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### Page 2085 of Mathematical Reviews Vol. , Issue 87d [page]

1987 Mathematical Reviews
D. (1-RTG-C); Kalantari, B. (1-RTG-C) A lower bound to the complexity of Euclidean and rectilinear matching algorithms. Inform. Process. Lett. 22 (1986), no. 2, 73-76.  ...  *, is shown to be bounded below by the infimum of the worst-case time complexities of all algorithms which sort n real numbers.  ...

### Page 4507 of Mathematical Reviews Vol. , Issue 97G [page]

1997 Mathematical Reviews
The algorithm works for both Euclidean and rectilinear metrics.  ...  For a fixed number k, the performance ratio of our algorithm is | + (35c/V3k) for the Euclidean metric and | + (9c/k) for the rectilinear metric.  ...

### Semi-bipartite Graph Visualization for Gene Ontology Networks [chapter]

Kai Xu, Rohan Williams, Seok-Hee Hong, Qing Liu, Ji Zhang
2010 Lecture Notes in Computer Science
We also present a lower bound on the area of rectilinear drawings, and a upper bound on the number of edges.  ...  Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally.  ...

### Page 1315 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews
“If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9—independent of m.  ...  “If the minimum distance to the target is now known in ad- vance, we show a lower bound on the competitive ratio of 1+ 2(k + 1)**!  ...

### Page 5114 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews
Existing related results already attain time complexity matching a natural lower bound, i.e., O(mm).  ...  Our parallel algorithm requires O(logn) time using a total of O(n) operations. The complexity of our algorithm matches that of the algorithm of K. M. McDonald and J. G. Peters [IEEE Trans. Comput.  ...

### Efficient Deployment of Base Stations in Wireless Communication Networks

Zimao Li, Yingying Wang, Maode Ma
2016 Journal of Communications
In this paper, we improve the time complexity of the approximation algorithms and conduct simulations to demonstrate the validness of our improvements.  ...  The problem has been proved to be NP-hard and cannot be approximated in the performance ratio 2 in polynomial time in both Euclidean and rectilinear plane and approximation algorithms in the best possible  ...  For the Euclidean plane, however, the gap between the lower bound and upper bound 2 is still big.  ...

### Topological Stability of Kinetic k-Centers [article]

Ivor van der Hoog, Marc van Kreveld, Wouter Meulemans, Kevin Verbeek, Jules Wulms
2021 arXiv   pre-print
We prove upper and lower bounds on the ratio between the radii of an optimal but unstable solution and the radii of a topologically stable solution – the topological stability ratio – considering various  ...  For k = 2 we provide tight bounds, and for small k > 2 we can obtain nontrivial lower and upper bounds.  ...  Since the lower bound construction is essentially one dimensional, the disks can be interchanged by squares, and thus works for both the Euclidean and the rectilinear case.  ...

### Page 2815 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews
Edge-packing is an efficient strategy to compute a lower bound on the probability.  ...  Then a turn in perspective is made and a nonoptimal process is analyzed: the greedy matching algorithm.  ...

### Page 1278 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews
We provide some explanation for this lack of success by presenting quadratic lower bounds on the certificate complexity of the seemingly difficult problems, in contrast to the known linear certificate  ...  The time complexity of the algorithm is O(k log* n).  ...

### Page 6946 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews
We also prove a lower bound n/2 for the majority function and a lower bound ,/n for the MidBit function.”  ...  Both bounds improve from n@) in [D. A. M. Barrington, R. Beigel and S. Rudich, Comput. Complexity 4 (1994), no. 4, 367-382] to Q(n).  ...

### A Fast Traversal Heuristic and Optimal Algorithm for Effective Environmental Coverage

L. Xu, T. Stentz
2010 Robotics: Science and Systems VI
This approach produces a tractable solution through the use of low complexity algorithms in a branch-and-bound framework.  ...  With a prior map, an optimal path can be computed using a graph to represent the environment and generating the solution using known graph algorithms.  ...  The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory  ...

### A Fast Traversal Heuristic and Optimal Algorithm for Effective Environmental Coverage [chapter]

2011 Robotics
This approach produces a tractable solution through the use of low complexity algorithms in a branch-and-bound framework.  ...  With a prior map, an optimal path can be computed using a graph to represent the environment and generating the solution using known graph algorithms.  ...  The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory  ...

### Lower and Upper Bounds for Tracking Mobile Users [chapter]

Sergei Bespamyatnikh, Binay Bhattacharya, David Kirkpatrick, Michael Segal
2002 Foundations of Information Technology in the Era of Network and Mobile Computing
This paper continues the research started in our joint paper where we present lower bounds and efficient algorithms for exact and approximate maintenance of the 1center for a set of moving points in the  ...  In this paper we prove improved lower and upper bounds for the location of mobile facilities (in Loo and L2 metrics) under the motion of clients when facility moves faster than clients.  ...  We present improved lower and upper bounds for mobile rectilinear and Euclidean 1-center when facility is allowed to move faster than the clients.  ...

### Chapter 7 A survey of computational geometry [chapter]

Joseph S.B. Mitchell, Subhash Suri
1995 Handbooks in Operations Research and Management Science
Both elds seem to have bene ted from each other: combinatorial bounds for certain structures have been obtained by analyzing an algorithm that enumerates them and, conversely, the analysis of algorithms  ...  The topic of path planning is a vast one, with problems ranging from nding shortest paths in a discrete graph to deciding the feasible motion of a complex robot in an environment full of complex obstacles  ...  Arkin et al. also give lower bounds on the number of possible matches and generalize the problem to allow arbitrary piecewise-linear cost functions for the matching.  ...

### The Geometric Maximum Traveling Salesman Problem [article]

Alexander Barvinok, Sandor P. Fekete, David S. Johnson, Arie Tamir, Gerhard J. Woeginger, Russ Woodroofe
2003 arXiv   pre-print
For the special case of two-dimensional metrics with f=4 (which includes the Rectilinear and Sup norms), we present a simple algorithm with O(n) running time.  ...  Thus for example we have O(n^2 log n) algorithms for the cases of points in the plane under the Rectilinear and Sup norms.  ...  Thanks to Henk Meijer, Estie Arkin, Volker Kaibel, Joe Mitchell, Bill Pulleyblank, Mauricio Resende, Peter Shor, and Peter Winkler for helpful discussions, and to an anonymous referee for useful comments  ...
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