Selection in Monotone Matrices and Computing kth Nearest Neighbors
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Cites background from "Selection in Monotone Matrices and ..."
...…distance (kNND) for each individual (idealized as a point) is found by computing the Euclidean distance from that point to all others (or some efficient subset), ranking the distances from low to high, and choosing the distance having the kth rank (Kovalenko 1993; Agarwal & Sen 1996)....
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Cites methods from "Selection in Monotone Matrices and ..."
...First, use the algorithm of Agarwal and Sen [1] to 3nd the j + i=2 th smallest entry in CT , which we call $....
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...We use the result of Agarwal and Sen [1], who show how to 3nd the dth smallest entry in an m × n totally monotone array in O((m+ n) √ n log n) time....
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...We use the algorithm of Agarwal and Sen [1] in our binary search, and for each dth smallest value T in the m×n cost array, we query to see if there exists a perfect matching that satis3es the threshold T ....
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...First, use the algorithm of Agarwal and Sen [1] to 3nd the j + i=2 th smallest entry in C , which we call $....
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Cites methods or result from "Selection in Monotone Matrices and ..."
...We use Agarwal and Sen [ 2 ] to find the � j+i 2 � -th smallest entryin CT , which we call τ . We use our queryalgorithm to test if the graph contains a k-edge 1 � → n path that uses onlyedges whose costs are less than or equal to τ . If the queryanswers “yes,” then we call Binary-Search(CT , i, � j+i 2 � ); otherwise we call Binary-...
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...Proof. We use the result of Agarwal and Sen [ 2 ], who show how to find the d-th smallest entryin an m × n totallymonotone arrayin O((m + n) √ n lg n) time....
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...Theorem 4.3. We use the algorithm of Agarwal and Sen [ 2 ] in our binarysearch, and for each d-th smallest value T in the m×n cost array, we query to see if there exists a perfect matching that satisfies the threshold T . The total running time is O(lg(mn)((m + n) √ n lg n + n)) = O(n 3/2 lg 2 n). Similarly, if the cost array is bitonic, then the running time is O(lg(mn)(m lg n + n)) = O(m lg 2 n + n lg n)....
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References
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"Selection in Monotone Matrices and ..." refers methods in this paper
...Dijkstra [ 15 ] gave a simple O(nlogn) time algorithm for computing the longest monotone subsequence U. Recently, Bar Yehuda and Fogel [11] presented an O(n 3/2) algorithm for partitioning U into a collection of at most 2v/n monotone subsequences....
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"Selection in Monotone Matrices and ..." refers methods in this paper
...We now apply the parametric searching technique, due to Megiddo [ 23 ], to the above data structure for answering queries of the following form: Given a point q E P and a parameter k < n, determine its k th nearest neighbor, ~k(q), in S. The basic idea is the same as described in [2], but we have to modify their technique a little....
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