arXiv subjects
Publications and source records attributed to Isaac M. Hair.
We revisit the single-source shortest paths (SSSP) problem on directed graphs with nonnegative real weights and give a deterministic parallel algorithm with $O(n^{1+o(1)}t^2 + m^{1+o(1)})$ work and $\tilde{O}(n/t)$ span, for any $t \in [1, n]$. This matches (up to subpolynomial factors) the tradeoff due to [Shi and Spencer '99] for undirected graphs with nonnegative real weights.
We consider the single-source shortest paths problem on a directed graph with real-valued (possibly negative) edge weights and solve this problem in $m^{1+o(1)}$ time.
Given two convex polygons $P$ and $Q$ with $n$ and $m$ edges, the maximum overlap problem is to find a translation of $P$ that maximizes the area of its intersection with $Q$. We give the first randomized algorithm for this problem with linear running time. Our result improves the previous two-and-a-half-decades-old algorithm by de Berg, Cheong, Devillers, van Kreveld, and Teillaud (1998), which ran in $O((n+m)\log(n+m))$ time, as well as multiple recent algorithms given for special cases of the problem.
We prove that SVP$_p$ is NP-hard to approximate within a factor of $2^{\log^{1 - \varepsilon} n}$, for all constants $\varepsilon > 0$ and $p > 2$, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVP$_p$ is NP-hard in a finite $\ell_p$ norm. Hardness for SVP$_p$ with $p$ finite was previously only known if NP $\not \subseteq$ RP, and under that assumption, hardness of approximation was only known for all constant factors. As a corollary to our main theorem, we show that under the Sliding Scale Conjecture, SVP$_p$ is NP-hard to approximate within a small polynomial factor, for all constants $p > 2$. Our proof techniques are surprisingly elementary; we reduce from a \emph{regularized} PCP instance directly to the shortest vector problem by using simple gadgets related to Vandermonde matrices and Hadamard matrices.
We revisit a standard polygon containment problem: given a convex $k$-gon $P$ and a convex $n$-gon $Q$ in the plane, find a placement of $P$ inside $Q$ under translation and rotation (if it exists), or more generally, find the largest copy of $P$ inside $Q$ under translation, rotation, and scaling. Previous algorithms by Chazelle (1983), Sharir and Toledo (1994), and Agarwal, Amenta, and Sharir (1998) all required $Ω(n^2)$ time, even in the simplest $k=3$ case. We present a significantly faster new algorithm for $k=3$ achieving $O(n$polylog $n)$ running time. Moreover, we extend the result for general $k$, achieving $O(k^{O(1/\varepsilon)}n^{1+\varepsilon})$ running time for any $\varepsilon>0$. Along the way, we also prove a new $O(k^{O(1)}n$polylog $n)$ bound on the number of similar copies of $P$ inside $Q$ that have 4 vertices of $P$ in contact with the boundary of $Q$ (assuming general position input), disproving a conjecture by Agarwal, Amenta, and Sharir (1998).