A Faster Distributed Single-Source Shortest Paths Algorithm

We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $\tilde \Omega (\sqrt{n} + D)$ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $n$ is the number of nodes in the network and $D$ is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs $\tilde O(n^{2/3} D^{1/3} + n^{5/6})$ rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing $\tilde O (\sqrt{n D})$ rounds and the second performing $\tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D)$ rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a $(1 + \epsilon)$-approximation $\tilde O ((\sqrt{n} D^{1/4} + D) / \epsilon)$-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018

Decoherence and Full Counting Statistics in a Mach-Zehnder Interferometer

We investigate the Full Counting Statistics of an electrical Mach-Zehnder interferometer penetrated by an Aharonov-Bohm flux, and in the presence of a classical fluctuating potential. Of interest is the suppression of the Aharonov-Bohm oscillations in the distribution function of the transmitted charge. For a Gaussian fluctuating field we calculate the first three cumulants. The fluctuating potential causes a modulation of the conductance leading in the third cumulant to a term cubic in voltage and to a contribution correlating modulation of current and noise. In the high voltage regime we present an approximation of the generating function.Comment: 10 pages, 6 figure

Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models

We present a method for solving the transshipment problem - also known as uncapacitated minimum cost flow - up to a multiplicative error of $1 + \varepsilon$ in undirected graphs with non-negative edge weights using a tailored gradient descent algorithm. Using $\tilde{O}(\cdot)$ to hide polylogarithmic factors in $n$ (the number of nodes in the graph), our gradient descent algorithm takes $\tilde O(\varepsilon^{-2})$ iterations, and in each iteration it solves an instance of the transshipment problem up to a multiplicative error of $\operatorname{polylog} n$. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a randomized rounding scheme, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining the following results: (1) Broadcast CONGEST model: $(1 + \varepsilon)$-approximate SSSP using $\tilde{O}((\sqrt{n} + D)\varepsilon^{-3})$ rounds, where $D$ is the (hop) diameter of the network. (2) Broadcast congested clique model: $(1 + \varepsilon)$-approximate transshipment and SSSP using $\tilde{O}(\varepsilon^{-2})$ rounds. (3) Multipass streaming model: $(1 + \varepsilon)$-approximate transshipment and SSSP using $\tilde{O}(n)$ space and $\tilde{O}(\varepsilon^{-2})$ passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative edge weights that are polynomially bounded in $n$; for general non-negative weights, running times scale with the logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC 2017. Abstract shortened to fit arXiv's limitation to 1920 character

Faster Cut Sparsification of Weighted Graphs

A cut sparsifier is a reweighted subgraph that maintains the weights of the cuts of the original graph up to a multiplicative factor of $(1\pm\epsilon)$. This paper considers computing cut sparsifiers of weighted graphs of size $O(n\log (n)/\epsilon^2)$. Our algorithm computes such a sparsifier in time $O(m\cdot\min(\alpha(n)\log(m/n),\log (n)))$, both for graphs with polynomially bounded and unbounded integer weights, where $\alpha(\cdot)$ is the functional inverse of Ackermann's function. This improves upon the state of the art by Bencz\'ur and Karger (SICOMP 2015), which takes $O(m\log^2 (n))$ time. For unbounded weights, this directly gives the best known result for cut sparsification. Together with preprocessing by an algorithm of Fung et al. (SICOMP 2019), this also gives the best known result for polynomially-weighted graphs. Consequently, this implies the fastest approximate min-cut algorithm, both for graphs with polynomial and unbounded weights. In particular, we show that it is possible to adapt the state of the art algorithm of Fung et al. for unweighted graphs to weighted graphs, by letting the partial maximum spanning forest (MSF) packing take the place of the Nagamochi-Ibaraki (NI) forest packing. MSF packings have previously been used by Abraham at al. (FOCS 2016) in the dynamic setting, and are defined as follows: an $M$-partial MSF packing of $G$ is a set $\mathcal{F}=\{F_1, \dots, F_M\}$, where $F_i$ is a maximum spanning forest in $G\setminus \bigcup_{j=1}^{i-1}F_j$. Our method for computing (a sufficient estimation of) the MSF packing is the bottleneck in the running time of our sparsification algorithm.Comment: To be presented at the 49th EATCS International Colloquium on Automata, Languages and Programming (ICALP 2022

An Improved Random Shift Algorithm for Spanners and Low Diameter Decompositions

Spanners have been shown to be a powerful tool in graph algorithms. Many spanner constructions use a certain type of clustering at their core, where each cluster has small diameter and there are relatively few spanner edges between clusters. In this paper, we provide a clustering algorithm that, given k ? 2, can be used to compute a spanner of stretch 2k-1 and expected size O(n^{1+1/k}) in k rounds in the CONGEST model. This improves upon the state of the art (by Elkin, and Neiman [TALG\u2719]) by making the bounds on both running time and stretch independent of the random choices of the algorithm, whereas they only hold with high probability in previous results. Spanners are used in certain synchronizers, thus our improvement directly carries over to such synchronizers. Furthermore, for keeping the total number of inter-cluster edges small in low diameter decompositions, our clustering algorithm provides the following guarantees. Given ? ? (0,1], we compute a low diameter decomposition with diameter bound O((log n)/?) such that each edge e ? E is an inter-cluster edge with probability at most ?? w(e) in O((log n)/?) rounds in the CONGEST model. Again, this improves upon the state of the art (by Miller, Peng, and Xu [SPAA\u2713]) by making the bounds on both running time and diameter independent of the random choices of the algorithm, whereas they only hold with high probability in previous results

Fast Algorithms for Energy Games in Special Cases

In this paper, we study algorithms for special cases of energy games, a class of turn-based games on graphs that show up in the quantitative analysis of reactive systems. In an energy game, the vertices of a weighted directed graph belong either to Alice or to Bob. A token is moved to a next vertex by the player controlling its current location, and its energy is changed by the weight of the edge. Given a fixed starting vertex and initial energy, Alice wins the game if the energy of the token remains nonnegative at every moment. If the energy goes below zero at some point, then Bob wins. The problem of determining the winner in an energy game lies in $\mathsf{NP} \cap \mathsf{coNP}$. It is a long standing open problem whether a polynomial time algorithm for this problem exists. We devise new algorithms for three special cases of the problem. The first two results focus on the single-player version, where either Alice or Bob controls the whole game graph. We develop an $\tilde{O}(n^\omega W^\omega)$ time algorithm for a game graph controlled by Alice, by providing a reduction to the All-Pairs Nonnegative Prefix Paths problem (APNP), where $W$ is the maximum weight and $\omega$ is the best exponent for matrix multiplication. Thus we study the APNP problem separately, for which we develop an $\tilde{O}(n^\omega W^\omega)$ time algorithm. For both problems, we improve over the state of the art of $\tilde O(mn)$ for small $W$. For the APNP problem, we also provide a conditional lower bound, which states that there is no $O(n^{3-\epsilon})$ time algorithm for any $\epsilon > 0$, unless the APSP Hypothesis fails. For a game graph controlled by Bob, we obtain a near-linear time algorithm. Regarding our third result, we present a variant of the value iteration algorithm, and we prove that it gives an $O(mn)$ time algorithm for game graphs without negative cycles

Deterministic Incremental APSP with Polylogarithmic Update Time and Stretch

We provide the first deterministic data structure that given a weighted undirected graph undergoing edge insertions, processes each update with polylogarithmic amortized update time and answers queries for the distance between any pair of vertices in the current graph with a polylogarithmic approximation in $O(\log \log n)$ time. Prior to this work, no data structure was known for partially dynamic graphs, i.e., graphs undergoing either edge insertions or deletions, with less than $n^{o(1)}$ update time except for dense graphs, even when allowing randomization against oblivious adversaries or considering only single-source distances

Fast Deterministic Fully Dynamic Distance Approximation

In this paper, we develop deterministic fully dynamic algorithms for computing approximate distances in a graph with worst-case update time guarantees. In particular, we obtain improved dynamic algorithms that, given an unweighted and undirected graph $G=(V,E)$ undergoing edge insertions and deletions, and a parameter $0 < \epsilon \leq 1$, maintain $(1+\epsilon)$-approximations of the $st$-distance between a given pair of nodes $s$ and $t$, the distances from a single source to all nodes ("SSSP"), the distances from multiple sources to all nodes ("MSSP"), or the distances between all nodes ("APSP"). Our main result is a deterministic algorithm for maintaining $(1+\epsilon)$-approximate $st$-distance with worst-case update time $O(n^{1.407})$ (for the current best known bound on the matrix multiplication exponent $\omega$). This even improves upon the fastest known randomized algorithm for this problem. Similar to several other well-studied dynamic problems whose state-of-the-art worst-case update time is $O(n^{1.407})$, this matches a conditional lower bound [BNS, FOCS 2019]. We further give a deterministic algorithm for maintaining $(1+\epsilon)$-approximate single-source distances with worst-case update time $O(n^{1.529})$, which also matches a conditional lower bound. At the core, our approach is to combine algebraic distance maintenance data structures with near-additive emulator constructions. This also leads to novel dynamic algorithms for maintaining $(1+\epsilon, \beta)$-emulators that improve upon the state of the art, which might be of independent interest. Our techniques also lead to improved randomized algorithms for several problems such as exact $st$-distances and diameter approximation.Comment: Changes to the previous version: improved bounds for approximate st distances using new algebraic data structure