All non-trivial variants of 3-LDT are equivalent

The popular 3-SUM conjecture states that there is no strongly subquadratic time algorithm for checking if a given set of integers contains three distinct elements that sum up to zero. A closely related problem is to check if a given set of integers contains distinct $x_1, x_2, x_3$ such that $x_1+x_2=2x_3$. This can be reduced to 3-SUM in almost-linear time, but surprisingly a reverse reduction establishing 3-SUM hardness was not known. We provide such a reduction, thus resolving an open question of Erickson. In fact, we consider a more general problem called 3-LDT parameterized by integer parameters $\alpha_1, \alpha_2, \alpha_3$ and $t$. In this problem, we need to check if a given set of integers contains distinct elements $x_1, x_2, x_3$ such that $\alpha_1 x_1+\alpha_2 x_2 +\alpha_3 x_3 = t$. For some combinations of the parameters, every instance of this problem is a NO-instance or there exists a simple almost-linear time algorithm. We call such variants trivial. We prove that all non-trivial variants of 3-LDT are equivalent under subquadratic reductions. Our main technical contribution is an efficient deterministic procedure based on the famous Behrend's construction that partitions a given set of integers into few subsets that avoid a chosen linear equation

Computing a flattest, undercut-free parting line for a convex polyhedron, with application to mold design

AbstractA parting line for a polyhedron is a closed curve on its surface, which identifies the two halves of the polyhedron for which mold-boxes must be made. A parting line is undercut-free if the two halves that it generates do not contain facets that obstruct the de-molding of the polyhedron. Computing an undercut-free parting line that is as “flat” as possible is an important problem in mold design. In this paper, algorithms are presented to compute such a parting line for a convex polyhedron, based on different flatness criteria

Data Structures Meet Cryptography: 3SUM with Preprocessing

This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat--Naor inversion, a technique with cryptographic origins, to obtain a data structure upper bound. In particular, our technique yields a suite of algorithms with space $S$ and (online) time $T$ for a preprocessing version of the $N$-input 3SUM problem where $S^3\cdot T = \widetilde{O}(N^6)$. This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for $S=N^{2-\delta}$ and $T = N^{1-\delta}$ for any constant $\delta>0$. Secondly, we show equivalence between lower bounds for a broad class of (static) data structure problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function $F: [N] \to [N]$ (accessed as an oracle) we show how to compile it into a function $G^F: [N^2] \to [N^2]$ which resists $S$-bit preprocessing attacks that run in query time $T$ where $ST=O(N^{2-\varepsilon})$ (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that $F$ itself can be more easily inverted, say with $N^{2/3}$-bit preprocessing in $N^{2/3}$ time. We also show that much stronger lower bounds follow from the hardness of kSUM. Our results can be equivalently interpreted as security against adversaries that are very non-uniform, or have large auxiliary input, or as security in the face of a powerfully backdoored random oracle. Thirdly, we give non-adaptive lower bounds for 3SUM and a range of geometric problems which match the best known lower bounds for static data structure problems

Geometric aspects of the casting process

Manufacturing is the process of converting raw materials into useful products. Among the most important manufacturing processes, casting is a commonly used manufacturing process for plastic and metal objects. The industrial casting process consists of two stages. First, liquid is filled into a cavity formed by two cast parts. After the liquid has hardened, one cast part retracts, carrying the object with it. Afterwards, the object is ejected from the retracted cast part. In both retraction and ejection steps, the cast parts and the object should not be damaged, so that the quality of final object is guaranteed and the cast parts can be reused to produce another object. This mode of production is called ``mass production''. Due to the geometric nature of the casting process, many geometric problems arise in the automation of casting. The problems we address here concern this aspect: Given a 3-dimensional object, is there a cast for it whose parts can be removed after the liquid has solidified? An object for which this is the case is called castable. We consider the castability problem in three different casting models with a two-part cast. In the first casting model, two parts must be removed in opposite directions. There are two cases depending on whether the removal direction is specified in advance or not. The second model is identical to the first casting model, except that the cast machinery has a certain level of uncertainty in its directional movement. In the third model, the two cast parts are to be removed in two given directions and these directions need not be opposite. For all three casting models, we give complete characterizations of castability, and obtain algorithms to verify these conditions for polyhedral parts. In manufacturing, features of an object imply manufacturing information, which facilitates the process of analyzing manufacturability and the automated design of a cast for the object. A small hole or a depression on the boundary of an object, for example, restricts the set of removal directions for which this object is castable, since the portion of the cast in the hole or in the depression must be removed from the object without breaking the object. We define a geometric feature, the cavity, which is related to the castability of objects, and provide algorithms to extract it from objects

Filling Polyhedral Molds

In the manufacturing industry, finding an orientation for a mold that eliminates surface defects and ensures a complete fill after termination of the gravity casting process, is an important and difficult problem. We study the problem of determining a favorable position of a mold (modeled as a polyhedron), such that when it is filled, no air pockets and ensuing surface defects arise. Given a polyhedron in a fixed orientation, we present a linear time algorithm that determines whether the mold can be filled from that orientation without forming air pockets

Filling polyhedral molds

In manufacturing industry, finding an orientation for a mold that eliminates surface defects and ensures a complete fill after termination of the gravity casting process is an important and difficult problem. We study the problem of determining a favorable position of a mold (modeled as a polyhedron) such that, when it is filled, no air pockets and ensuing surface defects arise. Given a polyhedron in a fixed orientation, we present a linear time algorithm that determines whether the mold can be filled from that orientation without forming air pockets. We also present an algorithm that determines the most favorable orientation for a polyhedral mold in O(n2) time. A reduction from a well-known problem indicates that improving the O(n2) bound is unlikely for general polyhedral molds. We relate fillability to some well known classes of polyhedra. For some of these classes of objects, an optimal direction of fillability can be determined in linear time. Finally, for molds that satisfy a local regularity condition, we give an improved algorithm that runs in time O(nklog2 nlog log(n/k)), where k is the number of venting holes needed to avoid air pockets in an optimal orientation

Filling Polyhedral Molds

In the manufacturing industry, finding an orientation for a mold that eliminates surface defects and insures a complete fill after termination of the gravity casting process, is an important and difficult problem. We study the problem of determining a favorable position of a mold (modeled as a polyhedron), such that when it is filled, no air bubbles and ensuing surface defects arise. Given a polyhedron in a fixed orientation, we present a linear time algorithm that determines whether the mold can be filled from that orientation without forming air bubbles. We also present an algorithm that determines the most favorable orientation for a polyhedral mold in O(n 2 ) time. A reduction from a well-known problem indicates that improving the O(n 2 ) bound is unlikely for general polyhedral molds. We relate fillability to some well known classes of polyhedra. For some of these classes of objects, an optimal direction of fillability can be determined in linear time. Finally, for molds that ..