We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in ℝ d that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., 2 –polylog( d ) ) fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. In other words, the point-hyperplane incidence graph for such configurations has a large complete bipartite subgraph. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank- d matrix containing at most O (1) distinct entries in each column contains a submatrix of fractional size 2 –polylog( d ) , in which each column is constant. We prove that our conjecture is equivalent to the log-rank conjecture; the crucial ingredient of this proof is a reduction from bounds for parallel k -partitions to bounds for parallel ( k -1)-partitions. We also introduce an (apparent) strengthening of the conjecture, which relaxes the requirements that the sets of hyperplanes be parallel. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density Ω (ε 2 d / d ) in any d -dimensional configuration with incidence density ε, qualitatively matching previous results proved using sophisticated geometric techniques. We also improve an upper-bound construction of Apfelbaum and Sharir [ 2 ], yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is Ω (1/√ d ). Finally, we discuss various constructions (due to others) of products of Boolean matrices which yield configurations with incidence density Ω (1) and complete bipartite subgraph density 2 -Ω (√ d ) , and pose several questions for this special case in the alternative language of extremal set combinatorics. Our framework and results may help shed light on the difficulty of improving Lovett’s Õ(√ rank( f )) bound [ 20 ] for the log-rank conjecture. In particular, any improvement on this bound would imply the first complete bipartite subgraph size bounds for parallel 3-partitioned configurations which beat our generic bounds for unstructured configurations.
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Parallel Nearest Neighbors in Low Dimensions with Batch Updates
We present a set of parallel algorithms for computing exact k-nearest neighbors in low dimensions. Many k-nearest neighbor algorithms use either a kd-tree or the Morton ordering of the point set; our algorithms combine these approaches using a data structure we call the zd-tree. We show that this combination is both theoretically efficient under common assumptions, and fast in practice. For point sets of size n with bounded expansion constant and bounded ratio, the zd-tree can be built in O(n) work with O(n^ε) span for constant ε < 1, and searching for the k-nearest neighbors of a point takes expected O(k log k) time. We benchmark our k-nearest neighbor algorithms against existing parallel k-nearest neighbor algorithms, showing that our implementations are generally faster than the state of the art as well as achieving 75x speedup on 144 hyperthreads. Furthermore, the zd-tree supports parallel batch-dynamic insertions and deletions; to our knowledge, it is the first k-nearest neighbor data structure to support such updates. On point sets with bounded expansion constant and bounded ratio, a batch-dynamic update of size k requires O(k log n/k) work with O(k^ε + polylog(n)) span.
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- NSF-PAR ID:
- 10416273
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- 2022 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX)
- Page Range / eLocation ID:
- 195 - 208
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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