We present theoretical algorithms for sorting and searching multikey data, and derive from them practical C implementations for applications in which keys are character strings. The sorting algorithm blends Quicksort and radix sort; it is competitive with the best known C sort codes. The searching algorithm blends tries and binary search trees; it is faster than hashing and other commonly used search methods. The basic ideas behind the algorithms date back at least to the 1960s, but their practical utility has been overlooked. We also present extensions to more complex string problems, such as partial-match searching. 1. Introduction Section 2 briefly reviews Hoare's [9] Quicksort and binary search trees. We emphasize a well-known isomorphism relating the two, and summarize other basic facts. The multikey algorithms and data structures are presented in Section 3. Multikey Quicksort orders a set of n vectors with k components each. Like regular Quicksort, it partitions its input into...

Introduction and Survey; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems --- Sorting and Searching; E.5 [Data]: Files --- Sorting/searching; G.3 [Mathematics of Computing]: Probability and Statistics --- Probabilistic algorithms; E.2 [Data Storage Representation]: Composite structures, linked representations. General Terms: Algorithms, Theory. Additional Key Words and Phrases: Adaptive sorting algorithms, Comparison trees, Measures of disorder, Nearly sorted sequences, Randomized algorithms. A Survey of Adaptive Sorting Algorithms 2 CONTENTS INTRODUCTION I.1 Optimal adaptivity I.2 Measures of disorder I.3 Organization of the paper 1.WORST-CASE ADAPTIVE (INTERNAL) SORTING ALGORITHMS 1.1 Generic Sort 1.2 Cook--Kim division 1.3 Partition Sort 1.4 Exponential Search 1.5 Adaptive Merging 2.EXPECTED-CASE ADAPTIV

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Cauchy-Euler differential equations surfaced naturally in a number of sorting and searching problems, notably in quicksort and binary search trees and their variations. Asymptotics of coefficients of functions satisfying such equations has been studied for several special cases in the literature. We study in this paper the most general framework for Cauchy-Euler equations and propose an asymptotic theory that covers almost all applications where Cauchy-Euler equations appear. Our approach is very general and requires almost no background on differential equations. Indeed the whole theory can be stated in terms of recurrences instead of functions. Old and new applications of the theory are given. New phase changes of limit laws of new variations of quicksort are systematically derived. We apply our theory to about a dozen of diverse examples in quicksort, binary search trees, urn models, increasing trees, etc.

Early aggregation is a technique for speeding up the processing of GROUP BY queries by reducing the amount of intermediate data transferred between main memory and disk. It can also be applied to duplicate elimination because duplicate elimination is equivalent to grouping with no aggregation functions. This paper describes six different algorithms for grouping and aggregation, shows how to incorporate early aggregation in each of them, and analyzes the resulting reduction in intermediate data. In addition to the grouping algorithm used, the reduction depends on several factors: the number of groups, the skew in group size distribution, the input size, and the amount of main memory available. All six algorithms considered benefit from early aggregation with grouping by hash partitioning producing the least amount of intermediate data. If the group size distribution is skewed, the overall reduction can be very significant, even with a modest amount of additional main memory. 1 Introduc...

SQL queries containing GROUPBY and aggregation occur frequently in decision support applications. Grouping with aggregation is typically done by first sorting the input and then performing the aggregation as part of the output phase of the sort. The most widely used external sorting algorithm is merge sort, consisting of a run formation phase followed by a (single) merge pass. The amount of data output from the run formation phase can be reduced by a technique that we call early grouping. The idea is straightforward: simply form groups and perform aggregation during run formation. Each run will now consist of partial groups instead of individual records. These partial groups are then combined during the merge phase. Early grouping always reduces the number of records output from the run formation phase. The relative output size depends on the amount of memory relative to the total number of groups and the distribution of records over groups. When the input data is uniformly distributed...

A current research topic in the area of relational databases is the design of systems based on the Non First Normal Form (NF ) data model. One particular development, the so-called extended NF data model, even permits structured values like lists and tuples to be included as attributes in relations. It is thus well suited to represent complex objects for non-standard database applications. A DBMS which uses this model, called the Advanced Information Management Prototype, is currently being implemented at the IBM Heidelberg Scientific Center. In this paper we examine the problem of detecting and deleting duplicates within this data model. Several alternative approaches are evaluated and a new method, based on sorting complex objects, is proposed, which is both time- and space-efficient.