. Dutton #1993# presents a further HEAPSORT variant called WEAK-HEAPSORT, which also contains a new data structure for priority queues. The sorting algorithm and the underlying data structure are analyzed showing that WEAK-HEAPSORT is the best HEAPSORT variant and that it has a lot of nice properties. It is shown that the worst case number of comparisons is ndlog ne# 2 dlog ne + n #dlog ne#nlog n +0:1nand weak heaps can be generated with n # 1 comparisons. A double-ended priority queue based on weakheaps can be generated in n + dn=2e#2 comparisons. Moreover, examples for the worst and the best case of WEAK-HEAPSORT are presented, the number of Weak-Heaps on f1;:::;ng is determined, and experiments on the average case are reported. 1

We present a simple, implicit data structure for implementing a double-ended priority queue. The data structure can be viewed as a natural generalization of the heap, and is different from a data structure for the same problem recently proposed by Atkineon et al. A number of applications to computational geometry are discussed. By generalizing the approach for d-dimensional data, a (dynamic) implicit data structure is obtained for complementary range searching in O(K) time per query and with O(log n) update times, for fixed d and K the number of answers of a query. Several related ideas and applications are also discussed.

Comparator networks for constructing binary heaps of size n are presented which have size O(n log log n) and depth O(log n). A lower bound of n log log n O(n) for the size of any heap construction network is also proven, implying that the networks presented are within a constant factor of optimal. We give a tight relation between the leading constants in the size of selection networks and in the size of heap construction networks.

During four years of arduous service, a Ph. D. student is expected to familiarise himself with his field of research, and, hopefully, contribute to this field. This is reflected by the division of this dissertation into two parts. Part I is a (partial) overview of the field of computational biology as I conceive it, an overview that is aimed at presenting the context for my contributions to the field of computational biology. These contributions are presented in part II as five independent articles

Abstract. In this paper we present a new data structure for implementing heapsort algorithm for pairs of which can be simultaneously stored and processed in a single register. Since time complexity of Carlsson type variants of heapsort has already achieved a leading coefficient of 1, concretely nlg n + nlg lg n, and lower bound theory asserts that no comparison based in-place sorting algorithm can sort n data in less than ⌈lg(n!) ⌉ ≈ n lg n − 1.44n comparisons on the average, any improvement in the number of comparisons can only be achieved in lower terms. Our new data structure results in improvement in the linear term of the time complexity function irrespective of the variant of the heapsort algorithm used. This improvement is important in the context that some of the variants of heapsort algorithm, for example weak heapsort although not in-place, are near optimal and is away from the theoretical bound on number of comparisons by only 1.54n.

Recently, many results on the computational complexity of sorting algorithms were obtained using Kolmogorov complexity (the incompressibility method). Especially, the usually hard average-case analysis is ammenable to this method. Here we survey such results about Bubblesort, Heapsort, Shellsort, Dobosiewicz-sort, Shakersort, and sorting with stacks and queues in sequential or parallel mode. Especially in the case of Shellsort the uses of Kolmogorov complexity surprisingly easily resolved problems that had stayed open for a long time despite strenuous attacks.

We present three variations of the following new sorting theme: Throughout the sort, the arrayismaintained in piles of sorted elements. Ateach step, the piles are split into two parts, so that the elements of the left piles are smaller than #or equal to# the elements of the right piles. Then, the two parts are each sorted, recursively. The theme, then, is a combination of Hoare's Quicksort idea, and the Pick algorithm, by Blum, et al., for linear selection. The variations arise from the possible choices of splitting method. Twovariations attempt to minimize the average number of comparisons. The better of these has an average performance of 1:075n lg n comparisons. The third variation sacri#ces the average case for a worst-case performance of 1:756n lg n, which is better than Heapsort. They all require minimal extra space and about as manydatamoves as comparisons. 1 Introduction The sorting problem is: Given an array a 1 ;a 2 ; ###;a n of elements, rearrange them so t...

We describe an approach for visually teaching important proofs in the Junior-Senior level course on the design and analysis of data structures and algorithms (CS7/DS&A). The main idea of this educational paradigm is to justify important claims about data structures and algorithms by using pictures that visualize proofs so clearly that the pictures can qualify as proofs themselves. The advantage of using this approach for DS&A is that it augments or even replaces inductive arguments that many students find difficult. Moreover, this paradigm communicates important algorithmic facts in a compelling way for students who are more visually-oriented. We illustrate this technique by giving examples of visual proofs of several key concepts in DS&A. 1 Introduction In this era of real-time video games and MTV, students these days seem more visually-oriented than ever. They learn most naturally by seeing a concept described with a picture, and they remember that concept by recalling the picture ...

An elementary approach is given to studying the recurrence relations associated with generalized heaps (or d-heaps), cost of optimal merge, and generalized divide-and-conquer minimization problems. We derive exact formulae for the solutions of all such recurrences and give some applications. In particular, we present a precise probabilistic analysis of Floyd's algorithm for constructing d-heaps when the input is randomly given. A variant of d-heap having some interesting combinatorial properties is also introduced.

In this paper we examine the problem of heap construction on a rooted tree T from a packet routing perspective. Each node of T initially contains a packet which has a key-value associated with it. The aim of the heap construction algorithm is to route the packets along the edges of the tree so that, at the end of the routing, the tree is heap ordered with respect to the key values associated with the packets. We consider the case where the routing is performed according to the matching model and we present and analyse an off-line algorithm that heap orders the tree within 2h(T) routing steps, where h(T) is the height of tree T. The main contribution of the paper is the novel analysis of the algorithm based on potential functions. It is our belief that potential functions will be the main vehicle in analysing fast non-recursive routing algorithms.