arXiv subjects

Pankaj K. Agarwal

Publications and source records attributed to Pankaj K. Agarwal.

At least 37 records · Page 2Linked to original sources

arXiv2023

Computing Optimal Kernels in Two Dimensions

Let $P$ be a set of $n$ points in $\Re^2$. For a parameter $\varepsilon\in (0,1)$, a subset $C\subseteq P$ is an \emph{$\varepsilon$-kernel} of $P$ if the projection of the convex hull of $C$ approximates that of $P$ within $(1-\varepsilon)$-factor in every direction. The set $C$ is a \emph{weak $\varepsilon$-kernel} of $P$ if its directional width approximates that of $P$ in every direction. Let $\mathsf{k}_{\varepsilon}(P)$ (resp.\ $\mathsf{k}^{\mathsf{w}}_{\varepsilon}(P)$) denote the minimum-size of an $\varepsilon$-kernel (resp. weak $\varepsilon$-kernel) of $P$. We present an $O(n\mathsf{k}_{\varepsilon}(P)\log n)$-time algorithm for computing an $\varepsilon$-kernel of $P$ of size $\mathsf{k}_{\varepsilon}(P)$, and an $O(n^2\log n)$-time algorithm for computing a weak $\varepsilon$-kernel of $P$ of size ${\mathsf{k}}^{\mathsf{w}}_{\varepsilon}(P)$. We also present a fast algorithm for the Hausdorff variant of this problem. In addition, we introduce the notion of \emph{$\varepsilon$-core}, a convex polygon lying inside $\mathsf{ch}(P)$, prove that it is a good approximation of the optimal $\varepsilon$-kernel, present an efficient algorithm for computing it, and use it to compute an $\varepsilon$-kernel of small size.

cs.CG↗

arXiv2022

All Politics is Local: Redistricting via Local Fairness

In this paper, we propose to use the concept of local fairness for auditing and ranking redistricting plans. Given a redistricting plan, a deviating group is a population-balanced contiguous region in which a majority of individuals are of the same interest and in the minority of their respective districts; such a set of individuals have a justified complaint with how the redistricting plan was drawn. A redistricting plan with no deviating groups is called locally fair. We show that the problem of auditing a given plan for local fairness is NP-complete. We present an MCMC approach for auditing as well as ranking redistricting plans. We also present a dynamic programming based algorithm for the auditing problem that we use to demonstrate the efficacy of our MCMC approach. Using these tools, we test local fairness on real-world election data, showing that it is indeed possible to find plans that are almost or exactly locally fair. Further, we show that such plans can be generated while sacrificing very little in terms of compactness and existing fairness measures such as competitiveness of the districts or seat shares of the plans.

cs.GT↗

arXiv2022

Deterministic, Near-Linear $\varepsilon$-Approximation Algorithm for Geometric Bipartite Matching

Given point sets $A$ and $B$ in $\mathbb{R}^d$ where $A$ and $B$ have equal size $n$ for some constant dimension $d$ and a parameter $\varepsilon>0$, we present the first deterministic algorithm that computes, in $n\cdot(\varepsilon^{-1} \log n)^{O(d)}$ time, a perfect matching between $A$ and $B$ whose cost is within a $(1+\varepsilon)$ factor of the optimal under any $\smash{\ell_p}$-norm. Although a Monte-Carlo algorithm with a similar running time is proposed by Raghvendra and Agarwal [J. ACM 2020], the best-known deterministic $\varepsilon$-approximation algorithm takes $Ω(n^{3/2})$ time. Our algorithm constructs a (refinement of a) tree cover of $\mathbb{R}^d$, and we develop several new tools to apply a tree-cover based approach to compute an $\varepsilon$-approximate perfect matching.

cs.DS↗

arXiv2021

Locally Fair Partitioning

We model the societal task of redistricting political districts as a partitioning problem: Given a set of $n$ points in the plane, each belonging to one of two parties, and a parameter $k$, our goal is to compute a partition $Π$ of the plane into regions so that each region contains roughly $σ= n/k$ points. $Π$ should satisfy a notion of ''local'' fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in $Π$ if it belongs to the minority party. A group $D$ of roughly $σ$ contiguous points is called a deviating group with respect to $Π$ if majority of points in $D$ are unhappy in $Π$. The partition $Π$ is locally fair if there is no deviating group with respect to $Π$. This paper focuses on a restricted case when points lie in $1$D. The problem is non-trivial even in this case. We consider both adversarial and ''beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are ''runs'' of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of $σ$, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists.

cs.DS↗

arXiv2021

Dynamic Enumeration of Similarity Joins

This paper considers enumerating answers to similarity-join queries under dynamic updates: Given two sets of $n$ points $A,B$ in $\mathbb{R}^d$, a metric $ϕ(\cdot)$, and a distance threshold $r > 0$, report all pairs of points $(a, b) \in A \times B$ with $ϕ(a,b) \le r$. Our goal is to store $A,B$ into a dynamic data structure that, whenever asked, can enumerate all result pairs with worst-case delay guarantee, i.e., the time between enumerating two consecutive pairs is bounded. Furthermore, the data structure can be efficiently updated when a point is inserted into or deleted from $A$ or $B$. We propose several efficient data structures for answering similarity-join queries in low dimension. For exact enumeration of similarity join, we present near-linear-size data structures for $\ell_1, \ell_\infty$ metrics with $\log^{O(1)} n$ update time and delay. We show that such a data structure is not feasible for the $\ell_2$ metric for $d \ge 4$. For approximate enumeration of similarity join, where the distance threshold is a soft constraint, we obtain a unified linear-size data structure for $\ell_p$ metric, with $\log^{O(1)} n$ delay and update time. In high dimensions, we present an efficient data structure with worst-case delay-guarantee using locality sensitive hashing (LSH).

cs.DS↗

arXiv2021

Efficiently Answering Durability Prediction Queries

We consider a class of queries called durability prediction queries that arise commonly in predictive analytics, where we use a given predictive model to answer questions about possible futures to inform our decisions. Examples of durability prediction queries include "what is the probability that this financial product will keep losing money over the next 12 quarters before turning in any profit?" and "what is the chance for our proposed server cluster to fail the required service-level agreement before its term ends?" We devise a general method called Multi-Level Splitting Sampling (MLSS) that can efficiently handle complex queries and complex models -- including those involving black-box functions -- as long as the models allow us to simulate possible futures step by step. Our method addresses the inefficiency of standard Monte Carlo (MC) methods by applying the idea of importance splitting to let one "promising" sample path prefix generate multiple "offspring" paths, thereby directing simulation efforts toward more promising paths. We propose practical techniques for designing splitting strategies, freeing users from manual tuning. Experiments show that our approach is able to achieve unbiased estimates and the same error guarantees as standard MC while offering an order-of-magnitude cost reduction.

cs.DB↗

arXiv2021

Durable Top-K Instant-Stamped Temporal Records with User-Specified Scoring Functions

A way of finding interesting or exceptional records from instant-stamped temporal data is to consider their "durability," or, intuitively speaking, how well they compare with other records that arrived earlier or later, and how long they retain their supremacy. For example, people are naturally fascinated by claims with long durability, such as: "On January 22, 2006, Kobe Bryant dropped 81 points against Toronto Raptors. Since then, this scoring record has yet to be broken." In general, given a sequence of instant-stamped records, suppose that we can rank them by a user-specified scoring function $f$, which may consider multiple attributes of a record to compute a single score for ranking. This paper studies "durable top-$k$ queries", which find records whose scores were within top-$k$ among those records within a "durability window" of given length, e.g., a 10-year window starting/ending at the timestamp of the record. The parameter $k$, the length of the durability window, and parameters of the scoring function (which capture user preference) can all be given at the query time. We illustrate why this problem formulation yields more meaningful answers in some practical situations than other similar types of queries considered previously. We propose new algorithms for solving this problem, and provide a comprehensive theoretical analysis on the complexities of the problem itself and of our algorithms. Our algorithms vastly outperform various baselines (by up to two orders of magnitude on real and synthetic datasets).

cs.DB↗

arXiv2021

An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications

In 2015, Guth proved that if $S$ is a collection of $n$ $g$-dimensional semi-algebraic sets in $\mathbb{R}^d$ and if $D\geq 1$ is an integer, then there is a $d$-variate polynomial $P$ of degree at most $D$ so that each connected component of $\mathbb{R}^d\setminus Z(P)$ intersects $O(n/D^{d-g})$ sets from $S$. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently -- the expected running time of our algorithm is linear in $|S|$. Our approach exploits the technique of quantifier elimination combined with that of $ε$-samples. We also present an extension of our construction to multi-level polynomial partitioning for semi-algebraic sets in $\mathbb{R}^d$. We present five applications of our result. The first is a data structure for answering point-enclosure queries among a family of semi-algebraic sets in $\mathbb{R}^d$ in $O(\log n)$ time, with storage complexity and expected preprocessing time of $O(n^{d+ε})$. The second is a data structure for answering range-searching queries with semi-algebraic ranges in $\mathbb{R}^d$ in $O(\log n)$ time, with $O(n^{t+ε})$ storage and expected preprocessing time, where $t > 0$ is an integer that depends on $d$ and the description complexity of the ranges. The third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in $\mathbb{R}^{d}$ in $O(\log^2 n)$ time, with $O(n^{d+ε})$ storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic curves in $\mathbb{R}^2$ into pseudo-segments. The fifth application is for eliminating depth cycles among triangles in $\mathbb{R}^3$, where we show a nearly-optimal algorithm to cut $n$ pairwise disjoint non-vertical triangles in $\mathbb{R}^3$ into pieces that form a depth order.

cs.CG↗

arXiv2020

Clustering under Perturbation Stability in Near-Linear Time

We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is $α$-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most $α$. Our main contribution is in presenting efficient exact algorithms for $α$-stable clustering instances whose running times depend near-linearly on the size of the data set when $α\ge 2 + \sqrt{3}$. For $k$-center and $k$-means problems, our algorithms also achieve polynomial dependence on the number of clusters, $k$, when $α\geq 2 + \sqrt{3} + ε$ for any constant $ε> 0$ in any fixed dimension. For $k$-median, our algorithms have polynomial dependence on $k$ for $α> 5$ in any fixed dimension; and for $α\geq 2 + \sqrt{3}$ in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures.

cs.DS↗

arXiv2020

1D and 2D Flow Routing on a Terrain

An important problem in terrain analysis is modeling how water flows across a terrain creating floods by forming channels and filling depressions. In this paper we study a number of \emph{flow-query} related problems: Given a terrain $Σ$, represented as a triangulated $xy$-monotone surface with $n$ vertices, a rain distribution $R$ which may vary over time, determine how much water is flowing over a given edge as a function of time. We develop internal-memory as well as I/O-efficient algorithms for flow queries. This paper contains four main results: (i) We present an internal-memory algorithm that preprocesses $Σ$ into a linear-size data structure that for a (possibly time varying) rain distribution $R$ can return the flow-rate functions of all edges of $Σ$ in $O(ρk+|ϕ| \log n)$ time, where $ρ$ is the number of sinks in $Σ$, $k$ is the number of times the rain distribution changes, and $|ϕ|$ is the total complexity of the flow-rate functions that have non-zero values; (ii) We also present an I/O-efficient algorithm for preprocessing $Σ$ into a linear-size data structure so that for a rain distribution $R$, it can compute the flow-rate function of all edges using $O(\text{Sort}(|ϕ|))$ I/Os and $O(|ϕ| \log |ϕ|)$ internal computation time. (iii) $Σ$ can be preprocessed into a linear-size data structure so that for a given rain distribution $R$, the flow-rate function of an edge $(q,r) \in Σ$ under the single-flow direction (SFD) model can be computed more efficiently. (iv) We present an algorithm for computing the two-dimensional channel along which water flows using Manning's equation; a widely used empirical equation that relates the flow-rate of water in an open channel to the geometry of the channel along with the height of water in the channel.

cs.CG↗

arXiv2020

Dynamic geometric set cover and hitting set

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our paper are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in $\mathbb{R}^1$ and $\mathbb{R}^2$. The first framework uses bootstrapping and gives a $(1+\varepsilon)$-approximate data structure for dynamic interval set cover in $\mathbb{R}^1$ with $O(n^α/\varepsilon)$ amortized update time for any constant $α> 0$; in $\mathbb{R}^2$, this method gives $O(1)$-approximate data structures for unit-square (and quadrant) set cover and hitting set with $O(n^{1/2+α})$ amortized update time. The second framework uses local modification, and leads to a $(1+\varepsilon)$-approximate data structure for dynamic interval hitting set in $\mathbb{R}^1$ with $\widetilde{O}(1/\varepsilon)$ amortized update time; in $\mathbb{R}^2$, it gives $O(1)$-approximate data structures for unit-square (and quadrant) set cover and hitting set in the \textit{partially} dynamic settings with $\widetilde{O}(1)$ amortized update time.

cs.CG↗

arXiv2019

Selecting Data to Clean for Fact Checking: Minimizing Uncertainty vs. Maximizing Surprise

We study the optimization problem of selecting numerical quantities to clean in order to fact-check claims based on such data. Oftentimes, such claims are technically correct, but they can still mislead for two reasons. First, data may contain uncertainty and errors. Second, data can be "fished" to advance particular positions. In practice, fact-checkers cannot afford to clean all data and must choose to clean what "matters the most" to checking a claim. We explore alternative definitions of what "matters the most": one is to ascertain claim qualities (by minimizing uncertainty in these measures), while an alternative is just to counter the claim (by maximizing the probability of finding a counterargument). We show whether the two objectives align with each other, with important implications on when fact-checkers should exercise care in selective data cleaning, to avoid potential bias introduced by their desire to counter claims. We develop efficient algorithms for solving the various variants of the optimization problem, showing significant improvements over naive solutions. The problem is particularly challenging because the objectives in the fact-checking context are complex, non-linear functions over data. We obtain results that generalize to a large class of functions, with potential applications beyond fact-checking.

cs.DB↗

arXiv2019

Efficient Algorithms for Geometric Partial Matching

Let $A$ and $B$ be two point sets in the plane of sizes $r$ and $n$ respectively (assume $r \leq n$), and let $k$ be a parameter. A matching between $A$ and $B$ is a family of pairs in $A \times B$ so that any point of $A \cup B$ appears in at most one pair. Given two positive integers $p$ and $q$, we define the cost of matching $M$ to be $c(M) = \sum_{(a, b) \in M}\|{a-b}\|_p^q$ where $\|{\cdot}\|_p$ is the $L_p$-norm. The geometric partial matching problem asks to find the minimum-cost size-$k$ matching between $A$ and $B$. We present efficient algorithms for geometric partial matching problem that work for any powers of $L_p$-norm matching objective: An exact algorithm that runs in $O((n + k^2) {\mathop{\mathrm{polylog}}} n)$ time, and a $(1 + \varepsilon)$-approximation algorithm that runs in $O((n + k\sqrt{k}) {\mathop{\mathrm{polylog}}} n \cdot \log\varepsilon^{-1})$ time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in $O(\min\{n^2, rn^{3/2}\} {\mathop{\mathrm{polylog}}} n)$ time.

cs.DS↗

arXiv2019

Faster Algorithms for the Geometric Transportation Problem

Let $R$ and $B$ be two point sets in $\mathbb{R}^d$, with $|R|+ |B| = n$ and where $d$ is a constant. Next, let $λ: R \cup B \to \mathbb{N}$ such that $\sum_{r \in R } λ(r) = \sum_{b \in B} λ(b)$ be demand functions over $R$ and $B$. Let $\|\cdot\|$ be a suitable distance function such as the $L_p$ distance. The transportation problem asks to find a map $τ: R \times B \to \mathbb{N}$ such that $\sum_{b \in B}τ(r,b) = λ(r)$, $\sum_{r \in R}τ(r,b) = λ(b)$, and $\sum_{r \in R, b \in B} τ(r,b) \|r-b\|$ is minimized. We present three new results for the transportation problem when $\|r-b\|$ is any $L_p$ metric: - For any constant $\varepsilon > 0$, an $O(n^{1+\varepsilon})$ expected time randomized algorithm that returns a transportation map with expected cost $O(\log^2(1/\varepsilon))$ times the optimal cost. - For any $\varepsilon > 0$, a $(1+\varepsilon)$-approximation in $O(n^{3/2}\varepsilon^{-d} \operatorname{polylog}(U) \operatorname{polylog}(n))$ time, where $U = \max_{p\in R\cup B} λ(p)$. - An exact strongly polynomial $O(n^2 \operatorname{polylog}n)$ time algorithm, for $d = 2$.

cs.DS↗

arXiv2019

Dynamic Maintenance of the Lower Envelope of Pseudo-Lines

We present a fully dynamic data structure for the maintenance of lower envelopes of pseudo-lines. The structure has $O(\log^2 n)$ update time and $O(\log n)$ vertical ray shooting query time. To achieve this performance, we devise a new algorithm for finding the intersection between two lower envelopes of pseudo-lines in $O(\log n)$ time, using \emph{tentative} binary search; the lower envelopes are special in that at $x=-\infty$ any pseudo-line contributing to the first envelope lies below every pseudo-line contributing to the second envelope. The structure requires $O(n)$ storage space.

cs.CG↗

arXiv2018

Approximate Minimum-Weight Matching with Outliers under Translation

Our goal is to compare two planar point sets by finding subsets of a given size such that a minimum-weight matching between them has the smallest weight. This can be done by a translation of one set that minimizes the weight of the matching. We give efficient algorithms (a) for finding approximately optimal matchings, when the cost of a matching is the $L_p$-norm of the tuple of the Euclidean distances between the pairs of matched points, for any $p\in [1,\infty]$, and (b)~for constructing small-size approximate minimization (or matching) diagrams: partitions of the translation space into regions, together with an approximate optimal matching for each region.

cs.CG↗

arXiv2018

Improved Dynamic Geodesic Nearest Neighbor Searching in a Simple Polygon

We present an efficient dynamic data structure that supports geodesic nearest neighbor queries for a set $S$ of point sites in a static simple polygon $P$. Our data structure allows us to insert a new site in $S$, delete a site from $S$, and ask for the site in $S$ closest to an arbitrary query point $q \in P$. All distances are measured using the geodesic distance, that is, the length of the shortest path that is completely contained in $P$. Our data structure achieves polylogarithmic update and query times, and uses $O(n\log^3n\log m + m)$ space, where $n$ is the number of sites in $S$ and $m$ is the number of vertices in $P$. The crucial ingredient in our data structure is an implicit representation of a vertical shallow cutting of the geodesic distance functions. We show that such an implicit representation exists, and that we can compute it efficiently.

cs.CG↗

arXiv2017

Computing the Gromov-Hausdorff Distance for Metric Trees

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is $\mathrm{NP}$-hard to approximate the Gromov-Hausdorff distance better than a factor of $3$ for geodesic metrics on a pair of trees. We complement this result by providing a polynomial time $O(\min\{n, \sqrt{rn}\})$-approximation algorithm for computing the GH distance between a pair of metric trees, where $r$ is the ratio of the longest edge length in both trees to the shortest edge length. For metric trees with unit length edges, this yields an $O(\sqrt{n})$-approximation algorithm.

cs.CG↗