On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is $1/2$, which is achieved by any greedy algorithm. D\"urr et al. recently presented a $2$-pass algorithm called Category-Advice that achieves approximation ratio $3/5$. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the $k$-pass Category-Advice algorithm for all $k \ge 1$, and show that the approximation ratio converges to the inverse of the golden ratio $2/(1+\sqrt{5}) \approx 0.618$ as $k$ goes to infinity. The convergence is extremely fast --- the $5$-pass Category-Advice algorithm is already within $0.01\%$ of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than $1-1/e$, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model

A Planarity Test via Construction Sequences

Optimal linear-time algorithms for testing the planarity of a graph are well-known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler linear-time tests. We give a simple reduction from planarity testing to the problem of computing a certain construction of a 3-connected graph. The approach is different from previous planarity tests; as key concept, we maintain a planar embedding that is 3-connected at each point in time. The algorithm runs in linear time and computes a planar embedding if the input graph is planar and a Kuratowski-subdivision otherwise

Simultaneous Orthogonal Planarity

We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the $k$ graphs? We show that the problem is NP-complete for $k \geq 3$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $k=2$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Gathering in Dynamic Rings

The gathering problem requires a set of mobile agents, arbitrarily positioned at different nodes of a network to group within finite time at the same location, not fixed in advanced. The extensive existing literature on this problem shares the same fundamental assumption: the topological structure does not change during the rendezvous or the gathering; this is true also for those investigations that consider faulty nodes. In other words, they only consider static graphs. In this paper we start the investigation of gathering in dynamic graphs, that is networks where the topology changes continuously and at unpredictable locations. We study the feasibility of gathering mobile agents, identical and without explicit communication capabilities, in a dynamic ring of anonymous nodes; the class of dynamics we consider is the classic 1-interval-connectivity. We focus on the impact that factors such as chirality (i.e., a common sense of orientation) and cross detection (i.e., the ability to detect, when traversing an edge, whether some agent is traversing it in the other direction), have on the solvability of the problem. We provide a complete characterization of the classes of initial configurations from which the gathering problem is solvable in presence and in absence of cross detection and of chirality. The feasibility results of the characterization are all constructive: we provide distributed algorithms that allow the agents to gather. In particular, the protocols for gathering with cross detection are time optimal. We also show that cross detection is a powerful computational element. We prove that, without chirality, knowledge of the ring size is strictly more powerful than knowledge of the number of agents; on the other hand, with chirality, knowledge of n can be substituted by knowledge of k, yielding the same classes of feasible initial configurations

Near-Optimal Directed Low-Diameter Decompositions

Low Diameter Decompositions (LDDs) are invaluable tools in the design ofcombinatorial graph algorithms. While historically they have been appliedmainly to undirected graphs, in the recent breakthrough for the negative-lengthSingle Source Shortest Path problem, Bernstein, Nanongkai, and Wulff-Nilsen[FOCS '22] extended the use of LDDs to directed graphs for the first time.Specifically, their LDD deletes each edge with probability at most$O(\frac{1}{D} \cdot \log^2 n)$, while ensuring that each strongly connectedcomponent in the remaining graph has a (weak) diameter of at most $D$. In this work, we make further advancements in the study of directed LDDs. Wereveal a natural and intuitive (in hindsight) connection to ExpanderDecompositions, and leveraging this connection along with additionaltechniques, we establish the existence of an LDD with an edge-cuttingprobability of $O(\frac{1}{D} \cdot \log n \log\log n)$. This improves theprevious bound by nearly a logarithmic factor and closely approaches the lowerbound of $\Omega(\frac{1}{D} \cdot \log n)$. With significantly more technicaleffort, we also develop two efficient algorithms for computing our LDDs: adeterministic algorithm that runs in time $\tilde O(m \cdot poly(D))$ and arandomized algorithm that runs in near-linear time $\tilde O(m)$. We believe that our work provides a solid conceptual and technical foundationfor future research relying on directed LDDs, which will undoubtedly followsoon.<br

An Introduction to Temporal Graphs: An Algorithmic Perspective

A \emph{temporal graph} is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence $G_1,G_2\ldots,G_l$ of static graphs over the same (static) set of nodes $V$. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community

Parallel and Distributed Exact Single-Source Shortest Paths with Negative Edge Weights

This paper presents parallel and distributed algorithms for single-sourceshortest paths when edges can have negative weights (negative-weight SSSP). Weshow a framework that reduces negative-weight SSSP in either setting to$n^{o(1)}$ calls to any SSSP algorithm that works with a virtual source. Morespecifically, for a graph with $m$ edges, $n$ vertices, undirected hop-diameter$D$, and polynomially bounded integer edge weights, we show randomizedalgorithms for negative-weight SSSP with (i) $W_{SSSP}(m,n)n^{o(1)}$ work and$S_{SSSP}(m,n)n^{o(1)}$ span, given access to an SSSP algorithm with$W_{SSSP}(m,n)$ work and $S_{SSSP}(m,n)$ span in the parallel model, (ii)$T_{SSSP}(n,D)n^{o(1)}$, given access to an SSSP algorithm that takes$T_{SSSP}(n,D)$ rounds in $\mathsf{CONGEST}$. This work builds off the recentresult of [Bernstein, Nanongkai, Wulff-Nilsen, FOCS'22], which gives anear-linear time algorithm for negative-weight SSSP in the sequential setting. Using current state-of-the-art SSSP algorithms yields randomized algorithmsfor negative-weight SSSP with (i) $m^{1+o(1)}$ work and $n^{1/2+o(1)}$ span inthe parallel model, (ii) $(n^{2/5}D^{2/5} + \sqrt{n} + D)n^{o(1)}$ rounds in$\mathsf{CONGEST}$. Our main technical contribution is an efficient reduction for computing alow-diameter decomposition (LDD) of directed graphs to computations of SSSPwith a virtual source. Efficiently computing an LDD has heretofore only beenknown for undirected graphs in both the parallel and distributed models. TheLDD is a crucial step of the algorithm in [Bernstein, Nanongkai, Wulff-Nilsen,FOCS'22], and we think that its applications to other problems in parallel anddistributed models are far from being exhausted.<br

DISPATCH: An Optimally-Competitive Algorithm for Maximum Online Perfect Bipartite Matching with i.i.d. Arrivals

This work presents an optimally-competitive algorithm for the problem of maximum weighted online perfect bipartite matching with i.i.d. arrivals. In this problem, we are given a known set of workers, a distribution over job types, and non-negative utility weights for each pair of worker and job types. At each time step, a job is drawn i.i.d. from the distribution over job types. Upon arrival, the job must be irrevocably assigned to a worker and cannot be dropped. The goal is to maximize the expected sum of utilities after all jobs are assigned. We introduce DISPATCH, a 0.5-competitive, randomized algorithm. We also prove that 0.5-competitive is the best possible. DISPATCH first selects a "preferred worker" and assigns the job to this worker if it is available. The preferred worker is determined based on an optimal solution to a fractional transportation problem. If the preferred worker is not available, DISPATCH randomly selects a worker from the available workers. We show that DISPATCH maintains a uniform distribution over the workers even when the distribution over the job types is non-uniform

Selênio como suplemento para bovinos intoxicados cronicamente por Pteridium sp. no Espirito Santo. 2017.

Pteridiumsp.(samambaia) é uma planta responsável por diversos quadros de intoxicação em animais e seres humanos. Em bovinos, um dos quadros comuns na região sul do Espírito Santo é a hematúria enzoótica bovina (HEB) que não possui tratamento. Assim, o objetivo do presente trabalho foi determinar os efeitos do selênio associado a vitamina E como suplemento em animais intoxicados cronicamente pelo Pteridium sp. Foram selecionados 21 animais intoxicados cronicamente pela planta e com HEB. Os animais foram examinados clinicamente e foi realizada a coleta da urina para a confirmação da hematúria. O delineamento experimental foi feito em quatro grupos divididos ao acaso (controle soro fisiológico; tratamento 1 0,05 mg/Kg do suplemento;tratamento20,10mg/Kgdosuplemento;tratamento30,20mg/Kgdo suplemento). Foi feita a suplementação parenteral, via intramuscular, uma vez por semana, durante 13 semanas. Quinzenalmente os animais foram avaliados clinicamente e foram coletadas amostras de sangue para dosagem do selêniosérico. A análise de selênio foi feita nos momentos inicial, antes da suplementação com selênio (M0), após quatro semanas de tratamento (M4), após oito semanas (M8) e após 12 semanas (M12), pelo método de espectrofotometria de absorção atômica. Utilizou-seaanálisedevariância(ANOVA)seguidadotestedeTukeya5%.Verificou-se que houve maior ganho de peso dos animais tratados com selênio em relação ao grupocontrolee,também,entreosgrupos.Aintensidadedahematúriareduziuapartir da sexta semana e houve diferença significativa entre os grupos tratados e o grupo controle, assim como entre os grupos. Houve diferença significativa da concentração sérica de selênio entre os tratamentos. Assim, conclui-se que o selênio associado a vitaminaEcomosuplementoparabovinosintoxicadoscronicamenteporPteridiumsp. no Espirito Santo com quadro de HEB teve efeito dose dependente sobre a melhora doquadroclínicocausandoreduçãodaintensidadedehematúriaeaumentodoganho de pes

Splaying Preorders and Postorders

Let $T$ be a binary search tree. We prove two results about the behavior of the Splay algorithm (Sleator and Tarjan 1985). Our first result is that inserting keys into an empty binary search tree via splaying in the order of either $T$'s preorder or $T$'s postorder takes linear time. Our proof uses the fact that preorders and postorders are pattern-avoiding: i.e. they contain no subsequences that are order-isomorphic to $(2,3,1)$ and $(3,1,2)$, respectively. Pattern-avoidance implies certain constraints on the manner in which items are inserted. We exploit this structure with a simple potential function that counts inserted nodes lying on access paths to uninserted nodes. Our methods can likely be extended to permutations that avoid more general patterns. Second, if $T'$ is any other binary search tree with the same keys as $T$ and $T$ is weight-balanced (Nievergelt and Reingold 1973), then splaying $T$'s preorder sequence or $T$'s postorder sequence starting from $T'$ takes linear time. To prove this, we demonstrate that preorders and postorders of balanced search trees do not contain many large "jumps" in symmetric order, and exploit this fact by using the dynamic finger theorem (Cole et al. 2000). Both of our results provide further evidence in favor of the elusive "dynamic optimality conjecture.