O(log n) bits, i.e., a constant number of variables. Stability is the property of a merge or sort which guarantees that the order of equal elements in t...
