Dan E. Willard

Trans-dichotomous algorithms for minimum spanning trees and shortest paths

The fusion tree method is extended to develop a linear-time algorithm for the minimum spanning tree problem and an O(m+n log n/log log n) implementation of Dijkstras shortest-path algorithm for a graph with n vertices and m edges. The shortest-path algorithm surpasses information-theoretic limitations. The extension of the fusion tree method involves the development of a new data structure, the atomic heap. The atomic heap accommodates heap (priority queue) operations in constant amortized time under suitable polylog restrictions on the heap size. The linear-time minimum spanning tree algorithm results from a direct application of the atomic heap. To obtain the shortest path algorithm, the atomic heap is used as a building block to construct a new data structure, the AF-heap, which has no size restrictions and surpasses information theoretic limitations. The AF-heap belongs to the Fibonacci heap family. >

Adding range restriction capability to dynamic data structures

A database is said to allow range restrictions if one may request that only records with some specified field in a specified range be considered when answering a given query. A transformation is presented that enables range restrictions to be added to an arbitrary dynamic data structure on <italic>n</italic> elements, provided that the problem satisfies a certain decomposability condition and that one is willing to allow increases by a factor of <italic>O</italic>(log <italic>n</italic>) in the worst-case time for an operation and in the space used. This is a generalization of a known transformation that works for static structures. This transformation is then used to produce a data structure for range queries in <italic>k</italic> dimensions with worst-case times of <italic>O</italic>(log<italic><supscrpt>k</supscrpt> n</italic>) for each insertion, deletion, or query operation.

Log-logarithmic selection resolution protocols in a multiple access channel

We propose two selection protols that run on multiple access channels in log-logarithmic expected time, and establish a complementary lower bound showing that the first protocols falls within an additive constant of optimality and that the second differs from optimality by less than any multiplicative factor infinitesimally greater than 1 as the size of the problem approaches infinity. It is difficult to second-guess the fast-changing electronics industry, but our mathematical analysis could be relevant outside the traditional interests of communications protocols to semaphore-like problems.

New data structures for orthogonal range queries

Consider a set of N records corresponding to points in k-dimensional space (

Examining Computational Geometry, Van Emde Boas Trees, and Hashing from the Perspective of the Fusion Tree

k \geqq 2

BLASTING through the information theoretic barrier with FUSION TREES

). This article introduces one new data structure which uses memory

Searching Unindexed and Nonuniformly Generated Files in

O(N\log ^{k - 1} N)

\log \log N

for supporting orthogonal range queries with worst-case complexity

Time

O(\log ^{k - 1} N)

Maintaining dense sequential files in a dynamic environment (Extended Abstract)

and several modifications of this proposal for a dynamic environment. These results are especially useful when