Jeoe Erickson. New lower bounds for convex hull problems in odd dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 19, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....vertices in polynomial time (see Problem 1) and check whether they are simple or not. Bremner, Fukuda, and Marzetta [8] noted that if P is given in V description the problem is polynomial time solvable: enumerate the edges (1 skeleton, see Problem 5) and apply the Lower Bound Theorem. Erickson [19] showed that in the worst case dd=2e1 m log m) sideness queries are required to test whether a polytope is simple. For odd d this matches the upper bound. A sideness query is a question of the following kind: given d 1 points p 0 ; p d in R , does p 0 lie above , below , or on ....

J. Erickson, New lower bounds for convex hull problems in odd dimensions, SIAM J. Comput., 28 (1999), pp. 1198-1214.

....status of the 2 d problem unresolved, our understanding of LP with violations in higher dimensions is even more tentative. A suspected lower bound for the general d dimensional problem is n k ) since detecting affine degeneracies among k points is conjectured to require ) time (see [31] for a proof in a restricted model) and is reducible to our problem for any n 2k 48 . Main results. Table 1 summarizes the previous results and our new results. We focus on the general case, since as we will demonstrate in Section 4, the feasible problem in d dimensions reduces to the ....

J. Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput., 28:1198--1214, 1999.

....to guarantee using computer arithmetic. Related works concern also new kinds of analysis. The intrinsic complexity of the geometric predicates can be studied [14, 3] or restricted models of computation were only a small number of well dened geometric predicates are allowed can be developed [10, 9]. Acknowledgments. The author would like to thank the committee of the sixth Discrete Geometry for Computer Imagery for inviting him to present this work. ....

Jeoe Erickson. New lower bounds for convex hull problems in odd dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 19, 1996.

.... the convex hull in: ffl 4 dimensions in O( n f) log 2 f) time and O(n f) space ( 9] ffl 5 dimensions in O( n f) log 3 f) time ( 1] Erickson showed that, in the worst case, Omega Gamma n dd=2e Gamma1 n log n) time is needed to determine the number of convex hull facets ([5]) For odd dimensions, this matches Chazelle s upper bound. 12 7 Two dimensions revisited This section discusses output sensitive algorithms for the planar convex problem. Kirkpatrick and Seidel proposed an O(n log f) time algorithm, that is both output sensitive and worst case optimal ( 6] ....

J. Erickson. New lower bounds for convex hull problems in odd dimensions. Proceedings of the 12th Annual ACM Symposium on Computational Geometry, pages 1-9, 1996.

....subtractions are allowed. His proof uses eigenvalue estimates for a suitable matrix based on known lower bounds for the combinatorial discrepancy of rectangles. Other papers considering lower bounds for geometric range searching and leading to very interesting geometric problems are [BCP93] and [Eri96] Upper bounds. Let us begin with a simple but nontrivial one dimensional rangesearching example, namely with intervals on the real line. Let P ae IR be an n point set, and let I be the family of all intervals in IR. For the purpose of canonical decompositions of sets of the form P I, I 2 I, we ....

J. Erickson. New lower bounds for convex hull problems in odd dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 1--9, 1996.

No context found.

J. Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput., 28:1198-1214, 1999.

....by Gajentaan and Overmars [11] to provide evidence in support of conjectured# ) lower bounds for several problems. The best known algorithm for 3sum runs in time #(n ) Quadratic lower bounds have been proven for 3sum and a few other 3sum hard problems in restricted models of computation [6, 7, 8], but the strongest lower bound for any of these problems in a general model of computation is # n log n) which follows from results of Ben Or [3] In Section 2, we consider static dihedral rotation queries, which determine whether a given dihedral rotation is feasible or not, without modifying ....

Je# Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput. 28(4):1198--1214, 1999.

....term n 2 hard ; see [BBG94] 2 problems can be reduced to a linear satisfiability problem with more variables. For example, deciding if a set of points in IR d contains d 1pointson a common hyperplane is at least as hard as the linear satisfiability problem for # = t 1 t 2 t d [Eri99] In this paper, we prove lower bounds on the complexity of linear satisfiability problems. We consider these problems under a restriction of the linear decision tree model of computation, called the r linear decision tree model, in which each decision is based on the sign of an arbitrary a#ne ....

....tree model, where every query polynomial has at most r 2 positive coe#cients and at most r 2 negative coe#cients. Our lower bound generalizes Dietzfelbinger s lower bound to arbitrary r linear decision trees. More recent techniques of Erickson and Seidel [ES95] but see also [ES97] and Erickson [Eri99] can be used to prove lower bounds for certain linear satisfiability problems, in a model of computation that allows only direct queries. However, there are still several such problems for which these techniques appear to be inadequate. Our lower bounds should be compared with the following ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput., 28:1198--1214, 1999.

....by either plane. Because of this error, the following results have incorrect proofs in [1] Lemma 3.1, Theorem 3.2, Theorem 3.3, Theorem 4.2, Corollary 4.3, Theorem 5.2, Theorem 5.3, Theorem 5.6, and Corollary 5.7. Correct (and much simpler) proofs of all but one of these results appear in [3]. The only theorem whose proof we cannot correct is Theorem 5.6. Proving an Omega n d 1 )lower bound for arbitrary spherical degeneracies in IR d remains an open problem. A similar error appears in Section 5 of [2] The proofs in that section are also invalid, although similar results are ....

....whose proof we cannot correct is Theorem 5.6. Proving an Omega n d 1 )lower bound for arbitrary spherical degeneracies in IR d remains an open problem. A similar error appears in Section 5 of [2] The proofs in that section are also invalid, although similar results are proven correctly in [3]. Figure 1: Acounterexample to a claim in [1] Acknowledgments Wewould like to thank one of the anonymous referees of the journal version of [2] for demanding a proof of the false claim. ....

J. Erickson. New lower bounds for convex hull problems in odd dimensions. Proc. 12th Ann. ACM Symp. on Computational Geometry, pp. 1--9, 1996. 2

....bounded by either plane. Because of this error, the following results have incorrect proofs in [1] Lemma 3.1, Theorem 3.2, Theorem 3.3, Theorem 4.2, Corollary 4.3, Theorem 5.2, Theorem 5.3, Theorem 5.6, and Corollary 5.7. Correct (and much simpler) proofs of all but one of these results appear in [3]. The only theorem whose proof we cannot correct is Theorem 5.6. Proving an Omega Gamma n d 1 ) lower bound for arbitrary spherical degeneracies in IR d remains an open problem. A similar error appears in Section 5 of [2] The proofs in that section are also invalid, although similar results ....

....proof we cannot correct is Theorem 5.6. Proving an Omega Gamma n d 1 ) lower bound for arbitrary spherical degeneracies in IR d remains an open problem. A similar error appears in Section 5 of [2] The proofs in that section are also invalid, although similar results are proven correctly in [3]. Figure 1: A counterexample to a claim in [1] Acknowledgments We would like to thank one of the anonymous referees of the journal version of [2] for demanding a proof of the false claim. ....

J. Erickson. New lower bounds for convex hull problems in odd dimensions. Proc. 12th Ann. ACM Symp. on Computational Geometry, pp. 1--9, 1996.

....of several of these problems can be reduced to linear satisfiability problem with more variables. For example, deciding if a set of points in IR d contains d 1 points on a common hyperplane is at least as hard as the linear satisfiability problem for OE = t 1 t 2 Delta Delta Delta t d [16]. In this paper, we prove lower bounds on the complexity of linear satisfiability problems. We consider these problems under a restriction of the linear decision tree model of computation, called the r linear decision tree model, in which each decision is based on the sign of an arbitrary affine ....

....tree model, where every query polynomial has most r=2 positive coefficients and at most r=2 negative coefficients. Our lower bound generalizes Dietzfelbinger s lower bound to arbitrary r linear decision trees. More recent techniques of Erickson and Seidel [17] but see also [18] and Erickson [16] can be used to prove lower bounds for certain linear satisfiability problems, in a model of computation that allows only direct queries. However, there are still several such problems for which these techniques appear to be inadequate. Our lower bounds should be compared with the following result ....

[Article contains additional citation context not shown here]

J. Erickson. New lower bounds for convex hull problems in odd dimensions. In Proc. 12th Ann. ACM Symp. Comput. Geom., pages 1--9, 1996.

No context found.

Jeoe Erickson. New lower bounds for convex hull problems in odd dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 19, 1996.

No context found.

J. Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput., 28:1198-1214, 1999.

No context found.

J. Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput., 28:1198-1214, 1999.

No context found.

J. Erickson. New lower bounds for convex hull problems in odd dimensions. SIAM J. Comput., 28:1198-- 1214, 1999.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC