Const-time Search & Linear-time Sorting

Const-time Search & Linear-time Sorting Shi-qing Xin & Guo-jin Wang 2006.10.25

Problem Description • At most n numbers, with each formatted as K=k1k2…kw (0110…) • Two dimensions to describe complexity • Max(a1,a2), Max(a1,a2,…an) • Linear-time sorting? • Maintain a priority queue without increasing time complexity?

Previous Methods • Previous search algorithms • Previous sorting algorithms • Stable sorting, comparison sorting • Is O(nlogn) the lower bound? • Radix sorting: from end; euqal word length; extra space; repeated copy; array-based and not dynamic; higher average time complexity; • Quick sorting and red-black tree

Dijkstra’s algorithm • Dijkstra’s algorithm on graphs • Time complexity O(ElogE) • advantages

An Ugly Method • An O(n2) method based on building a binary tree.

Contradiction • An O(E+nlogn) improved version • However, the improved algorithm isn’t essential. For example, Let wij += fij(min(di, dj)) for each edge, where fij(۰) is a non-decreasing function. • The definition is correct. • The improved version doesn’t work. • Another improved version by Tarjan;

Other Counter-intuitive Examples • Our work on 3d shortest path problem • What is the best sorting algorithm? • No one denies the possibility of const-time search and linear-time sorting; • Unreasonable array & hash table; • Unreasonable binary search & Fibonacci search; • How do we percept?

Goal of Our Algorithm • No empty position, linear space • A limited depth indepent of n • Dynamic search, deletion, insertion can be done in const time. • Extract the min/max element in const time. • Linear-time traversal • Visit k-th elem in const time • Reflect inherent similarity

Operations on RBT • Search an elem • Insert an elem • Delete a given elem • Extract the min/max element; • Read the k-th element • Traversal • Extract a group of similar elements

Applications • Complexity theory, graphs, computational geometry; for example, shortest path problem in 2D or 3D. • Design efficient algorithms • Computer design • Database & data structure & language • AI • And so on

Criticisms • Array, hash table, B- tree & B+ tree, balanced tree, huffman tree, read-black tree, suffix array sorting, binary search, k-ary heap, Fibonacci heap, etc. • Computer design: excessively digitized;

Thank you!

Const-time Search & Linear-time Sorting