A deterministic near-linear time approximation scheme for geometric transportation

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 $\sum_{(p,q)\in P^+\times P^-}\tau(p, q)\cdot ||q-p||_2$ is minimized. We present the first deterministic algorithm that computes, in near-linear 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 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, by removing the dependence on the dimension $d$ from the exponent in the running time's polylog.Comment: 23 page

Sparse Covers for Planar Graphs and Graphs that Exclude a Fixed Minor

We consider the construction of sparse covers for planar graphs and other graphs that exclude a fixed minor. We present an algorithm that gives a cover for the γ-neighborhood of each node. For planar graphs, the cover has radius less than 16γ and degree no more than 18. For every n node graph that excludes a minor of a fixed size, we present an algorithm that yields a cover with radius no more than 4γ and degree O(logn). This is a significant improvement over previous results for planar graphs and for graphs excluding a fixed minor; in order to obtain clusters with radius O(γ), it was required to have the degree polynomial in n. Our algorithms are based on a recursive application of a basic routine called shortest-path clustering, which seems to be a novel approach to the construction of sparse covers. Since sparse covers have many applications in distributed computing, including compact routing, distributed directories, synchronizers, and Universal TSP, our improved cover construction results in improved algorithms for all these problems, for the class of graphs that exclude a fixed minor

Measuring Effectiveness of Address Schemes for AS-level Graphs

This dissertation presents measures of efficiency and locality for Internet addressing schemes. Historically speaking, many issues, faced by the Internet, have been solved just in time, to make the Internet just work~\cite{justWork}. Consensus, however, has been reached that today\u27s Internet routing and addressing system is facing serious scaling problems: multi-homing which causes finer granularity of routing policies and finer control to realize various traffic engineering requirements, an increased demand for provider-independent prefix allocations which injects unaggregatable prefixes into the Default Free Zone (DFZ) routing table, and ever-increasing Internet user population and mobile edge devices. As a result, the DFZ routing table is again growing at an exponential rate. Hierarchical, topology-based addressing has long been considered crucial to routing and forwarding scalability. Recently, however, a number of research efforts are considering alternatives to this traditional approach. With the goal of informing such research, we investigated the efficiency of address assignment in the existing (IPv4) Internet. In particular, we ask the question: ``how can we measure the locality of an address scheme given an input AS-level graph?\u27\u27 To do so, we first define a notion of efficiency or locality based on the average number of bit-hops required to advertize all prefixes in the Internet. In order to quantify how far from ``optimal the current Internet is, we assign prefixes to ASes ``from scratch in a manner that preserves observed semantics, using three increasingly strict definitions of equivalence. Next we propose another metric that in some sense quantifies the ``efficiency of the labeling and is independent of forwarding/routing mechanisms. We validate the effectiveness of the metric by applying it to a series of address schemes with increasing randomness given an input AS-level graph. After that we apply the metric to the current Internet address scheme across years and compare the results with those of compact routing schemes