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Combinatorial Changes of Euclidean Minimum Spanning Tree of Moving Points in the Plane ∗
"... In this paper, we enumerate the number of combinatorial changes of the the Euclidean minimum spanning tree (EMST) of a set of n moving points in 2dimensional space. We assume that the motion of the points in the plane, is defined by algebraic functions of maximum degree s of time. We prove an upper ..."
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In this paper, we enumerate the number of combinatorial changes of the the Euclidean minimum spanning tree (EMST) of a set of n moving points in 2dimensional space. We assume that the motion of the points in the plane, is defined by algebraic functions of maximum degree s of time. We prove an upper bound of O(n 3 β2s(n 2)) for the number of the combinatorial changes of the EMST, where βs(n)=λs(n)/n and λs(n) is the maximum length of DavenportSchinzel sequences of order s on n symbols which is nearly linear in n. This result is an O(n) improvement over the previously trivial bound of O(n 4). 1
Kinetic Data Structures for the SemiYao Graph and All Nearest Neighbors in Rd
"... This paper presents kinetic data structures (KDS’s) for maintaining the SemiYao graph, all the nearest neighbors, and all the (1 + )nearest neighbors of a set of moving points in Rd. Our technique provides the first KDS for the SemiYao graph in Rd. It generalizes and improves on the previous wor ..."
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This paper presents kinetic data structures (KDS’s) for maintaining the SemiYao graph, all the nearest neighbors, and all the (1 + )nearest neighbors of a set of moving points in Rd. Our technique provides the first KDS for the SemiYao graph in Rd. It generalizes and improves on the previous work on maintaining the SemiYao graph in R2. Our KDS for all nearest neighbors is deterministic. The best previous KDS for all nearest neighbors in Rd is randomized. Our structure and analysis are simpler and improves on the previous work. Finally, we provide a KDS for all the (1 + )nearest neighbors, which in fact gives better performance than the exact KDS’s for all nearest neighbors. 1