Towards Space Efficient Two-Point Shortest Path Queries in a Polygonal Domain
arxiv(2023)
摘要
We devise a data structure that can answer shortest path queries for two
query points in a polygonal domain P on n vertices. For any ε >
0, the space complexity of the data structure is O(n^10+ε) and
queries can be answered in O(log n) time. Alternatively, we can achieve a
space complexity of O(n^9+ε) by relaxing the query time to
O(log^2 n). This is the first improvement upon a conference paper by Chiang
and Mitchell from 1999. They present a data structure with O(n^11) space
complexity and O(log n) query time. Our main result can be extended to
include a space-time trade-off. Specifically, we devise data structures with
O(n^9+ε/ℓ^4 + O(ε )) space complexity
and O(ℓlog^2 n ) query time, for any integer 1 ≤ℓ≤ n.
Furthermore, we present improved data structures with O(log n) query time
for the special case where we restrict one (or both) of the query points to lie
on the boundary of P. When one of the query points is restricted to lie on
the boundary, and the other query point is unrestricted, the space complexity
becomes O(n^6+ε). When both query points are on the boundary, the
space complexity is decreased further to O(n^4+ε), thereby
improving an earlier result of Bae and Okamoto.
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关键词
space efficient,polygonal
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