Partial match retrieval of multidimensional data
Journal of the ACM (JACM)
Branching processes in the analysis of the heights of trees
Acta Informatica
Applications of spatial data structures: Computer graphics, image processing, and GIS
Applications of spatial data structures: Computer graphics, image processing, and GIS
The design and analysis of spatial data structures
The design and analysis of spatial data structures
An analysis of random d-dimensional quad trees
SIAM Journal on Computing
Combinatorial variations on multidimensional quadtrees
Journal of Combinatorial Theory Series A
Mellin transforms and asymptotics: finite differences and Rice's integrals
Theoretical Computer Science - Special volume on mathematical analysis of algorithms (dedicated to D. E. Knuth)
On the distribution of the root arity of a d dimensional hyperquaternary branching
Proceedings of the 4th conference on Formal power series and algebraic combinatorics
Hypergeometrics and the cost structure of quadtrees
Random Structures & Algorithms
Study of universal constants for hyperquaternary research branches
FPSAC '93 Proceedings of the 5th conference on Formal power series and algebraic combinatorics
Patterns in random binary search trees
Random Structures & Algorithms
On convergence rates in the central limit theorems for combinatorial structures
European Journal of Combinatorics
Universal Limit Laws for Depths in Random Trees
SIAM Journal on Computing
On the internal path length of d-dimensional quad trees
Random Structures & Algorithms
Symbolic integration of meromorphic differential systems via Dirichlet functions
Discrete Mathematics
Phase changes in random m-ary search trees and generalized quicksort
Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
Phase Change of Limit Laws in the Quicksort Recurrence under Varying Toll Functions
SIAM Journal on Computing
Journal of Algorithms - Analysis of algorithms
An asymptotic theory for recurrence relations based on minimization and maximization
Theoretical Computer Science
m-ary search trees when m 27: a strong asymptotics for the space requirements
Random Structures & Algorithms
Limit distribution of the number of consecutive records
Random Structures & Algorithms
Partial Match Queries in Random k-d Trees
SIAM Journal on Computing
Combinatorics, Probability and Computing
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We show that a wide class of linear cost measures (such as the number of leaves) in random d-dimensional point quadtrees undergo a change in limit laws: If the dimension d = 1, …, 8, then the limit law is normal; if d ≥ 9 then there is no convergence to a fixed limit law. Stronger approximation results such as convergence rates and local limit theorems are also derived for the number of leaves, additional phase changes being unveiled. Our approach is new and very general, and also applicable to other classes of search trees. A brief discussion of Devroye's grid trees (covering m-ary search trees and quadtrees as special cases) is given. We also propose an efficient numerical procedure for computing the constants involved to high precision.