arXiv subjects

Jakob Bæk Tejs Knudsen

Publications and source records attributed to Jakob Bæk Tejs Knudsen.

4 recordsLinked to original sources

arXiv2021

On Sums of Monotone Random Integer Variables

We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,π]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random variables. In this note, we provide estimates for the probability that the sum of independent monotone integer variables attains precisely a specific value. We do not assume that the variables are identically distributed. Our estimates are sharp when the specific value is close to the mean, but they are not useful further out in the tail. By combining with the trick of \emph{exponential tilting}, we obtain sharp estimates for the point probabilities in the tail under a slightly stronger assumption on the random integer variables which we call strong monotonicity.

math.PR↗

arXiv2021

Load Balancing with Dynamic Set of Balls and Bins

In dynamic load balancing, we wish to distribute balls into bins in an environment where both balls and bins can be added and removed. We want to minimize the maximum load of any bin but we also want to minimize the number of balls and bins affected when adding or removing a ball or a bin. We want a hashing-style solution where we given the ID of a ball can find its bin efficiently. We are given a balancing parameter $c=1+ε$, where $ε\in (0,1)$. With $n$ and $m$ the current numbers of balls and bins, we want no bin with load above $C=\lceil c n/m\rceil$, referred to as the capacity of the bins. We present a scheme where we can locate a ball checking $1+O(\log 1/ε)$ bins in expectation. When inserting or deleting a ball, we expect to move $O(1/ε)$ balls, and when inserting or deleting a bin, we expect to move $O(C/ε)$ balls. Previous bounds were off by a factor $1/ε$. These bounds are best possible when $C=O(1)$ but for larger $C$, we can do much better: Let $f=εC$ if $C\leq \log 1/ε$, $f=ε\sqrt{C}\cdot \sqrt{\log(1/(ε\sqrt{C}))}$ if $\log 1/ε\leq C<\tfrac{1}{2ε^2}$, and $C=1$ if $C\geq \tfrac{1}{2ε^2}$. We show that we expect to move $O(1/f)$ balls when inserting or deleting a ball, and $O(C/f)$ balls when inserting or deleting a bin. For the bounds with larger $C$, we first have to resolve a much simpler probabilistic problem. Place $n$ balls in $m$ bins of capacity $C$, one ball at the time. Each ball picks a uniformly random non-full bin. We show that in expectation and with high probability, the fraction of non-full bins is $Θ(f)$. Then the expected number of bins that a new ball would have to visit to find one that is not full is $Θ(1/f)$. As it turns out, we obtain the same complexity in our more complicated scheme where both balls and bins can be added and removed.

cs.DS↗

arXiv2020

Subsets and Supermajorities: Optimal Hashing-based Set Similarity Search

We formulate and optimally solve a new generalized Set Similarity Search problem, which assumes the size of the database and query sets are known in advance. By creating polylog copies of our data-structure, we optimally solve any symmetric Approximate Set Similarity Search problem, including approximate versions of Subset Search, Maximum Inner Product Search (MIPS), Jaccard Similarity Search and Partial Match. Our algorithm can be seen as a natural generalization of previous work on Set as well as Euclidean Similarity Search, but conceptually it differs by optimally exploiting the information present in the sets as well as their complements, and doing so asymmetrically between queries and stored sets. Doing so we improve upon the best previous work: MinHash [J. Discrete Algorithms 1998], SimHash [STOC 2002], Spherical LSF [SODA 2016, 2017] and Chosen Path [STOC 2017] by as much as a factor $n^{0.14}$ in both time and space; or in the near-constant time regime, in space, by an arbitrarily large polynomial factor. Turning the geometric concept, based on Boolean supermajority functions, into a practical algorithm requires ideas from branching random walks on $\mathbb Z^2$, for which we give the first non-asymptotic near tight analysis. Our lower bounds follow from new hypercontractive arguments, which can be seen as characterizing the exact family of similarity search problems for which supermajorities are optimal. The optimality holds for among all hashing based data structures in the random setting, and by reductions, for 1 cell and 2 cell probe data structures. As a side effect, we obtain new hypercontractive bounds on the directed noise operator $T^{p_1 \to p_2}_ρ$.

cs.DS↗

arXiv2019

Classifying Convex Bodies by their Contact and Intersection Graphs

Suppose that $A$ is a convex body in the plane and that $A_1,\dots,A_n$ are translates of $A$. Such translates give rise to an intersection graph of $A$, $G=(V,E)$, with vertices $V=\{1,\dots,n\}$ and edges $E=\{uv\mid A_u\cap A_v\neq \emptyset\}$. The subgraph $G'=(V, E')$ satisfying that $E'\subset E$ is the set of edges $uv$ for which the interiors of $A_u$ and $A_v$ are disjoint is a unit distance graph of $A$. If furthermore $G'=G$, i.e., if the interiors of $A_u$ and $A_v$ are disjoint whenever $u\neq v$, then $G$ is a contact graph of $A$. In this paper we study which pairs of convex bodies have the same contact, unit distance, or intersection graphs. We say that two convex bodies $A$ and $B$ are equivalent if there exists a linear transformation $B'$ of $B$ such that for any slope, the longest line segments with that slope contained in $A$ and $B'$, respectively, are equally long. For a broad class of convex bodies, including all strictly convex bodies and linear transformations of regular polygons, we show that the contact graphs of $A$ and $B$ are the same if and only if $A$ and $B$ are equivalent. We prove the same statement for unit distance and intersection graphs.

cs.CG↗