# Sketched MinDist

## TL;DR

This paper introduces a sketch-based distance measure for geometric objects using minimal distance vectors, analyzing the size of point sets needed for accurate approximation across various shapes and scenarios.

## Contribution

It establishes bounds on the size of point sets required to preserve distances for different geometric shapes using sketch vectors, connecting to sensitivity sampling and enabling exact reconstruction.

## Key findings

- For hyperplanes, a sample size of O(d/ε^2) preserves relative error.
- For other shapes, a minimum distance parameter ρ and domain size L are needed.
- Trajectories with k pieces can be approximated and reconstructed with size depending on L, ρ, and k.

## Abstract

We consider sketch vectors of geometric objects $J$ through the \mindist function \[ v_i(J) = \inf_{p \in J} \|p-q_i\| \] for $q_i \in Q$ from a point set $Q$. Collecting the vector of these sketch values induces a simple, effective, and powerful distance: the Euclidean distance between these sketched vectors. This paper shows how large this set $Q$ needs to be under a variety of shapes and scenarios. For hyperplanes we provide direct connection to the sensitivity sample framework, so relative error can be preserved in $d$ dimensions using $Q = O(d/\varepsilon^2)$. However, for other shapes, we show we need to enforce a minimum distance parameter $\rho$, and a domain size $L$. For $d=2$ the sample size $Q$ then can be $\tilde{O}((L/\rho) \cdot 1/\varepsilon^2)$. For objects (e.g., trajectories) with at most $k$ pieces this can provide stronger \emph{for all} approximations with $\tilde{O}((L/\rho)\cdot k^3 / \varepsilon^2)$ points. Moreover, with similar size bounds and restrictions, such trajectories can be reconstructed exactly using only these sketch vectors.