Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform a computation on an arbitrary n-node (typically, n ≫ k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity, i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k 2 ) * Division of Mathematical Sciences, Nanyang Technological University, Singapore 637371 & Centre for Quantum Technologies, Singapore 117543. E-mail: hklauck@gmail.com. This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation.† KTH Royal Institute of Technology, Sweden, and University of Vienna, Austria E-mail: danupon@gmail.com. Work done while at ICERM, Brown University, USA, and Nanyang Technological University, Singapore. To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved inÕ(n/k) time (the notationÕ hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run inÕ(n/ √ k) time (for (1 + ǫ)-factor approximation) and inÕ(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

In this work we introduce, both for classical communication complexity and query complexity, a modification of the partition bound introduced by Jain and Klauck [JK10]. We call it the public-coin partition bound. We show that (the logarithm to the base two of) its communication complexity and query complexity versions form, for all relations, a quadratically tight lower bound on the public-coin randomized communication complexity and randomized query complexity respectively.The partition bound introduced by Jain and Klauck [JK10] is known to be one of the strongest lower bound methods in classical communication complexity and query complexity. It is known to be stronger than most other lower bound methods, both in communication complexity and query complexity, except its relationship with the information complexity lower bound method in communication complexity is unknown. It is an interesting open question, in both these settings, as to how tight this lower bound method is. We are not aware, to the best of our knowledge, of any function or relation where this method is asymptotically weaker either for communication complexity or for query complexity.In this work we introduce, both for communication complexity and query complexity, a modification of the partition bound which we call the public-coin partition bound. Analogous to the partition bound, our new bound is also a linear-programming based lower bound method. We show that (the logarithm to the base two of) its communication and query complexity versions continue to form a lower bound on the public-coin communication complexity and randomized query complexity respectively. In addition we show that the square of (the logarithm to the base two of) its communication and query complexity versions form an upper bound on the public-coin communication complexity and randomized query complexity respectively. Also it is easily seen via their linear programs that our new bound is stronger than the partition bound for all relations, both in communication complexity and query complexity.

A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function.Our direct product theorems imply a time-space tradeoff T 2 S = Ω N 3 for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication. * Supported by Canada's NSERC and MITACS, and by DFG grant KL 1470/1. † Supported in part by the EU fifth framework project RESQ, IST-2001-37559. expect a constant error on each instance and hence an exponentially small success probability for the k-vector as a whole. Such a statement is known as a weak direct product theorem:However, even if we give our algorithm roughly kt resources, on average it still has only t resources available per instance. So even here we expect a constant error per instance and an exponentially small success probability overall. Such a statement is known as a strong direct product theorem:Strong direct product theorems, though intuitively very plausible, are generally hard to prove and sometimes not even true. Shaltiel [Sha01] exhibits a general class of examples where strong direct product theorems fail. This applies for instance to query complexity, communication complexity, and circuit complexity. In his examples, success probability is taken under the uniform probability distribution on inputs. The function is chosen such that for most inputs, most of the k instances can be computed quickly and without any error probability. This leaves enough resources to solve the few hard instances with high success probability. Hence for his functions, with T ≈ tk, one can achieve average success probability close to 1.Accordingly, we can only establish direct product theorems in special cases. Examples are Nisan et al.'s [NRS94] strong direct product theorem for "decision forests", Parnafes et al.'s [PRW97] direct product theorem for "forests" of communication protocols, Shaltiel's strong direct product theorems for "fair" decision trees and his discrepancy bound for communication complexity [Sha01]. In the quantum case, Aaronson [Aar04, Theorem 10] established a result for the unordered search problem that lies in between the weak and the strong theorems: every T -query quantum algorithm for searching k marked items among N = kn input bits will have success probability σ ≤ O T 2 /N k .

A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR-function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the disjointness function. Our direct product theorems imply a time-space tradeoff T 2 S = Ω N 3 for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.