Sublinear-Time Distributed Algorithms for Detecting Small Cliques and Even Cycles

In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time. For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n^{5/6}) to O~(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Omega~(sqrt{n}) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant delta in (0,1/2) such that for any k, a Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity. For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(1/k)}) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018]

Large-Scale Distributed Algorithms for Facility Location with Outliers

This paper presents fast, distributed, O(1)-approximation algorithms for metric facility location problems with outliers in the Congested Clique model, Massively Parallel Computation (MPC) model, and in the k-machine model. The paper considers Robust Facility Location and Facility Location with Penalties, two versions of the facility location problem with outliers proposed by Charikar et al. (SODA 2001). The paper also considers two alternatives for specifying the input: the input metric can be provided explicitly (as an n x n matrix distributed among the machines) or implicitly as the shortest path metric of a given edge-weighted graph. The results in the paper are: - Implicit metric: For both problems, O(1)-approximation algorithms running in O(poly(log n)) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model. - Explicit metric: For both problems, O(1)-approximation algorithms running in O(log log log n) rounds in the Congested Clique and the MPC model and O(1)-approximation algorithms running in O~(n/k) rounds in the k-machine model. Our main contribution is to show the existence of Mettu-Plaxton-style O(1)-approximation algorithms for both Facility Location with outlier problems. As shown in our previous work (Berns et al., ICALP 2012, Bandyapadhyay et al., ICDCN 2018) Mettu-Plaxton style algorithms are more easily amenable to being implemented efficiently in distributed and large-scale models of computation

Efficient concurrent data structure access parallelism techniques for increasing scalability

Multi-core processors have revolutionised the way data structures are designed by bringing parallelism to mainstream computing. Key to exploiting hardware parallelism available in multi-core processors are concurrent data structures. However, some concurrent data structure abstractions are inherently sequential and incapable of harnessing the parallelism performance of multi-core processors. Designing and implementing concurrent data structures to harness hardware parallelism is challenging due to the requirement of correctness, efficiency and practicability under various application constraints. In this thesis, our research contribution is towards improving concurrent data structure access parallelism to increase data structure performance. We propose new design frameworks that improve access parallelism of already existing concurrent data structure designs. Also, we propose new concurrent data structure designs with significant performance improvements. To give an insight into the interplay between hardware and concurrent data structure access parallelism, we give a detailed analysis and model the performance scalability with varying parallelism.In the first part of the thesis, we focus on data structure semantic relaxation. By relaxing the semantics of a data structure, a bigger design space, that allows weaker synchronization and more useful parallelism, is unveiled. Investigating new data structure designs, capable of trading semantics for achieving better performance in a monotonic way, is a major challenge in the area. We algorithmically address this challenge in this part of the thesis. We present an efficient, lock-free, concurrent data structure design framework for out-of-order semantic relaxation. We introduce a new two-dimensional algorithmic design, that uses multiple instances of a given data structure to improve access parallelism. In the second part of the thesis, we propose an efficient priority queue that improves access parallelism by reducing the number of synchronization points for each operation. Priority queues are fundamental abstract data types, often used to manage limited resources in parallel systems. Typical proposed parallel priority queue implementations are based on heaps or skip lists. In recent literature, skip lists have been shown to be the most efficient design choice for implementing priority queues. Though numerous intricate implementations of skip list based queues have been proposed in the literature, their performance is constrained by the high number of global atomic updates per operation and the high memory consumption, which are proportional to the number of sub-lists in the queue. In this part of the thesis, we propose an alternative approach for designing lock-free linearizable priority queues, that significantly improve memory efficiency and throughput performance, by reducing the number of global atomic updates and memory consumption as compared to skip-list based queues. To achieve this, our new design combines two structures; a search tree and a linked list, forming what we call a Tree Search List Queue (TSLQueue). Subsequently, we analyse and introduce a model for lock-free concurrent data structure access parallelism. The major impediment to scaling concurrent data structures is memory contention when accessing shared data structure access points, leading to thread serialisation, and hindering parallelism. Aiming to address this challenge, a significant amount of work in the literature has proposed multi-access techniques that improve concurrent data structure parallelism. However, there is little work on analysing and modelling the execution behaviour of concurrent multi-access data structures especially in a shared memory setting. In this part of the thesis, we analyse and model the general execution behaviour of concurrent multi-access data structures in the shared memory setting. We study and analyse the behaviour of the two popular random access patterns: shared (Remote) and exclusive (Local) access, and the behaviour of the two most commonly used atomic primitives for designing lock-free data structures: Compare and Swap, and, Fetch and Add

Byzantine Connectivity Testing in the Congested Clique

We initiate the study of distributed graph algorithms under the presence of Byzantine nodes. We consider the fundamental problem of testing the connectivity of a graph in the congested clique model in a Byzantine setting. We are given a n-vertex (arbitrary) graph G embedded in a n-node congested clique where an arbitrary subset of B nodes of the clique of size up to (1/3-?)n (for any arbitrary small constant ? > 0) can be Byzantine. We consider the full information model where Byzantine nodes can behave arbitrarily, collude with each other, and have unlimited computational power and full knowledge of the states and actions of the honest nodes, including random choices made up to the current round. Our main result is an efficient randomized distributed algorithm that is able to correctly distinguish between two contrasting cases: (1) the graph G? B (i.e., the graph induced by the removal of the vertices assigned to the Byzantine nodes in the clique) is connected or (2) the graph G is far from connected, i.e., it has at least 2|B|+1 connected components. Our algorithm runs in O(polylog n) rounds in the congested clique model and guarantees that all honest nodes will decide on the correct case with high probability. Since Byzantine nodes can lie about the vertices assigned to them, we show that this is essentially the best possible that can be done by any algorithm. Our result can be viewed also in the spirit of property testing, where our algorithm is able to distinguish between two contrasting cases while giving no guarantees if the graph falls in the grey area (i.e., neither of the cases occur). Our work is a step towards robust and secure distributed graph computation that can output meaningful results even in the presence of a large number of faulty or malicious nodes