Tighter Connections Between Formula-SAT and Shaving Logs

A noticeable fraction of Algorithms papers in the last few decades improve the running time of well-known algorithms for fundamental problems by logarithmic factors. For example, the $O(n^2)$ dynamic programming solution to the Longest Common Subsequence problem (LCS) was improved to $O(n^2/\log^2 n)$ in several ways and using a variety of ingenious tricks. This line of research, also known as "the art of shaving log factors", lacks a tool for proving negative results. Specifically, how can we show that it is unlikely that LCS can be solved in time $O(n^2/\log^3 n)$? Perhaps the only approach for such results was suggested in a recent paper of Abboud, Hansen, Vassilevska W. and Williams (STOC'16). The authors blame the hardness of shaving logs on the hardness of solving satisfiability on Boolean formulas (Formula-SAT) faster than exhaustive search. They show that an $O(n^2/\log^{1000} n)$ algorithm for LCS would imply a major advance in circuit lower bounds. Whether this approach can lead to tighter barriers was unclear. In this paper, we push this approach to its limit and, in particular, prove that a well-known barrier from complexity theory stands in the way for shaving five additional log factors for fundamental combinatorial problems. For LCS, regular expression pattern matching, as well as the Fr\'echet distance problem from Computational Geometry, we show that an $O(n^2/\log^{7+\varepsilon} n)$ runtime would imply new Formula-SAT algorithms. Our main result is a reduction from SAT on formulas of size $s$ over $n$ variables to LCS on sequences of length $N=2^{n/2} \cdot s^{1+o(1)}$. Our reduction is essentially as efficient as possible, and it greatly improves the previously known reduction for LCS with $N=2^{n/2} \cdot s^c$, for some $c \geq 100$

Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet distance between polygonal curves in (Formula presented.) under polyhedral distance functions (e.g., (Formula presented.) and (Formula presented.)). We also get a (Formula presented.)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed (Formula presented.). For the exact Euclidean case, our framework currently yields an algorithm with running time (Formula presented.). However, we conjecture that it may eventually lead to a faster exact algorithm

Geodesic Fréchet Distance Inside a Simple Polygon

We present the first algorithm for the geodesic Fréchet distance between two polygonal curves A and B inside a simple polygon P. If A and B have total complexity N and P has complexity k, then the algorithm runs in O(k+N 2 log kN log N) expected time and O(k+N 2) space. This runtime is quite good as it is only a logarithmic factor larger than the non-geodesic Fréchet algorithm [2]. We also unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fréchet distance algorithms. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search

Geodesic Fréchet distance inside a simple polygon

We present an alternative to parametric search that applies to both the nongeodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We introduce the first algorithm to compute the geodesic Fréchet distance between two polygonal curves A and B inside a simple bounding polygon P. The geodesic Fréchet decision problem is solved almost as fast as its nongeodesic sibling in O(N2 log k) time and O(k+N) space after O(k) preprocessing, where N is the larger of the complexities of A and B and k is the complexity of P. The geodesic Fréchet optimization problem is solved by a randomized approach in O(k+N2 log kN log N) expected time and O(k+N2) space. This runtime is only a logarithmic factor larger than the standard nongeodesic Fréchet algorithm [Alt and Godau 1995]. Results are also presented for the geodesic Fréchet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon P.