Uniform Linked Lists Contraction

Abstract

We present a parallel algorithm (EREW PRAM algorithm) for linked lists contraction. We show that when we contract a linked list from size $n$ to size $n/c$ for a suitable constant $c$ we can pack the linked list into an array of size $n/d$ for a constant $1<d\le c$ in the time of 3 coloring the list. Thus for a set of linked lists with a total of $n$ elements and the longest list has $l$ elements our algorithm contracts them in $O(n\log i/p+({\log }^{(i)}n+\log i)\log \log l+\log l)$ time, for an arbitrary constructible integer $i$, with $p$ processors on the EREW PRAM, where ${\log }^{(1)}n=\log n$ and ${\log }^{(t)}n=\log {\log }^{(t-1)}n$ and ${\log }^{\ast }n=\min \{i|{\log }^{(i)}n<10\}$. When $i$ is a constant we get time $O(n/p+{\log }^{(i)}n\log \log l+\log l)$. Thus when $l=\Omega ({\log }^{(c)}n)$ for any constant $c$ we achieve $O(n/p+\log l)$ time. The previous best deterministic EREW PRAM algorithm has time $O(n/p+\log n)$ and best CRCW PRAM algorithm has time $O(n/p+\log n/\log \log n+\log l)$. Keywords: Parallel algorithms, linked list, linked list contraction, uniform linked list contraction, EREW PRAM.