arXiv subjects

Julian Portmann

Publications and source records attributed to Julian Portmann.

6 recordsLinked to original sources

arXiv2025

Constant Approximation of Arboricity in Near-Optimal Sublinear Time

We present a randomized algorithm that computes a constant approximation of a graph's arboricity, using $\tilde{O}(n/\lambda)$ queries to adjacency lists and in the same time bound. Here, $n$ and $\lambda$ denote the number of nodes and the graph's arboricity, respectively. The $\tilde{O}(n/\lambda)$ query complexity of our algorithm is nearly optimal. Our constant approximation settles a question of Eden, Mossel, and Ron [SODA'22], who achieved an $O(\log^2 n)$ approximation with the same query and time complexity and asked whether a better approximation can be achieved using near-optimal query complexity. A key technical challenge in the problem is due to recursive algorithms based on probabilistic samplings, each with a non-negligible error probability. In our case, many of the recursions invoked could have bad probabilistic samples and result in high query complexities. The particular difficulty is that those bad recursions are not easy or cheap to detect and discard. Our approach runs multiple recursions in parallel, to attenuate the error probability, using a careful \textit{scheduling mechanism} that manages the speed at which each of them progresses and makes our overall query complexity competitive with the single good recursion. We find this usage of parallelism and scheduling in a sublinear algorithm remarkable, and we are hopeful that similar ideas may find applications in a wider range of sublinear algorithms that rely on probabilistic recursions.

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arXiv2023

Distributed MIS with Low Energy and Time Complexities

We present randomized distributed algorithms for the maximal independent set problem (MIS) that, while keeping the time complexity nearly matching the best known, reduce the energy complexity substantially. These algorithms work in the standard CONGEST model of distributed message passing with $O(\log n)$ bit messages. The time complexity measures the number of rounds in the algorithm. The energy complexity measures the number of rounds each node is awake; during other rounds, the node sleeps and cannot perform any computation or communications. Our first algorithm has an energy complexity of $O(\log\log n)$ and a time complexity of $O(\log^2 n)$. Our second algorithm is faster but slightly less energy-efficient: it achieves an energy complexity of $O(\log^2 \log n)$ and a time complexity of $O(\log n \cdot \log\log n \cdot \log^* n)$. Thus, this algorithm nearly matches the $O(\log n)$ time complexity of the state-of-the-art MIS algorithms while significantly reducing their energy complexity from $O(\log n)$ to $O(\log^2 \log n)$.

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arXiv2023

Average Awake Complexity of MIS and Matching

Chatterjee, Gmyr, and Pandurangan [PODC 2020] recently introduced the notion of awake complexity for distributed algorithms, which measures the number of rounds in which a node is awake. In the other rounds, the node is sleeping and performs no computation or communication. Measuring the number of awake rounds can be of significance in many settings of distributed computing, e.g., in sensor networks where energy consumption is of concern. In that paper, Chatterjee et al. provide an elegant randomized algorithm for the Maximal Independent Set (MIS) problem that achieves an $O(1)$ node-averaged awake complexity. That is, the average awake time among the nodes is $O(1)$ rounds. However, to achieve that, the algorithm sacrifices the more standard round complexity measure from the well-known $O(\log n)$ bound of MIS, due to Luby [STOC'85], to $O(\log^{3.41} n)$ rounds. Our first contribution is to present a simple randomized distributed MIS algorithm that, with high probability, has $O(1)$ node-averaged awake complexity and $O(\log n)$ worst-case round complexity. Our second, and more technical contribution, is to show algorithms with the same $O(1)$ node-averaged awake complexity and $O(\log n)$ worst-case round complexity for $(1+\varepsilon)$-approximation of maximum matching and $(2+\varepsilon)$-approximation of minimum vertex cover, where $\varepsilon$ denotes an arbitrary small positive constant.

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arXiv2020

Tight Bounds for Deterministic High-Dimensional Grid Exploration

We study the problem of exploring an oriented grid with autonomous agents governed by finite automata. In the case of a 2-dimensional grid, the question how many agents are required to explore the grid, or equivalently, find a hidden treasure in the grid, is fully understood in both the synchronous and the semi-synchronous setting. For higher dimensions, Dobrev, Narayanan, Opatrny, and Pankratov [ICALP'19] showed very recently that, surprisingly, a (small) constant number of agents suffices to find the treasure, independent of the number of dimensions, thereby disproving a conjecture by Cohen, Emek, Louidor, and Uitto [SODA'17]. Dobrev et al. left as an open question whether their bounds on the number of agents can be improved. We answer this question in the affirmative for deterministic finite automata: we show that 3 synchronous and 4 semi-synchronous agents suffice to explore an $n$-dimensional grid for any constant $n$. The bounds are optimal and notably, the matching lower bounds already hold in the 2-dimensional case. Our techniques can also be used to make progress on other open questions asked by Dobrev et al.: we prove that 4 synchronous and 5 semi-synchronous agents suffice for polynomial-time exploration, and we show that, under a natural assumption, 3 synchronous and 4 semi-synchronous agents suffice to explore unoriented grids of arbitrary dimension (which, again, is tight).

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arXiv2020

k-means++: few more steps yield constant approximation

The k-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is a state-of-the-art algorithm for solving the k-means clustering problem and is known to give an O(log k)-approximation in expectation. Recently, Lattanzi and Sohler (ICML 2019) proposed augmenting k-means++ with O(k log log k) local search steps to yield a constant approximation (in expectation) to the k-means clustering problem. In this paper, we improve their analysis to show that, for any arbitrarily small constant $\eps > 0$, with only $\eps k$ additional local search steps, one can achieve a constant approximation guarantee (with high probability in k), resolving an open problem in their paper.

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arXiv2019

Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Network decompositions, as introduced by Awerbuch, Luby, Goldberg, and Plotkin [FOCS'89], are one of the key algorithmic tools in distributed graph algorithms. We present an improved deterministic distributed algorithm for constructing network decompositions of power graphs using small messages, which improves upon the algorithm of Ghaffari and Kuhn [DISC'18]. In addition, we provide a randomized distributed network decomposition algorithm, based on our deterministic algorithm, with failure probability exponentially small in the input size that works with small messages as well. Compared to the previous algorithm of Elkin and Neiman [PODC'16], our algorithm achieves a better success probability at the expense of its round complexity, while giving a network decomposition of the same quality. As a consequence of the randomized algorithm for network decomposition, we get a faster randomized algorithm for computing a Maximal Independent Set, improving on a result of Ghaffari [SODA'19]. Other implications of our improved deterministic network decomposition algorithm are: a faster deterministic distributed algorithms for constructing spanners and approximations of distributed set cover, improving results of Ghaffari, and Kuhn [DISC'18] and Deurer, Kuhn, and Maus [PODC'19]; and faster a deterministic distributed algorithm for constructing neighborhood covers, resolving an open question of Elkin [SODA'04].

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