Spatial join is an important operation for combining spatial data. Parallelization is essential for improving spatial join performance. However, load imbalance due to data skew limits the scalability of parallel spatial join. There are many work sharing techniques to address this problem in a parallel environment. One of the techniques is to use data and space partitioning and then scheduling the partitions among threads/processes with the goal of minimizing workload differences across threads/processes. However, load imbalance still exists due to differences in join costs of different pairs of input geometries in the partitions.
For the load imbalance problem, we have designed a work stealing spatial join system (WSSJ-DM) on a distributed memory environment. Work stealing is an approach for dynamic load balancing in which an idle processor steals computational tasks from other processors. This is the first work that uses work stealing concept (instead of work sharing) to parallelize spatial join computation on a large compute cluster. We have evaluated the scalability of the system on shared and distributed memory. Our experimental evaluation shows that work stealing is an effective strategy. We compared WSSJ-DM with work sharing implementations of spatial join on a high performance computing environment using partitioned and un-partitioned datasets. Static and dynamic load balancing approaches were used for comparison. We study the effect of memory affinity in work stealing operations involved in spatial join on a multi-core processor.
WSSJ-DM performed spatial join using ST_Intersection on Lakes (8.4M polygons) and Parks (10M polygons) in 30 seconds using 35 compute nodes on a cluster (1260 CPU cores). A work sharing Master-Worker implementation took 160 seconds in contrast.
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Hypergraph Partitioning with Embeddings
HYPERGRAPHS provide the formalism needed to solve
problems consisting of interconnected item sets. Similar
to a traditional graph, the hypergraph has the added
generalization that “hyperedges” may connect any number
of nodes. Domains such as very-large-scale integration for
creating integrated circuits [1], machine learning [2], [3],
[4], parallel algorithms [5], combinatorial scientific computing
[6], and social network analysis [7], [8] all contain significant
and challenging instances of hypergraph problems.
One important problem, Hypergraph partitioning, involves
dividing the nodes of a hypergraph among k similarly-sized
disjoint sets while reducing the number of hyperedges that
span multiple partitions. In the context of load balancing,
this is the problem of dividing logical threads (nodes) that
share data dependencies (hyperedges) among available machines
(partitions) in order to balance the number of threads
per machine and minimize communication overhead. However,
hypergraph partitioning is both NP-Hard to solve [9]
and approximate [10].
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- Award ID(s):
- 1725573
- NSF-PAR ID:
- 10205665
- Date Published:
- Journal Name:
- IEEE Transactions on Knowledge and Data Engineering
- ISSN:
- 1041-4347
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We consider the problem of space-efficiently estimating the number of simplices in a hypergraph stream. This is the most natural hypergraph generalization of the highly-studied problem of estimating the number of triangles in a graph stream. Our input is a k-uniform hypergraph H with n vertices and m hyperedges, each hyperedge being a k-sized subset of vertices. A k-simplex in H is a subhypergraph on k+1 vertices X such that all k+1 possible hyperedges among X exist in H. The goal is to process the hyperedges of H, which arrive in an arbitrary order as a data stream, and compute a good estimate of T_k(H), the number of k-simplices in H. We design a suite of algorithms for this problem. As with triangle-counting in graphs (which is the special case k = 2), sublinear space is achievable but only under a promise of the form T_k(H) ≥ T. Under such a promise, our algorithms use at most four passes and together imply a space bound of O(ε^{-2} log δ^{-1} polylog n ⋅ min{(m^{1+1/k})/T, m/(T^{2/(k+1)})}) for each fixed k ≥ 3, in order to guarantee an estimate within (1±ε)T_k(H) with probability ≥ 1-δ. We also give a simpler 1-pass algorithm that achieves O(ε^{-2} log δ^{-1} log n⋅ (m/T) (Δ_E + Δ_V^{1-1/k})) space, where Δ_E (respectively, Δ_V) denotes the maximum number of k-simplices that share a hyperedge (respectively, a vertex), which generalizes a previous result for the k = 2 case. We complement these algorithmic results with space lower bounds of the form Ω(ε^{-2}), Ω(m^{1+1/k}/T), Ω(m/T^{1-1/k}) and Ω(mΔ_V^{1/k}/T) for multi-pass algorithms and Ω(mΔ_E/T) for 1-pass algorithms, which show that some of the dependencies on parameters in our upper bounds are nearly tight. Our techniques extend and generalize several different ideas previously developed for triangle counting in graphs, using appropriate innovations to handle the more complicated combinatorics of hypergraphs.more » « less
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State-of-the-art hypergraph partitioners follow the multilevel paradigm that constructs multiple levels of progressively coarser hypergraphs that are used to drive cut refinements on each level of the hierarchy. Multilevel partitioners are subject to two limitations: (i) Hypergraph coarsening processes rely on local neighborhood structure without fully considering the global structure of the hypergraph. (ii) Refinement heuristics can stagnate on local minima. In this paper, we describe SpecPart, the first supervised spectral framework that directly tackles these two limitations. SpecPart solves a generalized eigenvalue problem that captures the balanced partitioning objective and global hypergraph structure in a low-dimensional vertex embedding while leveraging initial high-quality solutions from multilevel partitioners as hints. SpecPart further constructs a family of trees from the vertex embedding and partitions them with a tree-sweeping algorithm. Then, a novel overlay of multiple tree-based partitioning solutions, followed by lifting to a coarsened hypergraph, where an ILP partitioning instance is solved to alleviate local stagnation. We have validated SpecPart on multiple sets of benchmarks. Experimental results show that for some benchmarks, our SpecPart can substantially improve the cutsize by more than 50% with respect to the best published solutions obtained with leading partitioners hMETIS and KaHyPar.more » « less
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Abstract It is well known that, whenever divides , the complete ‐uniform hypergraph on vertices can be partitioned into disjoint perfect matchings. Equivalently, the set of ‐subsets of
n ‐set can be partitioned into parallel classes so that each parallel class is a partition of the ‐set. This result is known as Baranyai's theorem, which guarantees the existence ofBaranyai partitions . Unfortunately, the proof of Baranyai's theorem uses network flow arguments, making this result nonexplicit. In particular, there is no known method to produce Baranyai partitions in time and space that scale linearly with the number of hyperedges in the hypergraph. It is desirable for certain applications to have an explicit construction that generates Baranyai partitions in linear time. Such an efficient construction is known for and 3. In this paper, we present an explicit recursive quadrupling construction for and , where . In a follow‐up paper (Part II), the other values of , namely, , will be considered. -
null (Ed.)The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size Õ(n/ε²) such that the weight of every cut in G is preserved to within a (1 ± ε)-factor in G'. The graph G' is referred to as a (1 ± ε)-approximate cut sparsifier of G. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require Ω(n + m) time where n denotes the number of vertices and m denotes the number of hyperedges in the hypergraph. Since m can be exponentially large in n, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in n, independent of the number of edges. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph. Specifically, we design an algorithm that constructs a (1 ± ε)-approximate cut sparsifier of a hypergraph H(V,E) in polynomial time in n, independent of the number of hyperedges, when given access to the hypergraph using the following two queries: 1) given any cut (S, ̄S), return the size |δ_E(S)| (cut value queries); and 2) given any cut (S, ̄S), return a uniformly at random edge crossing the cut (cut edge sample queries). Our algorithm outputs a sparsifier with Õ(n/ε²) edges, which is essentially optimal. We then extend our results to show that cut value and cut edge sample queries can also be used to construct hypergraph spectral sparsifiers in poly(n) time, independent of the number of hyperedges. We complement the algorithmic results above by showing that any algorithm that has access to only one of the above two types of queries can not give a hypergraph cut sparsifier in time that is polynomial in n. Finally, we show that our algorithmic results also hold if we replace the cut edge sample queries with a pair neighbor sample query that for any pair of vertices, returns a random edge incident on them. In contrast, we show that having access only to cut value queries and queries that return a random edge incident on a given single vertex, is not sufficient.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TKDE.2020.3017120
