John Howat

Fast local searches and updates in bounded universes

Given a bounded universe { 0 , 1 , ? , U - 1 } , we show how to perform predecessor searches in O ( log log Δ ) expected time, where Δ is the difference between the element being searched for and its predecessor in the structure, while supporting updates in O ( log log Δ ) expected amortized time, as well. This unifies the results of traditional bounded universe structures (which support predecessor searches in O ( log log U ) time) and hashing (which supports membership queries in O ( 1 ) time). We also show how these results can be applied to approximate nearest neighbour queries and range searching.

Layered Working-Set Trees

The working-set bound [Sleator and Tarjan in J. ACM 32(3), 652–686, 1985] roughly states that searching for an element is fast if the element was accessed recently. Binary search trees, such as splay trees, can achieve this property in the amortized sense, while data structures that are not binary search trees are known to have this property in the worst case. We close this gap and present a binary search tree called a layered working-set tree that guarantees the working-set property in the worst case. The unified bound [Bădoiu et al. in Theor. Comput. Sci. 382(2), 86–96, 2007] roughly states that searching for an element is fast if it is near (in terms of rank distance) to a recently accessed element. We show how layered working-set trees can be used to achieve the unified bound to within a small additive term in the amortized sense while maintaining in the worst case an access time that is both logarithmic and within a small multiplicative factor of the working-set bound.

A distribution-sensitive dictionary with low space overhead

The time required for a sequence of operations on a data structure is usually measured in terms of the worst possible such sequence. This, however, is often an overestimate of the actual time required. Distribution-sensitive data structures attempt to take advantage of underlying patterns in a sequence of operations in order to reduce time complexity, since access patterns are non-random in many applications. Many of the distribution-sensitive structures in the literature require a great deal of space overhead in the form of pointers. We present a dictionary data structure that makes use of both randomization and existing space-efficient data structures to yield low space overhead while maintaining distribution sensitivity in the expected sense. We further show a modification that allows predecessor searches in a similar time bound.

Layered working-set trees

The working-set bound [Sleator and Tarjan, J. ACM, 1985] roughly states that searching for an element is fast if the element was accessed recently. Binary search trees, such as splay trees, can achieve this property in the amortized sense, while data structures that are not binary search trees are known to have this property in the worst case. We close this gap and present a binary search tree called a layered working-set tree that guarantees the working-set property in the worst case. The unified bound [Bădoiu et al., TCS, 2007] roughly states that searching for an element is fast if it is near (in terms of rank distance) to a recently accessed element. We show how layered working-set trees can be used to achieve the unified bound to within a small additive term in the amortized sense while maintaining in the worst case an access time that is both logarithmic and within a small multiplicative factor of the working-set bound.

A Distribution-Sensitive Dictionary with Low Space Overhead

The time required for a sequence of operations on a data structure is usually measured in terms of the worst possible such sequence. This, however, is often an overestimate of the actual time required. Distribution-sensitive data structures attempt to take advantage of underlying patterns in a sequence of operations in order to reduce time complexity, since access patterns are non-random in many applications. Unfortunately, many of the distribution-sensitive structures in the literature require a great deal of space overhead in the form of pointers. We present a dictionary data structure that makes use of both randomization and existing space-efficient data structures to yield very low space overhead while maintaining distribution sensitivity in the expected sense.

Biased Predecessor Search

We consider the problem of performing predecessor searches in a bounded universe while achieving query times that depend on the distribution of queries. We obtain several data structures with various properties: in particular, we give data structures that achieve expected query times logarithmic in the entropy of the distribution of queries but with space bounded in terms of universe size, as well as data structures that use only linear space but with query times that are higher (but still sublinear) functions of the entropy. For these structures, the distribution is assumed known. We also consider data structures with general weights on universe elements, as well as the case when the distribution is not known in advance.

A history of distribution-sensitive data structures

Distribution-sensitive data structures attempt to exploit patterns in query distributions in order to allow many sequences of queries execute faster than in traditional data structures. In this paper, we survey the history of such data structures, outline open problems in the area, and offer some new results.