Solving kk-SUM using few linear queries

The $k$-SUM problem is given $n$ input real numbers to determine whether any $k$ of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within $P$, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil \frac{k}{2} \rceil$. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth $O(n^3\log^3 n)$ solving $k$-SUM. Furthermore, we show that there exists a randomized algorithm that runs in $\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8})$ time, and performs $O(n^3\log^3 n)$ linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the $+8$) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of $k$. The $O(n^3\log^3 n)$ bound on the number of linear queries is also a tighter bound than any known algorithm solving $k$-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-$P$. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist $o(n)$-linear decision trees of depth $o(n^4)$

Mobility of bodies in contact. I. A 2nd-order mobility index formultiple-finger grasps

Using a configuration-space approach, the paper develops a 2nd-order mobility theory for rigid bodies in contact. A major component of this theory is a coordinate invariant 2nd-order mobility index for a body, B, in frictionless contact with finger bodies A1,...A k. The index is an integer that captures the inherent mobility of B in an equilibrium grasp due to second order, or surface curvature, effects. It differentiates between grasps which are deemed equivalent by classical 1st-order theories, but are physically different. We further show that 2nd-order effects can be used to lower the effective mobility of a grasped object, and discuss implications of this result for achieving new lower bounds on the number of contacting finger bodies needed to immobilize an object. Physical interpretation and stability analysis of 2nd-order effects are taken up in the companion pape

Threesomes, Degenerates, and Love Triangles

The 3SUM problem is to decide, given a set of $n$ real numbers, whether any three sum to zero. It is widely conjectured that a trivial $O(n^2)$-time algorithm is optimal and over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies $\Omega(n^2)$ lower bounds on numerous problems in computational geometry and a variant of the conjecture implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this paper we refute the 3SUM conjecture. We prove that the decision tree complexity of 3SUM is $O(n^{3/2}\sqrt{\log n})$ and give two subquadratic 3SUM algorithms, a deterministic one running in $O(n^2 / (\log n/\log\log n)^{2/3})$ time and a randomized one running in $O(n^2 (\log\log n)^2 / \log n)$ time with high probability. Our results lead directly to improved bounds for $k$-variate linear degeneracy testing for all odd $k\ge 3$. The problem is to decide, given a linear function $f(x_1,\ldots,x_k) = \alpha_0 + \sum_{1\le i\le k} \alpha_i x_i$ and a set $A \subset \mathbb{R}$, whether $0\in f(A^k)$. We show the decision tree complexity of this problem is $O(n^{k/2}\sqrt{\log n})$. Finally, we give a subcubic algorithm for a generalization of the $(\min,+)$-product over real-valued matrices and apply it to the problem of finding zero-weight triangles in weighted graphs. We give a depth-$O(n^{5/2}\sqrt{\log n})$ decision tree for this problem, as well as an algorithm running in time $O(n^3 (\log\log n)^2/\log n)$

An Optimal Algorithm for Higher-Order Voronoi Diagrams in the Plane: The Usefulness of Nondeterminism

We present the first optimal randomized algorithm for constructing the order-$k$ Voronoi diagram of $n$ points in two dimensions. The expected running time is $O(n\log n + nk)$, which improves the previous, two-decades-old result of Ramos (SoCG'99) by a $2^{O(\log^*k)}$ factor. To obtain our result, we (i) use a recent decision-tree technique of Chan and Zheng (SODA'22) in combination with Ramos's cutting construction, to reduce the problem to verifying an order-$k$ Voronoi diagram, and (ii) solve the verification problem by a new divide-and-conquer algorithm using planar-graph separators. We also describe a deterministic algorithm for constructing the $k$-level of $n$ lines in two dimensions in $O(n\log n + nk^{1/3})$ time, and constructing the $k$-level of $n$ planes in three dimensions in $O(n\log n + nk^{3/2})$ time. These time bounds (ignoring the $n\log n$ term) match the current best upper bounds on the combinatorial complexity of the $k$-level. Previously, the same time bound in two dimensions was obtained by Chan (1999) but with randomization.Comment: To appear in SODA 2024. 16 pages, 1 figur

Applications of incidence bounds in point covering problems

In the Line Cover problem a set of n points is given and the task is to cover the points using either the minimum number of lines or at most k lines. In Curve Cover, a generalization of Line Cover, the task is to cover the points using curves with d degrees of freedom. Another generalization is the Hyperplane Cover problem where points in d-dimensional space are to be covered by hyperplanes. All these problems have kernels of polynomial size, where the parameter is the minimum number of lines, curves, or hyperplanes needed. First we give a non-parameterized algorithm for both problems in O*(2^n) (where the O*(.) notation hides polynomial factors of n) time and polynomial space, beating a previous exponential-space result. Combining this with incidence bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant. Our result improves the previous best times O*((k/1.35)^k) for Line Cover (where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds for covering points by parabolas or conics. We also present an algorithm for Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving on the previous time of O*((k^2/1.3)^k).Comment: SoCG 201