Who Needs Crossings? Hardness of Plane Graph Rigidity

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard. One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete. The majority of the paper is devoted to proving an analog of Kempe\u27s Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well

EvolveGCN: Evolving Graph Convolutional Networks for Dynamic Graphs

Graph representation learning resurges as a trending research subject owing to the widespread use of deep learning for Euclidean data, which inspire various creative designs of neural networks in the non-Euclidean domain, particularly graphs. With the success of these graph neural networks (GNN) in the static setting, we approach further practical scenarios where the graph dynamically evolves. Existing approaches typically resort to node embeddings and use a recurrent neural network (RNN, broadly speaking) to regulate the embeddings and learn the temporal dynamics. These methods require the knowledge of a node in the full time span (including both training and testing) and are less applicable to the frequent change of the node set. In some extreme scenarios, the node sets at different time steps may completely differ. To resolve this challenge, we propose EvolveGCN, which adapts the graph convolutional network (GCN) model along the temporal dimension without resorting to node embeddings. The proposed approach captures the dynamism of the graph sequence through using an RNN to evolve the GCN parameters. Two architectures are considered for the parameter evolution. We evaluate the proposed approach on tasks including link prediction, edge classification, and node classification. The experimental results indicate a generally higher performance of EvolveGCN compared with related approaches. The code is available at \url{https://github.com/IBM/EvolveGCN}.Comment: AAAI 2020. The code is available at https://github.com/IBM/EvolveGC

A work-efficient parallel breadth-first search algorithm (or how to cope with the nondeterminism of reducers

We have developed a multithreaded implementation of breadth-first search (BFS) of a sparse graph using the Cilk++ extensions to C++. Our PBFS program on a single processor runs as quickly as a standard C++ breadth-first search implementation. PBFS achieves high work-efficiency by using a novel implementation of a multiset data structure, called a “bag, ” in place of the FIFO queue usually employed in serial breadth-first search algorithms. For a variety of benchmark input graphs whose diameters are significantly smaller than the number of vertices — a condition met by many real-world graphs — PBFS demonstrates good speedup with the number of processing cores. Since PBFS employs a nonconstant-time “reducer ” — a “hyperobject” feature of Cilk++ — the work inherent in a PBFS execution depends nondeterministically on how the underlying work-stealing scheduler load-balances the computation. We provide a general method for analyzing nondeterministic programs that use reducers. PBFS also is nondeterministic in that it contains benign races which affect its performance but not its correctness. Fixing these races with mutual-exclusion locks slows down PBFS empirically, but it makes the algorithm amenable to analysis. In particular, we show that for a graph G =(V,E) with diameter D and bounded outdegree, this data-race-free version of PBFS algorithm runs in time O((V + E)/P + Dlg3 (V /D)) on P processors, which means that it attains near-perfect linear speedup if P ≪ (V + E)/Dlg3 (V /D)

On-the-Fly Pipeline Parallelism

© 2015 ACM 2329-4949/2015/09-ART17 $15.00 Pipeline parallelism organizes a parallel program as a linear sequence of stages. Each stage processes elements of a data stream, passing each processed data element to the next stage, and then taking on a new element before the subsequent stages have necessarily completed their processing. Pipeline parallelism is used especially in streaming applications that perform video, audio, and digital signal processing. Three out of 13 benchmarks in PARSEC, a popular software benchmark suite designed for shared-memory multiprocessors, can be expressed as pipeline parallelism. Whereas most concurrency platforms that support pipeline parallelism use a “construct-and-run” approach, this article investigates “on-the-fly” pipeline parallelism, where the structure of the pipeline emerges as the program executes rather than being specified a priori. On-the-fly pipeline parallelism allows the number of stages to vary from iteration to iteration and dependencies to be data dependent. We propose simple linguistics for specifying on-the-fly pipeline parallelism and describe a provably efficient scheduling algorithm, the PIPER algorithm, which integrates pipeline parallelism into a work-stealing scheduler, allowing pipeline and fork-join parallelism to be arbitrarily nested. The PIPER algorithm automatically throttles the parallelism, precluding “runaway” pipelines. Given a pipeline computation with T1 work and T∞ span (critical-path length), PIPER executes the computation on P processors in TP ≤ T1/P+ O(T∞ +lg P) expected time. PIPER also limits stack space, ensuring that it does not grow unboundedly with running time. We have incorporated on-the-fly pipeline parallelism into a Cilk-based work-stealing runtime system. Our prototype Cilk-P implementation exploits optimizations such as “lazy enabling” and “dependency folding.” We have ported the three PARSEC benchmarks that exhibit pipeline parallelism to run on Cilk-P. One of these, x264, cannot readily be executed by systems that support only construct-and-run pipeline parallelism. Benchmark results indicate that Cilk-P has low serial overhead and good scalability. On x264, for example, Cilk-P exhibits a speedup of 13.87 over its respective serial counterpart when running on 16 processors

Communication-Efficient Graph Neural Networks with Probabilistic Neighborhood Expansion Analysis and Caching

Training and inference with graph neural networks (GNNs) on massive graphs has been actively studied since the inception of GNNs, owing to the widespread use and success of GNNs in applications such as recommendation systems and financial forensics. This paper is concerned with minibatch training and inference with GNNs that employ node-wise sampling in distributed settings, where the necessary partitioning of vertex features across distributed storage causes feature communication to become a major bottleneck that hampers scalability. To significantly reduce the communication volume without compromising prediction accuracy, we propose a policy for caching data associated with frequently accessed vertices in remote partitions. The proposed policy is based on an analysis of vertex-wise inclusion probabilities (VIP) during multi-hop neighborhood sampling, which may expand the neighborhood far beyond the partition boundaries of the graph. VIP analysis not only enables the elimination of the communication bottleneck, but it also offers a means to organize in-memory data by prioritizing GPU storage for the most frequently accessed vertex features. We present SALIENT++, which extends the prior state-of-the-art SALIENT system to work with partitioned feature data and leverages the VIP-driven caching policy. SALIENT++ retains the local training efficiency and scalability of SALIENT by using a deep pipeline and drastically reducing communication volume while consuming only a fraction of the storage required by SALIENT. We provide experimental results with the Open Graph Benchmark data sets and demonstrate that training a 3-layer GraphSAGE model with SALIENT++ on 8 single-GPU machines is 7.1 faster than with SALIENT on 1 single-GPU machine, and 12.7 faster than with DistDGL on 8 single-GPU machines.Comment: MLSys 2023. Code is available at https://github.com/MITIBMxGraph/SALIENT_plusplu