arXiv subjects

Micha Sharir

Publications and source records attributed to Micha Sharir.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Algorithms for the Bottleneck Path Problem in Geometric Graphs

We present efficient algorithms for the bottleneck path problem in two geometric settings that arise naturally in applications: directional-antenna graphs in the plane with antenna angles bounded from below by a constant, and visibility graphs whose vertices lie on or above a 1.5-dimensional terrain, both with Euclidean distances as edge weights. We provide near-linear algorithms for the corresponding decision problems, namely, determining whether the subgraph obtained by retaining all edges with weight at most some threshold ${\bf bn}$ contains a path from $s$ to $t$. We then use the decision procedures to obtain algorithms for the bottleneck path problem that run in $O^*(n^{8/7})$ randomized expected time, where $n$ is the input size and the $O^*(\cdot)$ notation hides subpolynomial factors. Within the same performance bounds, we can also solve the bounded-hop version, in which we only consider $s$-$t$ paths with at most $k$ edges, for a given integer $k < n$.

cs.CG↗

arXiv2026

Nearly-Tight Bounds for Vertical Decomposition in Three and Four Dimensions

Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in ${\mathbb R}^d$ into constant-complexity subcells. In this paper, we settle in the affirmative a few long-standing open problems involving the vertical decomposition of substructures of arrangements for $d = 3, 4$. For example, we obtain sharp bounds on the complexity of the vertical decomposition of the complement of the union of a set of semi-algebraic regions of constant complexity in ${\mathbb R}^3$, and of the minimization diagram of a set of trivariate functions. These results lead to efficient algorithms for a variety of problems involving vertical decompositions, including algorithms for constructing the decompositions themselves and for constructing $(1/r)$-cuttings of substructures of arrangements. They also lead to a data structure for answering point-enclosure queries amid semi-algebraic sets in ${\mathbb R}^3$ and ${\mathbb R}^4$.

cs.CG↗

arXiv2026

Implicit representations via the polynomial method

Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on $n$ vertices in this family, we can assign a label consisting of $O(n^{1-2/(d+1) + \varepsilon})$ bits to each vertex (where $\varepsilon > 0$ can be made arbitrarily small and the constant of proportionality depends on $\varepsilon$ and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of $O(n^{1/3 + \varepsilon})$ bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of $O(n^{1-1/d}\log n)$ for $d$-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-$d$ polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size $O(\log n)$. We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size $O(\log^3 n)$.

cs.CG↗

arXiv2026

Dynamic Nearest-Neighbor Searching Under General Metrics in ${\mathbb R}^3$ and Its Applications

Let $K$ be a compact, centrally-symmetric, strictly-convex region in ${\mathbb R}^3$, which is a semi-algebraic set of constant complexity, i.e. the unit ball of a corresponding metric, denoted as $\|\cdot\|_K$. Let ${\mathcal{K}}$ be a set of $n$ homothetic copies of $K$. This paper contains two main sets of results: (i) For a storage parameter $s\in[n,n^3]$, ${\mathcal{K}}$ can be preprocessed in $O^*(s)$ expected time into a data structure of size $O^*(s)$, so that for a query homothet $K_0$ of $K$, an intersection-detection query (determine whether $K_0$ intersects any member of ${\mathcal{K}}$, and if so, report such a member) or a nearest-neighbor query (return the member of ${\mathcal{K}}$ whose $\|\cdot\|_K$-distance from $K_0$ is smallest) can be answered in $O^*(n/s^{1/3})$ time; all $k$ homothets of ${\mathcal{K}}$ intersecting $K_0$ can be reported in additional $O(k)$ time. In addition, the data structure supports insertions/deletions in $O^*(s/n)$ amortized expected time per operation. Here the $O^*(\cdot)$ notation hides factors of the form $n^\varepsilon$, where $\varepsilon>0$ is an arbitrarily small constant, and the constant of proportionality depends on $\varepsilon$. (ii) Let $\mathcal{G}(\mathcal{K})$ denote the intersection graph of ${\mathcal{K}}$. Using the above data structure, breadth-first or depth-first search on $\mathcal{G}(\mathcal{K})$ can be performed in $O^*(n^{3/2})$ expected time. Combining this result with the so-called shrink-and-bifurcate technique, the reverse-shortest-path problem in a suitably defined proximity graph of ${\mathcal{K}}$ can be solved in $O^*(n^{62/39})$ expected time. Dijkstra's shortest-path algorithm, as well as Prim's MST algorithm, on a $\|\cdot\|_K$-proximity graph on $n$ points in ${\mathbb R}^3$, with edges weighted by $\|\cdot\|_K$, can also be performed in $O^*(n^{3/2})$ time.

cs.CG↗

arXiv2025

Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems

Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}^3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query object $γ$, which is also a plate, we can quickly answer various intersection queries, such as detecting whether $γ$ intersects any plate of $\mathcal{T}$, reporting all the plates intersected by $γ$, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree parametrized algebraic arcs in $\mathbb{R}^3$ (arcs, for short), or (ii) the input objects are arcs and the query objects are plates in $\mathbb{R}^3$. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case. By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we present many different data structures for intersection queries, which also provide trade-offs between their size and query time. For example, if $\mathcal{T}$ is a set of plates and the query objects are algebraic arcs, we obtain a data structure that uses $O^*(n^{4/3})$ storage (where the $O^*(\cdot)$ notation hides factors of the form $n^ε$, for an arbitrarily small $ε>0$) and answers an arc-intersection query in $O^*(n^{2/3})$ time. This result is significant since the exponents do not depend on the specific shape of the input and query objects. We generalize and slightly improve this result: for a parameter $s\in [n^{4/3}, n^{t_q}]$, where ${t_q}\ge 3$ is the number of real parameters needed to specify a query arc, the query time can be decreased to $O^*((n/s^{1/{t_q}})^{\tfrac{2/3}{1-1/{t_q}}})$ by increasing the storage to $O^*(s)$.

cs.CG↗

arXiv2024

Covering points by hyperplanes and related problems

For a set $P$ of $n$ points in $\mathbb R^d$, for any $d\ge 2$, a hyperplane $h$ is called $k$-rich with respect to $P$ if it contains at least $k$ points of $P$. Answering and generalizing a question asked by Peyman Afshani, we show that if the number of $k$-rich hyperplanes in $\mathbb R^d$, $d \geq 3$, is at least $Ω(n^d/k^α+ n/k)$, with a sufficiently large constant of proportionality and with $d\le α< 2d-1$, then there exists a $(d-2)$-flat that contains $Ω(k^{(2d-1-α)/(d-1)})$ points of $P$. We also present upper bound constructions that give instances in which the above lower bound is tight. An extension of our analysis yields similar lower bounds for $k$-rich spheres or $k$-rich flats.

math.CO↗

arXiv2024

Semi-Algebraic Off-line Range Searching and Biclique Partitions in the Plane

Let $P$ be a set of $m$ points in ${\mathbb R}^2$, let $Σ$ be a set of $n$ semi-algebraic sets of constant complexity in ${\mathbb R}^2$, let $(S,+)$ be a semigroup, and let $w: P \rightarrow S$ be a weight function on the points of $P$. We describe a randomized algorithm for computing $w(P\capσ)$ for every $σ\inΣ$ in overall expected time $O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$, where $s>0$ is a constant that bounds the maximum complexity of the regions of $Σ$, and where the $O^*(\cdot)$ notation hides subpolynomial factors. For $s\ge 3$, surprisingly, this bound is smaller than the best-known bound for answering $m$ such queries in an on-line manner. The latter takes $O^*(m^{\frac{s}{2s-1}}n^{\frac{2s-2}{2s-1}}+m+n)$ time. Let $Φ: Σ\times P \rightarrow \{0,1\}$ be the Boolean predicate (of constant complexity) such that $Φ(σ,p) = 1$ if $p\inσ$ and $0$ otherwise, and let $Σ\mathopΦ P = \{ (σ,p) \in Σ\times P \mid Φ(σ,p)=1\}$. Our algorithm actually computes a partition ${\mathcal B}_Φ$ of $Σ\mathopΦ P$ into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) $O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$. It is straightforward to compute $w(P\capσ)$ for all $σ\in Σ$ from ${\mathcal B}_Φ$. Similarly, if $η: Σ\rightarrow S$ is a weight function on the regions of $Σ$, $\sum_{σ\in Σ: p \in σ} η(σ)$, for every point $p\in P$, can be computed from ${\mathcal B}_Φ$ in a straightforward manner. A recent work of Chan et al. solves the online version of this dual point enclosure problem within the same performance bound as our off-line solution. We also mention a few other applications of computing ${\mathcal B}_Φ$.

cs.CG↗

arXiv2023

Vertical Decomposition in 3D and 4D with Applications to Line Nearest-Neighbor Searching in 3D

Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in $d$-space into constant-complexity subcells. In this paper, we settle in the affirmative a few long-standing open problems involving the vertical decomposition of substructures of arrangements for $d=3,4$: (i) Let $\mathcal{S}$ be a collection of $n$ semi-algebraic sets of constant complexity in 3D, and let $U(m)$ be an upper bound on the complexity of the union $\mathcal{U}(\mathcal{S}')$ of any subset $\mathcal{S}'\subseteq \mathcal{S}$ of size at most $m$. We prove that the complexity of the vertical decomposition of the complement of $\mathcal{U}(\mathcal{S})$ is $O^*(n^2+U(n))$ (where the $O^*(\cdot)$ notation hides subpolynomial factors). We also show that the complexity of the vertical decomposition of the entire arrangement $\mathcal{A}(\mathcal{S})$ is $O^*(n^2+X)$, where $X$ is the number of vertices in $\mathcal{A}(\mathcal{S})$. (ii) Let $\mathcal{F}$ be a collection of $n$ trivariate functions whose graphs are semi-algebraic sets of constant complexity. We show that the complexity of the vertical decomposition of the portion of the arrangement $\mathcal{A}(\mathcal{F})$ in 4D lying below the lower envelope of $\mathcal{F}$ is $O^*(n^3)$. These results lead to efficient algorithms for a variety of problems involving these decompositions, including algorithms for constructing the decompositions themselves, and for constructing $(1/r)$-cuttings of substructures of arrangements of the kinds considered above. One additional algorithm of interest is for output-sensitive point enclosure queries amid semi-algebraic sets in three or four dimensions. In addition, as a main domain of applications, we study various proximity problems involving points and lines in 3D.

cs.CG↗

arXiv2023

Near-Optimal Min-Sum Motion Planning for Two Square Robots in a Polygonal Environment

Let $\mathcal{W} \subset \mathbb{R}^2$ be a planar polygonal environment (i.e., a polygon potentially with holes) with a total of $n$ vertices, and let $A,B$ be two robots, each modeled as an axis-aligned unit square, that can translate inside $\mathcal{W}$. Given source and target placements $s_A,t_A,s_B,t_B \in \mathcal{W}$ of $A$ and $B$, respectively, the goal is to compute a \emph{collision-free motion plan} $\mathbfπ^*$, i.e., a motion plan that continuously moves $A$ from $s_A$ to $t_A$ and $B$ from $s_B$ to $t_B$ so that $A$ and $B$ remain inside $\mathcal{W}$ and do not collide with each other during the motion. Furthermore, if such a plan exists, then we wish to return a plan that minimizes the sum of the lengths of the paths traversed by the robots, $\left|\mathbfπ^*\right|$. Given $\mathcal{W}, s_A,t_A,s_B,t_B$ and a parameter $\varepsilon > 0$, we present an $n^2\varepsilon^{-O(1)} \log n$-time $(1+\varepsilon)$-approximation algorithm for this problem. We are not aware of any polynomial time algorithm for this problem, nor do we know whether the problem is NP-Hard. Our result is the first polynomial-time $(1+\varepsilon)$-approximation algorithm for an optimal motion planning problem involving two robots moving in a polygonal environment.

cs.RO↗

arXiv2023

The Unweighted and Weighted Reverse Shortest Path Problem for Disk Graphs

We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].

cs.CG↗

arXiv2022

Improved Algebraic Degeneracy Testing

In the classical linear degeneracy testing problem, we are given $n$ real numbers and a $k$-variate linear polynomial $F$, for some constant $k$, and have to determine whether there exist $k$ numbers $a_1,\ldots,a_k$ from the set such that $F(a_1,\ldots,a_k) = 0$. We consider a generalization of this problem in which $F$ is an arbitrary constant-degree polynomial, we are given $k$ sets of $n$ numbers, and have to determine whether there exist a $k$-tuple of numbers, one in each set, on which $F$ vanishes. We give the first improvement over the naïve $O^*(n^{k-1})$ algorithm for this problem (where the $O^*(\cdot)$ notation omits subpolynomial factors). We show that the problem can be solved in time $O^*\left( n^{k - 2 + \frac 4{k+2}}\right)$ for even $k$ and in time $O^*\left( n^{k - 2 + \frac{4k-8}{k^2-5}}\right)$ for odd $k$ in the real RAM model of computation. We also prove that for $k=4$, the problem can be solved in time $O^*(n^{2.625})$ in the algebraic decision tree model, and for $k=5$ it can be solved in time $O^*(n^{3.56})$ in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for $k$-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft's point-line incidence detection problem in any dimension.

cs.CG↗

arXiv2022

Intersection Searching amid Tetrahedra in Four Dimensions

We develop data structures for intersection queries in four dimensions that involve segments, triangles and tetrahedra. Specifically, we study three main problems: (i) Preprocess a set of $n$ tetrahedra in $\reals^4$ into a data structure for answering segment-intersection queries amid the given tetrahedra (referred to as \emph{segment-tetrahedron intersection queries}). (ii) Preprocess a set of $n$ triangles in $\reals^4$ into a data structure that supports triangle-intersection queries amid the input triangles (referred to as \emph{triangle-triangle intersection queries}). (iii) Preprocess a set of $n$ segments in $\reals^4$ into a data structure that supports tetrahedron-intersection queries amid the input segments (referred to as \emph{tetrahedron-segment intersection queries}). In each problem we want either to detect an intersection, or to count or report all intersections. As far as we can tell, these problems have not been previously studied. For problem (i), we first present a "standard" solution which, for any prespecified value $n \le s \le n^6$ of a so-called storage parameter $s$, yields a data structure with $O^*(s)$ storage and expected preprocessing, which answers an intersection query in $O^*(n/s^{1/6})$ time (here and in what follows, the $O^*(\cdot)$ notation hides subpolynomial factors). For problems (ii) and (iii), using similar arguments, we present a solution that has the same asymptotic performance bounds. We then improve the solution for problem (i), and present a more intricate data structure that uses $O^*(n^{2})$ storage and expected preprocessing, and answers a segment-tetrahedron intersection query in $O^*(n^{1/2})$ time.

cs.CG↗

arXiv2022

Efficient algorithms for optimization problems involving semi-algebraic range searching

We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of semi-algebraic geometric objects of constant complexity. The common feature of these problems is that they involve a `growth parameter' $r$ and a semi-algebraic predicate $Π(o,o';r)$ of constant complexity on pairs of input objects, which depends on $r$ and is monotone in $r$. One then defines a graph $G(r)$ whose edges are all the pairs $(o,o')$ for which $Π(o,o';r)$ is true, and seeks the smallest value of $r$ for which some monotone property holds for $G(r)$. Problems that fit into this context include (i) the reverse shortest path problem in unit-disk graphs, recently studied by Wang and Zhao, (ii) the same problem for weighted unit-disk graphs, with a decision procedure recently provided by Wang and Xue, (iii) extensions of these problems to three and higher dimensions, (iv) the discrete Fréchet distance with one-sided shortcuts in higher dimensions, extending the study by Ben Avraham et al., (v) perfect matchings in intersection graphs: given, e.g., a set of fat ellipses of roughly the same size, find the smallest value $r$ such that if we expand each of the ellipses by $r$, the resulting intersection graph contains a perfect matching, (vi) generalized distance selection problems: given, e.g., a set of disjoint segments, find the $k$'th smallest distance among the pairwise distances determined by the segments, for a given (sufficiently small but superlinear) parameter $k$, and (vii) the maximum-height independent towers problem, in which we want to erect vertical towers of maximum height over a 1.5-dimensional terrain so that no pair of tower tips are mutually visible. We obtain significantly improved solutions for problems (i), (ii) and (vi), and new efficient solutions to the other problems.

cs.CG↗

arXiv2022

Bottleneck Matching in the Plane

We present an algorithm for computing a bottleneck matching in a set of $n=2\ell$ points in the plane, which runs in $O(n^{ω/2}\log n)$ deterministic time, where $ω\approx 2.37$ is the exponent of matrix multiplication.

cs.CG↗

arXiv2022

On rich points and incidences with restricted sets of lines in 3-space

Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plücker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show: \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+ε}/r^2)$, for $r \ge 3$ and for any $ε>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+ε})$ $2$-rich points. For $r$ larger than some sufficiently large constant, the number of $r$-rich points is also $O(n/r)$. As an application, we deduce (with an $ε$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances determined by $n$ points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. \medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$. \medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O\left(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n \right)$, provided that no plane contains more than $s$ of the points. When $s = O(\min\{n^{3/5}/m^{2/5}, m^{1/2}\})$, the bound becomes $O(m^{3/5}n^{3/5}+m+n)$. As an application, we prove that the number of incidences between $m$ points and $n$ lines in $R^4$ contained in a quadratic hypersurface (which does not contain a hyperplane) is $O(m^{3/5}n^{3/5} + m + n)$. The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.

math.CO↗

arXiv2021

Subquadratic Algorithms for Some \textsc{3Sum}-Hard Geometric Problems in the Algebraic Decision Tree Model

We present subquadratic algorithms in the algebraic decision-tree model for several \textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the plane, we want to count, for each triangle $Δ\in C$, the number of intersection points between the segments of $A$ and those of $B$ that lie in $Δ$. The problems considered in this paper have been studied by Chan~(2020), who gave algorithms that solve them, in the standard real-RAM model, in $O((n^2/\log^2n)\log^{O(1)}\log n)$ time. We present solutions in the algebraic decision-tree model whose cost is $O(n^{60/31+\varepsilon})$, for any $\varepsilon>0$. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl~(2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the \emph{order type} of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.

cs.CG↗

arXiv2021

Stabbing Pairwise Intersecting Disks by Five Points

Suppose we are given a set $\mathcal{D}$ of $n$ pairwise intersecting disks in the plane. A planar point set $P$ stabs $\mathcal{D}$ if and only if each disk in $\mathcal{D}$ contains at least one point from $P$. We present a deterministic algorithm that takes $O(n)$ time to find five points that stab $\mathcal{D}$. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. Moreover, we present a simple argument showing that eight disks can be stabbed by at most three points. This provides a simple-albeit slightly weaker-algorithmic version of a classical result by Danzer that such a set $\mathcal{D}$ can always be stabbed by four points.

cs.CG↗

arXiv2021

Throwing a Sofa Through the Window

We study several variants of the problem of moving a convex polytope $K$, with $n$ edges, in three dimensions through a flat rectangular (and sometimes more general) window. Specifically: $\bullet$ We study variants where the motion is restricted to translations only, discuss situations where such a motion can be reduced to sliding (translation in a fixed direction), and present efficient algorithms for those variants, which run in time close to $O(n^{8/3})$. $\bullet$ We consider the case of a `gate' (an unbounded window with two parallel infinite edges), and show that $K$ can pass through such a window, by any collision-free rigid motion, if and only if it can slide through it. $\bullet$ We consider arbitrary compact convex windows, and show that if $K$ can pass through such a window $W$ (by any motion) then $K$ can slide through a gate of width equal to the diameter of $W$. $\bullet$ We study the case of a circular window $W$, and show that, for the regular tetrahedron $K$ of edge length $1$, there are two thresholds $1 > δ_1\approx 0.901388 > δ_2\approx 0.895611$, such that (a) $K$ can slide through $W$ if the diameter $d$ of $W$ is $\ge 1$, (b) $K$ cannot slide through $W$ but can pass through it by a purely translational motion when $δ_1\le d < 1$, (c) $K$ cannot pass through $W$ by a purely translational motion but can do it when rotations are allowed when $δ_2 \le d < δ_1$, and (d) $K$ cannot pass through $W$ at all when $d < δ_2$. $\bullet$ Finally, we explore the general setup, where we want to plan a general motion (with all six degrees of freedom) for $K$ through a rectangular window $W$, and present an efficient algorithm for this problem, with running time close to $O(n^4)$.

cs.CG↗