Thirty nine years of stratified trees

Abstract

The stratified tree, also called van Emde Boas tree, is a data structure implementing the full repertoire of instructions manipulating a single subset $A$of a finite ordered Universe $U = [0 ... u-1]$. Instructions include $member$, $insert$, $delete$, $min$, $max$, $predecessor$ and $successor$, as well as composite ones like $extract-min$. The processing time per instruction is $O(loglog(u))$. Hence it improves upon the traditional comparison based tree structures for dense subsets $A$; if $A$ is sparse, meaning that the size $n = # A = O(log(u))$ the improvement vanishes.

Examples exist where this improvement helps to speed-up algorithmic solutions of real problems; such applications can be found for example in graph algorithms, computational geometry and forwarding of packets on the internet.

The structure was invented during a short postdoc residence at Cornell University in the fall of 1974. In the sequel of this paper I will use the original name Stratified Trees which was used in my own papers on this data structure.

There are two strategies for understanding how this $O(loglog(u))$ improvement can be obtained. Today a direct recursive approach is used where the universe is divided into a cluster of $sqrt{u}$ galaxies each of size $sqrt{u}$; the set manipulation instructions decompose accordingly in a instruction at the cluster and galaxy level, but one of these two instructions is always of a special trivial type. The processing complexity thus satisfies a recurrence $T(u) = T(sqrt{u}) + O(1)$. Consequently $T(u) = O(loglog(u))$.

However, this recursive approach requires address calculations on the arguments which use multiplicative arithmetical instructions. These instructions are not allowed in the Random Access Machine model (RAM) which was the standard model in the developing research area of design and analysis of algorithms in 1974. Therefore the early implementations of the stratified trees are based on a different approach which best is described as a binary-search-on-levels strategy. In this approach the address calculations are not required, and the structure can be implemented using pointers. The downside of this approach is that it leads to rather complex algorithms, which are still hard to present correctly even today. Another bad consequence was the super linear space consumption of the data structure, which was only eliminated three years later.

In this paper I want to describe the historical backgrounds against which the stratified trees were discovered and implemented. I do not give complete code fragments implementing the data structure and the operations; they can be found in the various textbooks and papers mentioned, including a Wikipedia page. Code fragments appearing in this paper are copied verbatim from the original sources; the same holds for the figures.