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1.
Pankaj K. Agarwal Sandeep Sen 《Journal of Algorithms in Cognition, Informatics and Logic》1996,20(3):581-601
Anm×nmatrix
=(ai, j), 1≤i≤mand 1≤j≤n, is called atotally monotonematrix if for alli1, i2, j1, j2, satisfying 1≤i1<i2≤m, 1≤j1<j2≤n.[formula]We present an[formula]time algorithm to select thekth smallest item from anm×ntotally monotone matrix for anyk≤mn. This is the first subquadratic algorithm for selecting an item from a totally monotone matrix. Our method also yields an algorithm of the same time complexity for ageneralized row-selection problemin monotone matrices. Given a setS={p1,…, pn} ofnpoints in convex position and a vectork={k1,…, kn}, we also present anO(n4/3logc n) algorithm to compute thekith nearest neighbor ofpifor everyi≤n; herecis an appropriate constant. This algorithm is considerably faster than the one based on a row-selection algorithm for monotone matrices. If the points ofSare arbitrary, then thekith nearest neighbor ofpi, for alli≤n, can be computed in timeO(n7/5 logc n), which also improves upon the previously best-known result. 相似文献
2.
Wilderotter Klaus 《Journal of Complexity》1995,11(4)
The n-widths of the unit ball Ap of the Hardy space Hp in Lq( −1, 1) are determined asymptotically. It is shown that for 1 ≤ q < p ≤∞ there exist constants k1 and k2 such that [formula]≤ dn(Ap, Lq(−1, 1)),dn(Ap, Lq(−1, 1)), δn(Ap, Lq(−1, 1))[formula]where dn, dn, and δn denote the Kolmogorov, Gel′fand and linear n-widths, respectively. This result is an improvement of estimates previously obtained by Burchard and Höllig and by the author. 相似文献
3.
Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D0(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q0(n) to be the minimum of D0(G) over all graphs G of order n.We prove that for all n we have
log*n−log*log*n−2≤q0(n)≤log*n+22,