2006 Fiscal Year Final Research Report Summary
Solving Computationally Hard Problems Based on Fast Algorithms for Fixed-Parameter Problems
| Project/Area Number |
15300003
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Fundamental theory of informatics
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| Research Institution | JapanAdvanced Institute of Science and Technology |
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Principal Investigator |
ASANO Testuo JapanAdvanced Institute of Science and Technology, 情報科学研究科, 教授 (90113133)
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| Project Period (FY) |
2003 – 2006
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| Keywords | algorithm / computational geometry / fixed-parameter problem / asymptotic analysis / computational complexity |
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| Research Abstract |
The purpose of this research is to establish methodology for solving fixed parameter problems in an efficient way under latest computer environment. For the purpose we mathematically evaluate some aspects of programming which has not been reflected to analysis as just simple programming techniques and then analyze computational performance from a completely different standpoint from the existing ones.
In this year we spent much time for the study of distance trisector curves. Given two points in the plane, it is easy to draw perpendicular bisector, but it is hard to draw two curves equidistant from each other. More exactly, we can approximate points on the curves at any precision, but it is impossible to compute their coordinates exactly without any error. In fact we conjecture that the curves are non-algebraic. In this research we proved that such curves exist and they are unique, mathematically in a constructive manner. We also found many interesting properties of the curves. The resu
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lts were presented at an international symposium STOC, one of the top conference in the world in this area and also published in a top mathematical journal, Advances in Mathematics. It is rather surprising that it is quite simple and fundamental problem while there is no study on the curves. We also applied the idea to Voronoi diagrams, which is one of the most important research topics in computational geometry. In this research we defined various Voronoi diagrams based on criteria on goodness of triangles by generalizing the traditional Voronoi diagrams. More concretely, given a set of line segments in the plane, an angular Voronoi diagram is a partition of the plane into regions by the relation on which line segment gives the smallest visual angle. We have shown that this Voronoi diagram has properties which are quite different from those of the exisiting ones. We also gave an efficient algorithm for finding a point that maximizes the smallest visual angle. The results were presented at an international symposium on Voronoi diagrams We are now preparing journal version of those papers to submit them to international journals. Less
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