arXiv subjects

Jara Uitto

Publications and source records attributed to Jara Uitto.

At least 37 records · Page 2Linked to original sources

arXiv2021

Massively Parallel Correlation Clustering in Bounded Arboricity Graphs

Identifying clusters of similar elements in a set is a common task in data analysis. With the immense growth of data and physical limitations on single processor speed, it is necessary to find efficient parallel algorithms for clustering tasks. In this paper, we study the problem of correlation clustering in bounded arboricity graphs with respect to the Massively Parallel Computation (MPC) model. More specifically, we are given a complete graph where the edges are either positive or negative, indicating whether pairs of vertices are similar or dissimilar. The task is to partition the vertices into clusters with as few disagreements as possible. That is, we want to minimize the number of positive inter-cluster edges and negative intra-cluster edges. Consider an input graph $G$ on $n$ vertices such that the positive edges induce a $λ$-arboric graph. Our main result is a 3-approximation ($\textit{in expectation}$) algorithm to correlation clustering that runs in $\mathcal{O}(\log λ\cdot \textrm{poly}(\log \log n))$ MPC rounds in the $\textit{strongly sublinear memory regime}$. This is obtained by combining structural properties of correlation clustering on bounded arboricity graphs with the insights of Fischer and Noever (SODA '18) on randomized greedy MIS and the $\texttt{PIVOT}$ algorithm of Ailon, Charikar, and Newman (STOC '05). Combined with known graph matching algorithms, our structural property also implies an exact algorithm and algorithms with $\textit{worst case}$ $(1+ε)$-approximation guarantees in the special case of forests, where $λ=1$.

cs.DC↗

arXiv2021

Efficient CONGEST Algorithms for the Lovasz Local Lemma

We present a poly $\log \log n$ time randomized CONGEST algorithm for a natural class of Lovasz Local Lemma (LLL) instances on constant degree graphs. This implies, among other things, that there are no LCL problems with randomized complexity between $\log n$ and poly $\log \log n$. Furthermore, we provide extensions to the network decomposition algorithms given in the recent breakthrough by Rozhon and Ghaffari [STOC2020] and the follow up by Ghaffari, Grunau, and Rozhon [SODA2021]. In particular, we show how to obtain a large distance separated weak network decomposition with a negligible dependency on the range of unique identifiers.

cs.DC↗

arXiv2021

Efficient Load-Balancing through Distributed Token Dropping

We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for stable orientations and more generally for locally optimal semi-matchings. The prior work by Czygrinow et al. (DISC 2012) finds a stable orientation in $O(Δ^5)$ rounds in graphs of maximum degree $Δ$, while we improve it to $O(Δ^4)$ and also prove a lower bound of $Ω(Δ)$.

cs.DC↗

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).

cs.DC↗

arXiv2019

A Sharp Threshold Phenomenon for the Distributed Complexity of the Lovász Local Lemma

The Lovász Local Lemma (LLL) says that, given a set of bad events that depend on the values of some random variables and where each event happens with probability at most $p$ and depends on at most $d$ other events, there is an assignment of the variables that avoids all bad events if the LLL criterion $ep(d+1)<1$ is satisfied. In this paper, we study the dependency of the distributed complexity of the LLL problem on the chosen LLL criterion. We show that for the fundamental case of each random variable of the considered LLL instance being associated with an edge of the input graph, that is, each random variable influences at most two events, a sharp threshold phenomenon occurs at $p = 2^{-d}$: we provide a simple deterministic (!) algorithm that matches a known $Ω(\log^* n)$ lower bound in bounded degree graphs, if $p < 2^{-d}$, whereas for $p \geq 2^{-d}$, a known $Ω(\log \log n)$ randomized and a known $Ω(\log n)$ deterministic lower bounds hold. In many applications variables affect more than two events; our main contribution is to extend our algorithm to the case where random variables influence at most three different bad events. We show that, surprisingly, the sharp threshold occurs at the exact same spot, providing evidence for our conjecture that this phenomenon always occurs at $p = 2^{-d}$, independent of the number $r$ of events that are affected by a variable. Almost all steps of the proof framework we provide for the case $r=3$ extend directly to the case of arbitrary $r$; consequently, our approach serves as a step towards characterizing the complexity of the LLL under different exponential criteria.

cs.DS↗

arXiv2019

Navigating an Infinite Space with Unreliable Movements

We consider a search problem on a $2$-dimensional infinite grid with a single mobile agent. The goal of the agent is to find her way home, which is located in a grid cell chosen by an adversary. Initially, the agent is provided with an infinite sequence of instructions, that dictate the movements performed by the agent. Each instruction corresponds to a movement to an adjacent grid cell and the set of instructions can be a function of the initial locations of the agent and home. The challenge of our problem stems from faults in the movements made by the agent. In every step, with some constant probability $0 \leq p \leq 1$, the agent performs a random movement instead of following the current instruction. This paper provides two results on this problem. First, we show that for some values of $p$, there does not exist any set of instructions that guide the agent home in finite expected time. Second, we complement this impossibility result with an algorithm that, for sufficiently small values of $p$, yields a finite expected hitting time for home. In particular, we show that for any $p < 1$, our approach gives a hitting rate that decays polynomially as a function of time. In that sense, our approach is far superior to a standard random walk in terms of hitting time. The main contribution and take-home message of this paper is to show that, for some value of $0.01139\dots < p < 0.6554\ldots$, there exists a phase transition on the solvability of the problem.

cs.DM↗

arXiv2019

On the Complexity of Distributed Splitting Problems

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph $G=(V,E)$ has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient $\text{poly}\log n$-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if $δ$ and $Δ$ are the minimum and maximum degrees of $G$ and if $δ=Ω(\log n)$, weak splitting can be solved deterministically in time $O\big(\fracΔδ\cdot\text{poly}(\log n)\big)$. Further, if $δ= Ω(\log\log n)$ and $Δ\leq 2^{\varepsilonδ}$, there is a randomized algorithm with time complexity $O\big(\fracΔδ\cdot\text{poly}(\log\log n)\big)$.

cs.DC↗

arXiv2019

Breaking the Linear-Memory Barrier in MPC: Fast MIS on Trees with Strongly Sublinear Memory

Recently, studying fundamental graph problems in the \emph{Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow $\widetildeΩ(n)$ memory per machine, where $n$ is the number of nodes in the graph and $\widetildeΩ$ hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory. In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality. In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in $O(\log^3 \log n)$ rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in $O(\sqrt{\log n})$-algorithm in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from $\widetildeΩ(n)$ required by the recent $O(\log \log n)$-round MIS algorithm of Ghaffari et al. [PODC'18] to $n^α$ for any $α>0$, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the $\mathsf{LOCAL}$ and $\mathsf{PRAM}$ models by Lenzen and Wattenhofer [PODC'11].

cs.DS↗

arXiv2019

Distributed Recoloring

Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the notion of distributed recoloring: The input graph represents a network of computers that needs to be recolored. Initially, each node is aware of its own input color and target color. The nodes can exchange messages with each other, and eventually each node has to stop and output its own recoloring schedule, indicating when and how the node changes its color. The recoloring schedules have to be globally consistent so that the graph remains properly colored at each point, and we require that adjacent nodes do not change their colors simultaneously. We are interested in the following questions: How many communication rounds are needed (in the LOCAL model of distributed computing) to find a recoloring schedule? What is the length of the recoloring schedule? And how does the picture change if we can use extra colors to make recoloring easier? The main contributions of this work are related to distributed recoloring with one extra color in the following graph classes: trees, $3$-regular graphs, and toroidal grids.

cs.DC↗

arXiv2019

Improved Distributed Degree Splitting and Edge Coloring

The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy. We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su [SODA'17]: our algorithms are significantly simpler and faster, and have a much smaller discrepancy. This also leads to a faster and simpler deterministic algorithm for $(2+o(1))Δ$-edge-coloring, improving on that of Ghaffari and Su.

cs.DC↗

arXiv2018

The Complexity of $(Δ+ 1)$Coloring inCongested Clique, Massively Parallel Computation,and Centralized Local Computation

We present new randomized algorithms that improve the complexity of the classic $(Δ+1)$-coloring problem, and its generalization $(Δ+1)$-list-coloring, in three well-studied models of distributed, parallel, and centralized computation: Distributed Congested Clique: We present an $O(1)$-round randomized algorithm for $(Δ+1)$-list coloring in the congested clique model of distributed computing. This settles the asymptotic complexity of this problem. It moreover improves upon the $O(\log^\ast Δ)$-round randomized algorithms of Parter and Su [DISC'18] and $O((\log\log Δ)\cdot \log^\ast Δ)$-round randomized algorithm of Parter [ICALP'18]. Massively Parallel Computation: We present a $(Δ+1)$-list coloring algorithm with round complexity $O(\sqrt{\log\log n})$ in the Massively Parallel Computation (MPC) model with strongly sublinear memory per machine. This algorithm uses a memory of $O(n^α)$ per machine, for any desirable constant $α>0$, and a total memory of $\widetilde{O}(m)$, where $m$ is the size of the graph. Notably, this is the first coloring algorithm with sublogarithmic round complexity, in the sublinear memory regime of MPC. For the quasilinear memory regime of MPC, an $O(1)$-round algorithm was given very recently by Assadi et al. [SODA'19]. Centralized Local Computation: We show that $(Δ+1)$-list coloring can be solved with $Δ^{O(1)} \cdot O(\log n)$ query complexity, in the centralized local computation model. The previous state-of-the-art for $(Δ+1)$-list coloring in the centralized local computation model are based on simulation of known LOCAL algorithms.

cs.DS↗

arXiv2018

Matching and MIS for Uniformly Sparse Graphs in the Low-Memory MPC Model

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale parallel computation frameworks and has recently gained a lot of importance, especially in the context of classic graph problems. Unsatisfactorily, all current $\text{poly} (\log \log n)$-round MPC algorithms seem to get fundamentally stuck at the linear-memory barrier: their efficiency crucially relies on each machine having space at least linear in the number $n$ of nodes. As this might not only be prohibitively large, but also allows for easy if not trivial solutions for sparse graphs, we are interested in the low-memory MPC model, where the space per machine is restricted to be strongly sublinear, that is, $n^δ$ for any $0<δ<1$. We devise a degree reduction technique that reduces maximal matching and maximal independent set in graphs with arboricity $λ$ to the corresponding problems in graphs with maximum degree $\text{poly}(λ)$ in $O(\log^2 \log n)$ rounds. This gives rise to $O\left(\log^2\log n + T(\text{poly} λ)\right)$-round algorithms, where $T(Δ)$ is the $Δ$-dependency in the round complexity of maximal matching and maximal independent set in graphs with maximum degree $Δ$. A concurrent work by Ghaffari and Uitto shows that $T(Δ)=O(\sqrt{\log Δ})$. For graphs with arboricity $λ=\text{poly}(\log n)$, this almost exponentially improves over Luby's $O(\log n)$-round PRAM algorithm [STOC'85, JALG'86], and constitutes the first $\text{poly} (\log \log n)$-round maximal matching algorithm in the low-memory MPC model, thus breaking the linear-memory barrier. Previously, the only known subpolylogarithmic algorithm, due to Lattanzi et al. [SPAA'11], required strongly superlinear, that is, $n^{1+Ω(1)}$, memory per machine.

cs.DS↗

arXiv2018

Sparsifying Distributed Algorithms with Ramifications in Massively Parallel Computation and Centralized Local Computation

We introduce a method for sparsifying distributed algorithms and exhibit how it leads to improvements that go past known barriers in two algorithmic settings of large-scale graph processing: Massively Parallel Computation (MPC), and Local Computation Algorithms (LCA). - MPC with Strongly Sublinear Memory: Recently, there has been growing interest in obtaining MPC algorithms that are faster than their classic $O(\log n)$-round parallel counterparts for problems such as MIS, Maximal Matching, 2-Approximation of Minimum Vertex Cover, and $(1+ε)$-Approximation of Maximum Matching. Currently, all such MPC algorithms require $\tildeΩ(n)$ memory per machine. Czumaj et al. [STOC'18] were the first to handle $\tildeΩ(n)$ memory, running in $O((\log\log n)^2)$ rounds. We obtain $\tilde{O}(\sqrt{\log Δ})$-round MPC algorithms for all these four problems that work even when each machine has memory $n^α$ for any constant $α\in (0, 1)$. Here, $Δ$ denotes the maximum degree. These are the first sublogarithmic-time algorithms for these problems that break the linear memory barrier. - LCAs with Query Complexity Below the Parnas-Ron Paradigm: Currently, the best known LCA for MIS has query complexity $Δ^{O(\log Δ)} poly(\log n)$, by Ghaffari [SODA'16]. As pointed out by Rubinfeld, obtaining a query complexity of $poly(Δ\log n)$ remains a central open question. Ghaffari's bound almost reaches a $Δ^{Ω\left(\frac{\log Δ}{\log\log Δ}\right)}$ barrier common to all known MIS LCAs, which simulate distributed algorithms by learning the local topology, à la Parnas-Ron [TCS'07]. This barrier follows from the $Ω(\frac{\log Δ}{\log\log Δ})$ distributed lower bound of Kuhn, et al. [JACM'16]. We break this barrier and obtain an MIS LCA with query complexity $Δ^{O(\log\log Δ)} poly(\log n)$.

cs.DS↗

arXiv2018

Tight Bounds for Asynchronous Collaborative Grid Exploration

Consider a small group of mobile agents whose goal is to locate a certain cell in a two-dimensional infinite grid. The agents operate in an asynchronous environment, where in each discrete time step, an arbitrary subset of the agents execute one atomic look-compute-move cycle. The protocol controlling each agent is determined by a (possibly distinct) finite automaton. The only means of communication is to sense the states of the agents sharing the same grid cell. Whenever an agent moves, the destination cell of the movement is chosen by the agent's automaton from the set of neighboring grid cells. We study the minimum number of agents required to locate the target cell within finite time and our main result states a tight lower bound for agents endowed with a global compass. Furthermore, we show that the lack of such a compass makes the problem strictly more difficult and present tight upper and lower bounds for this case.

cs.DC↗

arXiv2017

Deterministic Distributed Edge-Coloring with Fewer Colors

We present a deterministic distributed algorithm, in the LOCAL model, that computes a $(1+o(1))Δ$-edge-coloring in polylogarithmic-time, so long as the maximum degree $Δ=\tildeΩ(\log n)$. For smaller $Δ$, we give a polylogarithmic-time $3Δ/2$-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of $2Δ-1$ colors, and they improve significantly on the recent polylogarithmic-time $(2Δ-1)(1+o(1))$-edge-coloring of Ghaffari and Su [SODA'17] and the $(2Δ-1)$-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS'17], positively answering the main open question of the latter. The key technical ingredient of our algorithm is a simple and novel gradual packing of judiciously chosen near-maximum matchings, each of which becomes one of the color classes.

cs.DS↗

arXiv2017

Dynamic Networks of Finite State Machines

Like distributed systems, biological multicellular processes are subject to dynamic changes and a biological system will not pass the survival-of-the-fittest test unless it exhibits certain features that enable fast recovery from these changes. In particular, a question that is crucial in the context of biological cellular networks, is whether the system can keep the changing components \emph{confined} so that only nodes in their vicinity may be affected by the changes, but nodes sufficiently far away from any changing component remain unaffected. Based on this notion of confinement, we propose a new metric for measuring the dynamic changes recovery performance in distributed network algorithms operating under the \emph{Stone Age} model (Emek \& Wattenhofer, PODC 2013), where the class of dynamic topology changes we consider includes inserting/deleting an edge, deleting a node together with its incident edges, and inserting a new isolated node. Our main technical contribution is a distributed algorithm for maximal independent set (MIS) in synchronous networks subject to these topology changes that performs well in terms of the aforementioned new metric. Specifically, our algorithm guarantees that nodes which do not experience a topology change in their immediate vicinity are not affected and that all surviving nodes (including the affected ones) perform $\mathcal{O}((C + 1) \log^{2} n)$ computationally-meaningful steps, where $C$ is the number of topology changes; in other words, each surviving node performs $\mathcal{O}(\log^{2} n)$ steps when amortized over the number of topology changes. This is accompanied by a simple example demonstrating that the linear dependency on $C$ cannot be avoided.

cs.DC↗

arXiv2017

Exploring an Infinite Space with Finite Memory Scouts

Consider a small number of scouts exploring the infinite $d$-dimensional grid with the aim of hitting a hidden target point. Each scout is controlled by a probabilistic finite automaton that determines its movement (to a neighboring grid point) based on its current state. The scouts, that operate under a fully synchronous schedule, communicate with each other (in a way that affects their respective states) when they share the same grid point and operate independently otherwise. Our main research question is: How many scouts are required to guarantee that the target admits a finite mean hitting time? Recently, it was shown that $d + 1$ is an upper bound on the answer to this question for any dimension $d \geq 1$ and the main contribution of this paper comes in the form of proving that this bound is tight for $d \in \{ 1, 2 \}$.

math.PR↗

arXiv2015

A Lower Bound for the Distributed Lovász Local Lemma

We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires $Ω(\log \log n)$ communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of $d = O(1)$, where $d$ is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of $O(\log n)$ rounds in bounded-degree graphs, and the best lower bound before our work was $Ω(\log^* n)$ rounds [Chung et al. 2014].

cs.DC↗