Given a set of points $P = (P^+ \sqcup P^) \subset \mathbb{R}^d$ for some constant $d$ and a supply function $\mu:P\to \mathbb{R}$ such that $\mu(p) > 0~\forall p \in P^+$, $\mu(p) < 0~\forall p \in P^$, and $\sum_{p\in P}{\mu(p)} = 0$, the geometric transportation problem asks one to find a transportation map $\tau: P^+\times P^\to \mathbb{R}_{\ge 0}$ such that $\sum_{q\in P^}{\tau(p, q)} = \mu(p)~\forall p \in P^+$, $\sum_{p\in P^+}{\tau(p, q)} = \mu(q) \forall q \in P^$, and the weighted sum of Euclidean distances for the pairs $\sum_{(p,q)\in P^+\times P^}\tau(p, q)\cdot qp_2$ is minimized. We present the first deterministic algorithm that computes, in nearlinear time, a transportation map whose cost is within a $(1 + \varepsilon)$ factor of optimal. More precisely, our algorithm runs in $O(n\varepsilon^{(d+2)}\log^5{n}\log{\log{n}})$ time for any constant $\varepsilon > 0$. While a randomized $n\varepsilon^{O(d)}\log^{O(d)}{n}$ time algorithm for this problem was discovered in the last few years, all previously known deterministic $(1 + \varepsilon)$approximation algorithms run in~$\Omega(n^{3/2})$ time. A similar situation existed for geometric bipartite matching, the special case of geometric transportation where all supplies are unit, until a deterministic $n\varepsilon^{O(d)}\log^{O(d)}{n}$ time $(1 + \varepsilon)$approximation algorithm was presented at STOC 2022. Surprisingly, our result is not only a generalization of the bipartite matching one to arbitrary instances of geometric transportation, but it also reduces the running time for all previously known $(1 + \varepsilon)$approximation algorithms, randomized or deterministic, even for geometric bipartite matching. In particular, we give the first $(1 + \varepsilon)$approximate deterministic algorithm for geometric bipartite matching and the first $(1 + \varepsilon)$approximate deterministic or randomized algorithm for geometric transportation with no dependence on $d$ in the exponent of the running time's polylog. As an additional application of our main ideas, we also give the first randomized nearlinear $O(\varepsilon^{2} m \log^{O(1)} n)$ time $(1 + \varepsilon)$approximation algorithm for the uncapacitated minimum cost flow (transshipment) problem in undirected graphs with arbitrary \emph{real} edge costs.
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Massively Parallel Algorithms and Hardness for SingleLinkage Clustering under ℓpDistances
We present first massively parallel (MPC) algorithms and hardness of approximation results for computing SingleLinkage Clustering of $n$ input $d$dimensional vectors under Hamming, $\ell_1, \ell_2$ and $\ell_\infty$ distances. All our algorithms run in $O(\log n)$ rounds of MPC for any fixed $d$ and achieve $(1+\epsilon)$approximation for all distances (except Hamming for which we show an exact algorithm).
We also show constantfactor inapproximability results for $o(\log n)$round algorithms under standard MPC hardness assumptions (for sufficiently large dimension depending on the distance used). Efficiency of implementation of our algorithms in Apache Spark is demonstrated through experiments on the largest available vector datasets from the UCI machine learning repository exhibiting speedups of several orders of magnitude.
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 Award ID(s):
 1657477
 NSFPAR ID:
 10088159
 Date Published:
 Journal Name:
 35th International Conference on Machine Learning (ICML'18)
 Page Range / eLocation ID:
 55965605
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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