Abstract
Algorithms for range partial sum query on high dimensional integer grids typically focus on orthogonal ranges, which by definition demands fixed right triangles between all adjacent boundary edges. We extend the algorithm to solve 2D homothetic triangular range queries in \(\langle O(N \alpha (N)), O(\alpha ^2 (N)) \rangle \) (\(\langle \)preprocessing bound, query bound\(\rangle \) of both time and space since they are identical.), where N is the total number of grid points and \(\alpha (\cdot )\) is a functional equivalence of the inverse Ackermann function. This asymmetric bound improves over the existing bound for orthogonal ranges. By the property of homotheticity, we mean that the angles between any two adjacent boundaries are arbitrarily fixed constants. The technique and bounds of our work can be extended to even higher dimensional grids.
Y. Tang—Supported in part by the Open Funding (No. CARCH201606).
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Notes
- 1.
For convenience, we use the term “grid point” and “point” interchangeably in the rest of paper.
- 2.
In the RAM model, we can locate a proper memory cell in a recursive divide-and-conquer tree by finding the Lowest Common Ancestor (LCA) of the end vertices i of the query range. The LCA problem can in turn be solved by a \(\pm 1\) RMQ algorithm with linear preprocessing time and O(1) query time [6, 7].
- 3.
“Hyper-zoid” is a trapezoidal analogue in a \((d+1)\)-dimensional grid, where \(d > 1\). The \((d+1)\)-dimensional grid composes of a d-dimensional spatial grid plus a 1-dimensional temporal axis.
- 4.
We define \(f^* (n) = 0\) if \(n \le 1\), otherwise \(f^* (n) = 1 + f^*(f(n))\).
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Tang, Y., Kan, H. (2018). Non-orthogonal Homothetic Range Partial-Sum Query on Integer Grids. In: Chen, J., Lu, P. (eds) Frontiers in Algorithmics. FAW 2018. Lecture Notes in Computer Science(), vol 10823. Springer, Cham. https://doi.org/10.1007/978-3-319-78455-7_21
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