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Temporal graphs are graphs where the edge set can change in each time step, and the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been widely studied. We extend the concept of graph automorphisms from static graphs to temporal graphs and show that symmetries enable faster exploration: We prove that a connected temporal graph with $n$ vertices and orbit number $r$ (i.e., $r$ is the number of automorphism orbits) can be explored in $O(r n^{1+\epsilon})$ time steps, for any fixed $\epsilon>0$. For $r=O(n^c)$ for constant $c<1$, this is a significant improvement over the known tight worst-case bound of $\Theta(n^2)$ time steps for arbitrary connected temporal graphs. We also give two lower bounds for exploration, showing that $\Omega(n \log n)$ time steps are required for some inputs with $r=O(1)$ and that $\Omega(rn)$ time steps are required for some inputs for any $r$ with $1\le r\le n$. The techniques we develop for fast exploration are used to derive the following result for rendezvous in connected temporal graphs: Two agents are placed by an adversary at arbitrary vertices and given full information about the temporal graph, except that they do not have consistent vertex labels. The agents can meet at a common vertex after $O(n^{1+\epsilon})$ time steps, for any $\epsilon>0$. For some connected temporal graphs with constant orbit number we present a complementary lower bound of $\Omega(n\log n)$ time steps. Finally, we give a randomized algorithm to construct a temporal walk $W$ that visits all vertices of a given orbit with probability at least $1-\epsilon$ for any $0<\epsilon<1$ such that $W$ spans $O((n^{5/3}+rn)\log n)$ time steps. The runtime of this algorithm consists of $O(n^{1/3} \log (n/\epsilon))$ linear-time scans of the snapshots that exist in this time span.

A singularly (near) optimal distributed algorithm is one that is (near) optimal in \emph{two} criteria, namely, its time and message complexities. For \emph{synchronous} CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general \emph{asynchronous} CONGEST networks. We present a randomized distributed MST algorithm that, with high probability, computes an MST in \emph{asynchronous} CONGEST networks and takes $\tilde{O}(D^{1+\epsilon} + \sqrt{n})$ time and $\tilde{O}(m)$ messages, where $n$ is the number of nodes, $m$ the number of edges, $D$ is the diameter of the network, and $\epsilon >0$ is an arbitrarily small constant (both time and message bounds hold with high probability). Our algorithm is message optimal (up to a polylog$(n)$ factor) and almost time optimal (except for a $D^{\epsilon}$ factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all $D = O(n^{1-\epsilon})$) and uses $\tilde{O}(m)$ messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the $KT_1$ CONGEST model. A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth $\tilde{O}(D^{1+\epsilon})$ (for an arbitrarily small constant $\epsilon > 0$) in $\tilde{O}(D^{1+\epsilon})$ time and $\tilde{O}(m)$ messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting., Comment: 27 pages, accepted to DISC 2022

We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST. To complement our trade-off lower bound, we present a parameterized family of distributed algorithms that gives an essentially optimal trade-off between the awake complexity and the round complexity., Comment: 29 pages, 1 table, 5 figures, abstract modified to fit arXiv constraints

Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best-known (randomized) MIS algorithms have $O(\log{n})$ round complexity on general graphs [Luby, STOC 1986] (where $n$ is the number of nodes), while the best-known lower bound is $\Omega(\sqrt{\log{n}/\log\log{n}})$ [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the $O(\log{n})$ round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of $O(\log n)$ and also bypassing its fundamental $\Omega(\sqrt{\log{n}/\log\log{n}})$ round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in $O(\log\log{n} )$ awake complexity with high probability (i.e., with probability at least $1 - n^{-1}$). This algorithm has a round complexity of $O((\log^7 n) \log \log n)$. We also show that we can improve the round complexity at the cost of a slight increase in awake complexity, by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability, computes an MIS in $O((\log\log{n})\log^*n)$ awake complexity and $O((\log^3 n) (\log \log n) \log^*n)$ round complexity. Our algorithms work in the CONGEST model where messages of size $O(\log n)$ bits can be sent per edge per round., Comment: Abstract shortened to fit arXiv constraints

This paper concerns designing distributed algorithms that are {\em singularly optimal}, i.e., algorithms that are {\em simultaneously} time and message {\em optimal}, for the fundamental leader election problem in {\em asynchronous} networks. Kutten et al. (JACM 2015) presented a singularly near optimal randomized leader election algorithm for general {\em synchronous} networks that ran in $O(D)$ time and used $O(m \log n)$ messages (where $D$, $m$, and $n$ are the network's diameter, number of edges and number of nodes, respectively) with high probability.\footnote{Throughout, "with high probability" means "with probability at least $1-1/n^c$, for constant $c$."} Both bounds are near optimal (up to a logarithmic factor), since $\Omega(D)$ and $\Omega(m)$ are the respective lower bounds for time and messages for leader election even for synchronous networks and even for (Monte-Carlo) randomized algorithms. On the other hand, for general asynchronous networks, leader election algorithms are only known that are either time or message optimal, but not both. Kutten et al. (DISC 2020) presented a randomized asynchronous leader election algorithm that is singularly near optimal for \emph{complete networks}, but left open the problem for general networks. This paper shows that singularly near optimal (up to polylogarithmic factors) bounds can be achieved for general {\em asynchronous} networks. We present a randomized singularly near optimal leader election algorithm that runs in $O(D + \log^2n)$ time and $O(m\log^2 n)$ messages with high probability. Our result is the first known distributed leader election algorithm for asynchronous networks that is near optimal with respect to both time and message complexity and improves over a long line of results including the classical results of Gallager et al. (ACM TOPLAS, 1983), Peleg (JPDC, 1989), and Awerbuch (STOC 89)., Comment: 22 pages. Accepted to DISC 2021

It was suggested that a programmable matter system (composed of multiple computationally weak mobile particles) should remain connected at all times since otherwise, reconnection is difficult and may be impossible. At the same time, it was not clear that allowing the system to disconnect carried a significant advantage in terms of time complexity. We demonstrate for a fundamental task, that of leader election, an algorithm where the system disconnects and then reconnects automatically in a non-trivial way (particles can move far away from their former neighbors and later reconnect to others). Moreover, the runtime of the temporarily disconnecting deterministic leader election algorithm is linear in the diameter. Hence, the disconnecting -- reconnecting algorithm is as fast as previous randomized algorithms. When comparing to previous deterministic algorithms, we note that some of the previous work assumed weaker schedulers. Still, the runtime of all the previous deterministic algorithms that did not assume special shapes of the particle system (shapes with no holes) was at least quadratic in $n$, where $n$ is the number of particles in the system. (Moreover, the new algorithm is even faster in some parameters than the deterministic algorithms that did assume special initial shapes.) Since leader election is an important module in algorithms for various other tasks, the presented algorithm can be useful for speeding up other algorithms under the assumption of a strong scheduler. This leaves open the question: "can a deterministic algorithm be as fast as the randomized ones also under weaker schedulers?", Comment: PODC 2021

This paper considers the problem of Byzantine dispersion and extends previous work along several parameters. The problem of Byzantine dispersion asks: given $n$ robots, up to $f$ of which are Byzantine, initially placed arbitrarily on an $n$ node anonymous graph, design a terminating algorithm to be run by the robots such that they eventually reach a configuration where each node has at most one non-Byzantine robot on it. Previous work solved this problem for rings and tolerated up to $n-1$ Byzantine robots. In this paper, we investigate the problem on more general graphs. We first develop an algorithm that tolerates up to $n-1$ Byzantine robots and works for a more general class of graphs. We then develop an algorithm that works for any graph but tolerates a lesser number of Byzantine robots. We subsequently turn our focus to the strength of the Byzantine robots. Previous work considers only ``weak" Byzantine robots that cannot fake their IDs. We develop an algorithm that solves the problem when Byzantine robots are not weak and can fake IDs. Finally, we study the situation where the number of the robots is not $n$ but some $k$. We show that in such a scenario, the number of Byzantine robots that can be tolerated is severely restricted. Specifically, we show that it is impossible to deterministically solve Byzantine dispersion when $\lceil k/n \rceil > \lceil (k-f)/n \rceil$.

This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in networks. Our main result is a randomized distributed leader election algorithm for asynchronous complete networks that is essentially (up to a polylogarithmic factor) singularly optimal. Our algorithm uses $O(n)$ messages with high probability and runs in $O(\log^2 n)$ time (with high probability) to elect a unique leader. The $O(n)$ message complexity should be contrasted with the $\Omega(n \log n)$ lower bounds for the deterministic message complexity of leader election algorithms (regardless of time), proven by Korach, Moran, and Zaks (TCS, 1989) for asynchronous algorithms and by Afek and Gafni (SIAM J. Comput., 1991) for synchronous networks. Hence, our result also separates the message complexities of randomized and deterministic leader election. More importantly, our (randomized) time complexity of $O(\log^2 n)$ for obtaining the optimal $O(n)$ message complexity is significantly smaller than the long-standing $\tilde{\Theta}(n)$ time complexity obtained by Afek and Gafni and by Singh (SIAM J. Comput., 1997) for message optimal (deterministic) election in asynchronous networks. In synchronous complete networks, Afek and Gafni showed an essentially singularly optimal deterministic algorithm with $O(\log n)$ time and $O(n \log n)$ messages. Ramanathan et al. (Distrib. Comput. 2007) used randomization to improve the message complexity, and showed a randomized algorithm with $O(n)$ messages and $O(\log n)$ time (with failure probability $O(1 / \log^{\Omega(1)}n)$). Our second result is a tightly singularly optimal randomized algorithm, with $O(1)$ time and $O(n)$ messages, for this setting, whose time bound holds with certainty and message bound holds with high probability., Comment: 24 pages. Full version of paper accepted at DISC 2020

The problem of dispersion of mobile robots on a graph asks that $n$ robots initially placed arbitrarily on the nodes of an $n$-node anonymous graph, autonomously move to reach a final configuration where exactly each node has at most one robot on it. This problem is of significant interest due to its relationship to other fundamental robot coordination problems, such as exploration, scattering, load balancing, relocation of self-driving electric cars to recharge stations, etc. The robots have unique IDs, typically in the range $[1,poly(n)]$ and limited memory, whereas the graph is anonymous, i.e., the nodes do not have identifiers. The objective is to simultaneously minimize two performance metrics: (i) time to achieve dispersion and (ii) memory requirement at each robot. This problem has been relatively well-studied when robots are non-faulty. In this paper, we introduce the notion of Byzantine faults to this problem, i.e., we formalize the problem of dispersion in the presence of up to $f$ Byzantine robots. We then study the problem on a ring while simultaneously optimizing the time complexity of algorithms and the memory requirement per robot. Specifically, we design deterministic algorithms that attempt to match the time lower bound ($\Omega(n)$ rounds) and memory lower bound ($\Omega(\log n)$ bits per robot). Our main result is a deterministic algorithm that is both time and memory optimal, i.e., $O(n)$ rounds and $O(\log n)$ bits of memory required per robot, subject to certain constraints. We subsequently provide results that require less assumptions but are either only time or memory optimal but not both. We also provide a primitive, utilized often, that takes robots initially gathered at a node of the ring and disperses them in a time and memory optimal manner without additional assumptions required., Comment: 20 pages

The graph exploration problem requires a group of mobile robots, initially placed arbitrarily on the nodes of a graph, to work collaboratively to explore the graph such that each node is eventually visited by at least one robot. One important requirement of exploration is the {\em termination} condition, i.e., the robots must know that exploration is completed. The problem of live exploration of a dynamic ring using mobile robots was recently introduced in [Di Luna et al., ICDCS 2016]. In it, they proposed multiple algorithms to solve exploration in fully synchronous and semi-synchronous settings with various guarantees when $2$ robots were involved. They also provided guarantees that with certain assumptions, exploration of the ring using two robots was impossible. An important question left open was how the presence of $3$ robots would affect the results. In this paper, we try to settle this question in a fully synchronous setting and also show how to extend our results to a semi-synchronous setting. In particular, we present algorithms for exploration with explicit termination using $3$ robots in conjunction with either (i) unique IDs of the robots and edge crossing detection capability (i.e., two robots moving in opposite directions through an edge in the same round can detect each other), or (ii) access to randomness. The time complexity of our deterministic algorithm is asymptotically optimal. We also provide complementary impossibility results showing that there does not exist any explicit termination algorithm for $2$ robots. The theoretical analysis and comprehensive simulations of our algorithm show the effectiveness and efficiency of the algorithm in dynamic rings. We also present an algorithm to achieve exploration with partial termination using $3$ robots in the semi-synchronous setting., Comment: 13 pages

Addressing a fundamental problem in programmable matter, we present the first deterministic algorithm to elect a unique leader in a system of connected amoebots assuming only that amoebots are initially contracted. Previous algorithms either used randomization, made various assumptions (shapes with no holes, or known shared chirality), or elected several co-leaders in some cases. Some of the building blocks we introduce in constructing the algorithm are of interest by themselves, especially the procedure we present for reaching common chirality among the amoebots. Given the leader election and the chirality agreement building block, it is known that various tasks in programmable matter can be performed or improved. The main idea of the new algorithm is the usage of the ability of the amoebots to move, which previous leader election algorithms have not used., Comment: 33 pages, 16 figures, to appear in the proceedings of ICALP 2019

We consider cooperation among insects, modeled as cooperation between mobile robots on a graph. Within this setting, we consider the problem of mobile robot dispersion on graphs. The study of mobile robots on a graph is an interesting paradigm with many interesting problems and applications. The problem of dispersion in this context, introduced by Augustine and Moses Jr., asks that $n$ robots, initially placed arbitrarily on an $n$ node graph, work together to quickly reach a configuration with exactly one robot at each node. Previous work on this problem has looked at the trade-off between the time to achieve dispersion and the amount of memory required by each robot. However, the trade-off was analyzed for \textit{deterministic algorithms} and the minimum memory required to achieve dispersion was found to be $\Omega(\log n)$ bits at each robot. In this paper, we show that by harnessing the power of \textit{randomness}, one can achieve dispersion with $O(\log \Delta)$ bits of memory at each robot, where $\Delta$ is the maximum degree of the graph. Furthermore, we show a matching lower bound of $\Omega(\log \Delta)$ bits for any \textit{randomized algorithm} to solve dispersion. We further extend the problem to a general $k$-dispersion problem where $k> n$ robots need to disperse over $n$ nodes such that at most $\lceil k/n \rceil$ robots are at each node in the final configuration., Comment: 20 pages, 1 table. Accepted at TAMC 2019: Theory & Applications of Models of Computation. The final authenticated version is available online at https://doi.org/10.1007/978-3-030-14812-6_30

In this work, we study the problem of dispersion of mobile robots on dynamic rings. The problem of dispersion of $n$ robots on an $n$ node graph, introduced by Augustine and Moses Jr. [1], requires robots to coordinate with each other and reach a configuration where exactly one robot is present on each node. This problem has real world applications and applies whenever we want to minimize the total cost of $n$ agents sharing $n$ resources, located at various places, subject to the constraint that the cost of an agent moving to a different resource is comparatively much smaller than the cost of multiple agents sharing a resource (e.g. smart electric cars sharing recharge stations). The study of this problem also provides indirect benefits to the study of scattering on graphs, the study of exploration by mobile robots, and the study of load balancing on graphs. We solve the problem of dispersion in the presence of two types of dynamism in the underlying graph: (i) vertex permutation and (ii) 1-interval connectivity. We introduce the notion of vertex permutation dynamism and have it mean that for a given set of nodes, in every round, the adversary ensures a ring structure is maintained, but the connections between the nodes may change. We use the idea of 1-interval connectivity from Di Luna et al. [10], where for a given ring, in each round, the adversary chooses at most one edge to remove. We assume robots have full visibility and present asymptotically time optimal algorithms to achieve dispersion in the presence of both types of dynamism when robots have chirality. When robots do not have chirality, we present asymptotically time optimal algorithms to achieve dispersion subject to certain constraints. Finally, we provide impossibility results for dispersion when robots have no visibility., Comment: 21 pages, 10 figures, concise version of paper to appear in ICDCN 2018

We introduce a new problem in the domain of mobile robots, which we term dispersion. In this problem, $n$ robots are placed in an $n$ node graph arbitrarily and must coordinate with each other to reach a final configuration such that exactly one robot is at each node. We study this problem through the lenses of minimizing the memory required by each robot and of minimizing the number of rounds required to achieve dispersion. Dispersion is of interest due to its relationship to the problems of scattering on a graph, exploration using mobile robots, and load balancing on a graph. Additionally, dispersion has an immediate real world application due to its relationship to the problem of recharging electric cars, as each car can be considered a robot and recharging stations and the roads connecting them nodes and edges of a graph respectively. Since recharging is a costly affair relative to traveling, we want to distribute these cars amongst the various available recharge points where communication should be limited to car-to-car interactions. We provide lower bounds on both the memory required for robots to achieve dispersion and the minimum running time to achieve dispersion on any type of graph. We then analyze the trade-offs between time and memory for various types of graphs. We provide time optimal and memory optimal algorithms for several types of graphs and show the power of a little memory in terms of running time., Comment: 18 pages, conference version appeared in ICDCN 2018

For a given network, a backbone is an overlay network consisting of a connected dominating set with additional accessibility properties. Once a backbone is created for a network, it can be utilized for fast communication amongst the nodes of the network. The Signal-to-Interference-plus-Noise-Ratio (SINR) model has become the standard for modeling communication among devices in wireless networks. For this model, the community has pondered what the most realistic solutions for communication problems in wireless networks would look like. Such solutions would have the characteristic that they would make the least number of assumptions about the availability of information about the participating nodes. Solving problems when nothing at all is known about the network and having nodes just start participating would be ideal. However, this is quite challenging and most likely not feasible. The pragmatic approach is then to make meaningful assumptions about the available information and present efficient solutions based on this information. We present a solution for creation of backbone in the SINR model, when nodes do not have access to their physical coordinates or the coordinates of other nodes in the network. This restriction models the deployment of nodes in various situations for sensing hurricanes, cyclones, and so on, where only information about nodes prior to their deployment may be known but not their actual locations post deployment. We assume that nodes have access to knowledge of their label, the labels of nodes within their neighborhood, the range from which labels are taken $[N]$ and the total number of participating nodes $n$. We also assume that nodes wake up spontaneously. We present an efficient deterministic protocol to create a backbone with a round complexity of $O(\Delta \lg^2 N)$., Comment: 12 pages, 1 table

Much work has been developed for studying the classical broadcasting problem in the SINR (Signal-to-Interference-plus-Noise-Ratio) model for wireless device transmission. The setting typically studied is when all radio nodes transmit a signal of the same strength. This work studies the challenging problem of devising a distributed algorithm for multi-broadcasting, assuming a subset of nodes are initially awake, for the SINR model when each device only has access to knowledge about the total number of nodes in the network $n$, the range from which each node's label is taken $\lbrace 1,\dots,N \rbrace$, and the label of the device itself. Specifically, we assume no knowledge of the physical coordinates of devices and also no knowledge of the neighborhood of each node. We present a deterministic protocol for this problem in $O(n \lg N \lg n)$ rounds. There is no known polynomial time deterministic algorithm in literature for this setting, and it remains the principle open problem in this domain. A lower bound of $\Omega(n \lg N)$ rounds is known for deterministic broadcasting without local knowledge. In addition to the above result, we present algorithms to achieve multi-broadcast in $O(n \lg N)$ rounds and create a backbone in $O(n \lg N)$ rounds, assuming that all nodes are initially awake. For a given backbone, messages can be exchanged between every pair of connected nodes in the backbone in $O(\lg N)$ rounds and between any node and its designated contact node in the backbone in $O(\Delta \lg N)$ rounds., Comment: This is the author version of the paper which will appear in the Journal of Computer and System Sciences. 36 pages, 1 table, 4 figures; v3 improves the presentation, style, and some technical matter of the paper

Load balancing is a well-studied problem, with balls-in-bins being the primary framework. The greedy algorithm $\mathsf{Greedy}[d]$ of Azar et al. places each ball by probing $d > 1$ random bins and placing the ball in the least loaded of them. With high probability, the maximum load under $\mathsf{Greedy}[d]$ is exponentially lower than the result when balls are placed uniformly randomly. V\"ocking showed that a slightly asymmetric variant, $\mathsf{Left}[d]$, provides a further significant improvement. However, this improvement comes at an additional computational cost of imposing structure on the bins. Here, we present a fully decentralized and easy-to-implement algorithm called $\mathsf{FirstDiff}[d]$ that combines the simplicity of $\mathsf{Greedy}[d]$ and the improved balance of $\mathsf{Left}[d]$. The key idea in $\mathsf{FirstDiff}[d]$ is to probe until a different bin size from the first observation is located, then place the ball. Although the number of probes could be quite large for some of the balls, we show that $\mathsf{FirstDiff}[d]$ requires only at most $d$ probes on average per ball (in both the standard and the heavily-loaded settings). Thus the number of probes is no greater than either that of $\mathsf{Greedy}[d]$ or $\mathsf{Left}[d]$. More importantly, we show that $\mathsf{FirstDiff}[d]$ closely matches the improved maximum load ensured by $\mathsf{Left}[d]$ in both the standard and heavily-loaded settings. We further provide a tight lower bound on the maximum load up to $O(\log \log \log n)$ terms. We additionally give experimental data that $\mathsf{FirstDiff}[d]$ is indeed as good as $\mathsf{Left}[d]$, if not better, in practice., Comment: 26 pages, preliminary version accepted at SODA 2016

Motivated by the increasing need for fast processing of large-scale graphs, we study a number of fundamental graph problems in a message-passing model for distributed computing, called $k$-machine model, where we have $k$ machines that jointly perform computations on $n$-node graphs. The graph is assumed to be partitioned in a balanced fashion among the $k$ machines, a common implementation in many real-world systems. Communication is point-to-point via bandwidth-constrained links, and the goal is to minimize the round complexity, i.e., the number of communication rounds required to finish a computation. We present a generic methodology that allows to obtain efficient algorithms in the $k$-machine model using distributed algorithms for the classical CONGEST model of distributed computing. Using this methodology, we obtain algorithms for various fundamental graph problems such as connectivity, minimum spanning trees, shortest paths, maximal independent sets, and finding subgraphs, showing that many of these problems can be solved in $\tilde{O}(n/k)$ rounds; this shows that one can achieve speedup nearly linear in $k$. To complement our upper bounds, we present lower bounds on the round complexity that quantify the fundamental limitations of solving graph problems distributively. We first show a lower bound of $\Omega(n/k)$ rounds for computing a spanning tree of the input graph. This result implies the same bound for other fundamental problems such as computing a minimum spanning tree, breadth-first tree, or shortest paths tree. We also show a $\tilde \Omega(n/k^2)$ lower bound for connectivity, spanning tree verification and other related problems. The latter lower bounds follow from the development and application of novel results in a random-partition variant of the classical communication complexity model., Comment: Preliminary version appeared at SODA 2015