arXiv subjects

Gabriel Nivasch

Publications and source records attributed to Gabriel Nivasch.

At least 19 recordsLinked to original sources

arXiv2025

On the complexity of the free space of a translating box in R^3

Consider a convex polyhedral robot $B$ that can translate (without rotating) amidst a finite set of non-moving polyhedral obstacles in $\mathbb R^3$. The "free space" $\mathcal F$ of $B$ is the set of all positions in which $B$ is disjoint from the interior of every obstacle. Aronov and Sharir (1997) derived an upper bound of $O(n^2\log n)$ for the combinatorial complexity of $\mathcal F$, where $n$ is the total number of vertices of the obstacles, and the complexity of $B$ is assumed constant. Halperin and Yap (1993) showed that, if $B$ is either a box or a "flat" convex polygon, then a tighter bound of $O(n^2\alpha(n))$ holds. Here $\alpha(n)$ is the inverse Ackermann function. In this paper we prove that if $B$ is a box, then the complexity of $\mathcal F$ is $O(n^2)$. Furthermore, if $B$ is a convex polygon whose edges come in parallel pairs, then the complexity of $\mathcal F$ is $O(n^2)$ as well. These results settle the question of the asymptotical worst-case complexity of $\mathcal F$ for a box, as well as for all convex polygons.

cs.CG↗

arXiv2025

A convergence technique for the game i-Mark

The game of i-Mark is an impartial combinatorial game introduced by Sopena (2016). The game is parametrized by two sets of positive integers $S$, $D$, where $\min D\ge 2$. From position $n\ge 0$ one can move to any position $n-s$, $s\in S$, as long as $n-s\ge 0$, as well as to any position $n/d$, $d\in D$, as long as $n>0$ and $d$ divides $n$. The game ends when no more moves are possible, and the last player to move is the winner. Sopena, and subsequently Friman and Nivasch (2021), characterized the Sprague-Grundy sequences of many cases of i-Mark$(S,D)$ with $|D|=1$. Friman and Nivasch also obtained some partial results for the case i-Mark$(\{1\},\{2,3\})$. In this paper we present a convergence technique that gives polynomial-time algorithms for the Sprague-Grundy sequence of many instances of i-Mark with $|D|>1$. In particular, we prove our technique works for all games i-Mark$(\{1\},\{d_1,d_2\})$. Keywords: Combinatorial game, impartial game, Sprague-Grundy function, convergence, dynamic programming.

math.CO↗

arXiv2023

Ordinals and recursively defined functions on the reals

We determine sufficient conditions under which certain recursively defined functions are well defined for all real inputs. Given a function $f:\mathbb R\to\mathbb R$, call a decreasing sequence $x_1>x_2>x_3>\cdots$ "$f$-bad" if $f(x_1)>f(x_2)>f(x_3)>\cdots$, and call the function $f$ "ordinal decreasing" if there exist no infinite $f$-bad sequences. We prove the following result: Given ordinal decreasing functions $f,g_1,\ldots,g_k,s$ that are everywhere larger than $0$, define the recursive algorithm "$M(x)$: if $x<0$ return $f(x)$, else return $g_1(-M(x-g_2(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$". Then $M(x)$ halts and is ordinal decreasing for all $x \in \mathbb{R}$. The recursive algorithms $M$ and $M_n$ previously studied in the context of fusible numbers by Ericskon et al. (2022) and Bufetov et al. (2024), respectively, are special cases of this scheme. Moreover, given an ordinal decreasing function $f$, denote by $o(f)$ the ordinal height of the root of the tree of $f$-bad sequences. Then we prove that, for $k\ge 2$, the function $M(x)$ defined by the above algorithm satisfies $o(M)\le\varphi_{k-1}(\gamma+o(s)+1)$, where $\gamma$ is the smallest ordinal such that $\max\{o(s),o(f),o(g_1), \ldots, o(g_k)\} <\varphi_{k-1}(\gamma)$.

math.LO↗

arXiv2022

Generalized fusible numbers and their ordinals

Erickson defined the fusible numbers as a set $\mathcal F$ of reals generated by repeated application of the function $\frac{x+y+1}{2}$. Erickson, Nivasch, and Xu showed that $\mathcal F$ is well ordered, with order type $\varepsilon_0$. They also investigated a recursively defined function $M\colon \mathbb{R}\to\mathbb{R}$. They showed that the set of points of discontinuity of $M$ is a subset of $\mathcal F$ of order type $\varepsilon_0$. They also showed that, although $M$ is a total function on $\mathbb R$, the fact that the restriction of $M$ to $\mathbb{Q}$ is total is not provable in first-order Peano arithmetic $\mathsf{PA}$. In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $\mathcal F$ of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function $g:\mathbb R^n\to\mathbb R$. The most straightforward generalization of $\frac{x+y+1}{2}$ to an $n$-ary function is the function $\frac{x_1+\cdots+x_n+1}{n}$. We show that this function generates a set $\mathcal F_n$ whose order type is just $\varphi_{n-1}(0)$. For this, we develop recursively defined functions $M_n\colon \mathbb{R}\to\mathbb{R}$ naturally generalizing the function $M$. Furthermore, we prove that for any linear function $g:\mathbb R^n\to\mathbb R$, the order type of the resulting $\mathcal F$ is at most $\varphi_{n-1}(0)$. Finally, we show that there do exist continuous functions $g:\mathbb R^n\to\mathbb R$ for which the order types of the resulting sets $\mathcal F$ approach the small Veblen ordinal.

math.CO↗

arXiv2021

Learning convex polyhedra with margin

We present an improved algorithm for {\em quasi-properly} learning convex polyhedra in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polyhedron as an intersection of about $t \log t$ halfspaces with constant-size margins in time polynomial in $t$ (where $t$ is the number of halfspaces forming an optimal polyhedron). We also identify distinct generalizations of the notion of margin from hyperplanes to polyhedra and investigate how they relate geometrically; this result may have ramifications beyond the learning setting.

cs.LG↗

arXiv2021

Some i-Mark games

Let $S$ be a set of positive integers, and let $D$ be a set of integers larger than $1$. The game $i$-Mark$(S,D)$ is an impartial combinatorial game introduced by Sopena (2016), which is played with a single pile of tokens. In each turn, a player can subtract $s \in S$ from the pile, or divide the size of the pile by $d \in D$, if the pile size is divisible by $d$. Sopena partially analyzed the games with $S=[1, t-1]$ and $D=\{d\}$ for $d \not\equiv 1 \pmod t$, but left the case $d \equiv 1 \pmod t$ open. We solve this problem by calculating the Sprague-Grundy function of $i$-Mark$([1,t-1],\{d\})$ for $d \equiv 1 \pmod t$, for all $t,d \geq 2$. We also calculate the Sprague-Grundy function of $i$-Mark$(\{2\},\{2k + 1\})$ for all $k$, and show that it exhibits similar behavior. Finally, following Sopena's suggestion to look at games with $|D|>1$, we derive some partial results for the game $i$-Mark$(\{1\}, \{2, 3\})$, whose Sprague-Grundy function seems to behave erratically and does not show any clean pattern. We prove that each value $0,1,2$ occurs infinitely often in its SG sequence, with a maximum gap length between consecutive appearances.

math.CO↗

arXiv2021

Upper bounds for stabbing simplices by a line

It is known that for every dimension $d\ge 2$ and every $k 0$ such that for every $n$-point set $X\subset \mathbb R^d$ there exists a $k$-flat that intersects at least $c_{d,k} n^{d+1-k} - o(n^{d+1-k})$ of the $(d-k)$-dimensional simplices spanned by $X$. However, the optimal values of the constants $c_{d,k}$ are mostly unknown. The case $k=0$ (stabbing by a point) has received a great deal of attention. In this paper we focus on the case $k=1$ (stabbing by a line). Specifically, we try to determine the upper bounds yielded by two point sets, known as the "stretched grid" and the "stretched diagonal". Even though the calculations are independent of $n$, they are still very complicated, so we resort to analytical and numerical software methods. We provide strong evidence that, surprisingly, for $d=4,5,6$ the stretched grid yields better bounds than the stretched diagonal (unlike for all cases $k=0$ and for the case $(d,k)=(3,1)$, in which both point sets yield the same bound). Our experiments indicate that the stretched grid yields $c_{4,1}\leq 0.00457936$, $c_{5,1}\leq 0.000405335$, and $c_{6,1}\leq 0.0000291323$.

cs.CG↗

arXiv2020

Fusible numbers and Peano Arithmetic

Inspired by a mathematical riddle involving fuses, we define the "fusible numbers" as follows: $0$ is fusible, and whenever $x,y$ are fusible with $|y-x|<1$, the number $(x+y+1)/2$ is also fusible. We prove that the set of fusible numbers, ordered by the usual order on $\mathbb R$, is well-ordered, with order type $\varepsilon_0$. Furthermore, we prove that the density of the fusible numbers along the real line grows at an incredibly fast rate: Letting $g(n)$ be the largest gap between consecutive fusible numbers in the interval $[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant $c$, where $F_\alpha$ denotes the fast-growing hierarchy. Finally, we derive some true statements that can be formulated but not proven in Peano Arithmetic, of a different flavor than previously known such statements: PA cannot prove the true statement "For every natural number $n$ there exists a smallest fusible number larger than $n$." Also, consider the algorithm "$M(x)$: if $x<0$ return $-x$, else return $M(x-M(x-1))/2$." Then $M$ terminates on real inputs, although PA cannot prove the statement "$M$ terminates on all natural inputs."

cs.LO↗

arXiv2020

Nested Barycentric Coordinate System as an Explicit Feature Map

We propose a new embedding method which is particularly well-suited for settings where the sample size greatly exceeds the ambient dimension. Our technique consists of partitioning the space into simplices and then embedding the data points into features corresponding to the simplices' barycentric coordinates. We then train a linear classifier in the rich feature space obtained from the simplices. The decision boundary may be highly non-linear, though it is linear within each simplex (and hence piecewise-linear overall). Further, our method can approximate any convex body. We give generalization bounds based on empirical margin and a novel hybrid sample compression technique. An extensive empirical evaluation shows that our method consistently outperforms a range of popular kernel embedding methods.

cs.LG↗

arXiv2019

Homotopic curve shortening and the affine curve-shortening flow

We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call "homotopic curve shortening" (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $P\subset \mathbb R^2$ of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and $P$ is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between ACSF and HCS generalizes the link between ACSF and convex-layer decomposition (Eppstein et al., 2017; Calder and Smart, 2020), which is restricted to convex curves. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.

cs.CG↗

arXiv2018

Grid peeling and the affine curve-shortening flow

In this paper we study an experimentally-observed connection between two seemingly unrelated processes, one from computational geometry and the other from differential geometry. The first one (which we call "grid peeling") is the convex-layer decomposition of subsets $G\subset \mathbb Z^2$ of the integer grid, previously studied for the particular case $G=\{1,\ldots,m\}^2$ by Har-Peled and Lidický (2013). The second one is the affine curve-shortening flow (ACSF), first studied by Alvarez et al. (1993) and Sapiro and Tannenbaum (1993). We present empirical evidence that, in a certain well-defined sense, grid peeling behaves at the limit like ACSF on convex curves. We offer some theoretical arguments in favor of this conjecture. We also pay closer attention to the simple case where $G=\mathbb N^2$ is a quarter-infinite grid. This case corresponds to ACSF starting with an infinite L-shaped curve, which when transformed using the ACSF becomes a hyperbola for all times $t>0$. We prove that, in the grid peeling of $\mathbb N^2$, (1) the number of grid points removed up to iteration $n$ is $Θ(n^{3/2}\log n)$; and (2) the boundary at iteration $n$ is sandwiched between two hyperbolas that are separated from each other by a constant factor.

cs.CG↗

arXiv2017

Classifying unavoidable Tverberg partitions

Let $T(d,r) = (r-1)(d+1)+1$ be the parameter in Tverberg's theorem, and call a partition $\mathcal I$ of $\{1,2,\ldots,T(d,r)\}$ into $r$ parts a "Tverberg type". We say that $\mathcal I$ "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition. We say that $\mathcal I$ is "unavoidable" if it occurs in every sufficiently long point sequence. In this paper we study the problem of determining which Tverberg types are unavoidable. We conjecture a complete characterization of the unavoidable Tverberg types, and we prove some cases of our conjecture for $d\le 4$. Along the way, we study the avoidability of many other geometric predicates. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. This lends further support for Sierksma's conjecture on the number of Tverberg partitions.

cs.CG↗

arXiv2017

On the zone of a circle in an arrangement of lines

Let $\mathcal L$ be a set of $n$ lines in the plane, and let $C$ be a convex curve in the plane, like a circle or a parabola. The "zone" of $C$ in $\mathcal L$, denoted $\mathcal Z(C,\mathcal L)$, is defined as the set of all cells in the arrangement $\mathcal A(\mathcal L)$ that are intersected by $C$. Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of $\mathcal Z(C,\mathcal L)$ is at most $O(nα(n))$, where $α$ is the inverse Ackermann function. They did this by translating the sequence of edges of $\mathcal Z(C,\mathcal L)$ into a sequence $S$ that avoids the subsequence $ababa$. Whether the worst-case complexity of $\mathcal Z(C,\mathcal L)$ is only linear is a longstanding open problem. Since the relaxation of the problem to pseudolines does have a $Θ(nα(n))$ bound, any proof of $O(n)$ for the case of straight lines must necessarily use geometric arguments. In this paper we present some such geometric arguments. We show that, if $C$ is a circle, then certain configurations of straight-line segments with endpoints on $C$ are impossible. In particular, we show that there exists a Hart-Sharir sequence that cannot appear as a subsequence of $S$. The Hart-Sharir sequences are essentially the only known way to construct $ababa$-free sequences of superlinear length. Hence, if it could be shown that every family of $ababa$-free sequences of superlinear-length eventually contains all Hart-Sharir sequences, it would follow that the complexity of $\mathcal Z(C,\mathcal L)$ is $O(n)$ whenever $C$ is a circle.

cs.CG↗

arXiv2016

One-sided epsilon-approximants

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show that, in contrast with the usual (two-sided) weak $\varepsilon$-approximants, for every set $P\subset \mathbb{R}^d$ there exists a one-sided weak $\varepsilon$-approximant of size bounded by a function of $\varepsilon$ and $d$.

math.CO↗

arXiv2016

Rainbow matchings and algebras of sets

Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $\mathfrak v = \mathfrak v(n)$ such that, if $A_1, \ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$, such that for each $i$ there are at least $\mathfrak v$ elements of $X$ that belong to $A_i$-equivalence classes of size larger than $1$, then $X$ has a rainbow matching---a set of $2n$ distinct elements $a_1, b_1, \ldots, a_n, b_n$, such that $a_i$ is $A_i$-equivalent to $b_i$ for each $i$? Grinblat has shown that $\mathfrak v(n) \le 10n/3 + O(\sqrt{n})$. He asks whether $\mathfrak v(n) = 3n-2$ for all $n\ge 4$. In this paper we improve the upper bound (for all large enough $n$) to $\mathfrak v(n) \le 16n/5 + O(1)$.

math.CO↗

arXiv2014

A variant of the Hadwiger-Debrunner (p,q)-problem in the plane

Let $X$ be a convex curve in the plane (say, the unit circle), and let $\mathcal S$ be a family of planar convex bodies, such that every two of them meet at a point of $X$. Then $\mathcal S$ has a transversal $N\subset\mathbb R^2$ of size at most $1.75\cdot 10^9$. Suppose instead that $\mathcal S$ only satisfies the following "$(p,2)$-condition": Among every $p$ elements of $\mathcal S$ there are two that meet at a common point of $X$. Then $\mathcal S$ has a transversal of size $O(p^8)$. For comparison, the best known bound for the Hadwiger--Debrunner $(p, q)$-problem in the plane, with $q=3$, is $O(p^6)$. Our result generalizes appropriately for $\mathbb R^d$ if $X\subset \mathbb R^d$ is, for example, the moment curve.

cs.CG↗

arXiv2013

The visible perimeter of an arrangement of disks

Given a collection of n opaque unit disks in the plane, we want to find a stacking order for them that maximizes their visible perimeter---the total length of all pieces of their boundaries visible from above. We prove that if the centers of the disks form a dense point set, i.e., the ratio of their maximum to their minimum distance is O(n^1/2), then there is a stacking order for which the visible perimeter is Omega(n^2/3). We also show that this bound cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform grid. On the other hand, if the set of centers is dense and the maximum distance between them is small, then the visible perimeter is O(n^3/4) with respect to any stacking order. This latter bound cannot be improved either. Finally, we address the case where no more than c disks can have a point in common. These results partially answer some questions of Cabello, Haverkort, van Kreveld, and Speckmann.

cs.CG↗

arXiv2013

The number of distinct distances from a vertex of a convex polygon

Erdős conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on this result to (13/36 + eps)n - O(1) for eps ~= 1/23000. Our main ingredient is an improved bound on the maximum number of isosceles triangles determined by P.

cs.CG↗