Parallel Algorithms for Geometric Graph Problems (1401.0042v2)

Abstract: We give algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes a $(1+\epsilon)$-approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms). In contrast, for general graphs, achieving the same result for MST (or even connectivity) remains a challenging open problem, despite drawing significant attention in recent years. We develop a general algorithmic framework that, besides MST, also applies to Earth-Mover Distance (EMD) and the transportation cost problem. Our algorithmic framework has implications beyond the MapReduce model. For example it yields a new algorithm for computing EMD cost in the plane in near-linear time, $n^{1+o_\epsilon(1)}$. We note that while recently Sharathkumar and Agarwal developed a near-linear time algorithm for $(1+\epsilon)$-approximating EMD, our algorithm is fundamentally different, and, for example, also solves the transportation (cost) problem, raised as an open question in their work. Furthermore, our algorithm immediately gives a $(1+\epsilon)$-approximation algorithm with $n^{\delta}$ space in the streaming-with-sorting model with $1/\delta^{O(1)}$ passes. As such, it is tempting to conjecture that the parallel models may also constitute a concrete playground in the quest for efficient algorithms for EMD (and other similar problems) in the vanilla streaming model, a well-known open problem.

Parallel Algorithms for Geometric Graph Problems: An Analysis

The paper "Parallel Algorithms for Geometric Graph Problems" represents a significant contribution to the field of algorithmic design for parallel computing architectures, specifically within the context of modern paradigms such as MapReduce. The authors provide algorithms that operate on geometric graph problems with an emphasis on efficiency concerning communication rounds, space complexity, and approximation factors in a parallel computing environment.

Algorithmic Framework and Key Innovations

The paper introduces a Solve-And-Sketch framework, which the authors seamlessly integrate into the design of parallel algorithms for Minimum Spanning Tree (MST) and Earth-Mover Distance (EMD) problems. This framework divides computation into "local computation chunks" organized hierarchically, allowing efficient interaction between data points processed locally and globally.

In the context of geometric graphs in low-dimensional Euclidean spaces, the authors propose a random hierarchical spatial partitioning scheme as the foundation for their algorithms. This spatial partitioning is conducive to scalable parallelization. For MST, an innovative approach enables the calculation of a $(1+\epsilon)$-approximate MST through Kruskal's algorithm-like processes that incorporate local spanning forest construction while employing only "short edges." This ensures computational efficiency by restricting irrelevant edge processing.

Strong Numerical Results and Implications

The results demonstrate notable advances in handling geometric graph problems with bounded doubling dimension via the parallel models described. The algorithms achieve an approximation factor of $(1+\epsilon)$ with space lower bounded by $(\delta^{-1} \log n)^{O(1)}$. From a practical standpoint, this provides substantial improvements over existing methods in both communication efficiency and computational complexity, particularly in cloud computing environments where data locality and minimization of round-based interactions are critical.

Bold Claims and Theoretical Underpinning

The authors make bold assertions regarding the versatility of the proposed framework, which is designed to offer near-linear time solutions for well-known computational challenges in geometric graph domains. They highlight that these frameworks might bridge the gap between a need for efficient geometric graph problem-solving and constraints inherent to real-world distributed systems, paving the way for new investigation domains in streaming models and sub-linear space computations.

Furthermore, they propose a conjecture regarding the potential extension of the streaming-with-sorting model to encompass efficient algorithms for problems like EMD, thereby opening discussions on what might be possible if certain limitations in the streaming models are addressed through parallel computation paradigms.

Future Developments and Speculations

The framework could potentially be adapted to various other geometric problems, suggesting that future research could yield new algorithms that further extend the capabilities of parallel systems in processing spatial data, implementing clustering algorithms, and more. The results encourage speculations around the role of dimension reduction and query complexity in achieving improved scalability and efficacy of these algorithms.

The paper also articulates possibilities for future work in optimizing sketch structures for sub-linear space representation, extension to other graph-based metric problems, and understanding lower bounds in geometric graph approximations.

Overall, "Parallel Algorithms for Geometric Graph Problems" offers robust solutions to longstanding challenges in processing geometric graphs through parallel architectures, presenting an avenue for future exploration in algorithmic design for complex system environments.