arXiv subjects
Publications and source records attributed to Yijie Han.
We preprocess the input subdivision with $n$ points on the plane in $O(n\sqrt{\log n})$ time to facilitate point location in constant time. Previously the preprocessing time is $O(n\log n)$ and point location takes $O(\log n)$ time.
We show, for the input vectors $(a_0, a_1, ..., a_{n-1})$ and $(b_0, b_1, ..., b_{n-1})$, where $a_i$'s and $b_j$'s are real numbers, after $O(n\log^4 n)$ time preprocessing for each of them, the vector multiplication $(a_0, a_1, ..., a_{n-1})(b_0, b_1, ..., b_{n-1})^T $ can be computed in $O(\log^4 n \log \log n)$ time. This enables the matrix multiplication for two $n\times n$ matrices to be computed in $O(n^2 \log^4 n\log \log n)$ time.
We show that $n$ real numbers can be stored in a constant number of real numbers such that each original real number can be fetched in $O(\log n)$ time. Although our result has implications for many computational geometry problems, we show here, combined with Han's $O(n\sqrt{\log n})$ time real number sorting algorithm [3, arXiv:1801.00776], we can improve the complexity of Kirkpatrick's point location algorithm [8] to $O(n\sqrt{\log n})$ preprocessing time, a constant number of real numbers for storage and $O(\log n)$ point location time. Kirkpatrick's algorithm uses $O(n\log n)$ preprocessing time, $O(n)$ storage and $O(\log n)$ point location time. The complexity results in Kirkpatrick's algorithm was the previous best result. Although Lipton and Tarjan's algorithm [10] predates Kirkpatrick's algorithm and has the same complexity, Kirkpatrick's algorithm is simpler and has a better structure. This paper can be viewed as a companion paper of paper [3, arXiv:1801.00776].
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 $1 < d\leq c$ 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 $\log^*n=\min \{ i|\log^{(i)} n < 10\}$. When $i$ is a constant we get time $O(n/p+\log^{(i)}n\log \log l+\log l)$. Thus when $l=Ω(\log^{(c)}n)$ 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.
We present an $O(n\sqrt{\log n})$ time and linear space algorithm for sorting real numbers. This breaks the long time illusion that real numbers have to be sorted by comparison sorting and take $Ω(n\log n)$ time to be sorted.