## Geometric Applications of a Randomized Optimization Technique (1999)

Venue: | Discrete Comput. Geom |

Citations: | 51 - 6 self |

### BibTeX

@ARTICLE{Chan99geometricapplications,

author = {Timothy M. Chan},

title = {Geometric Applications of a Randomized Optimization Technique},

journal = {Discrete Comput. Geom},

year = {1999},

volume = {22},

pages = {547--567}

}

### Years of Citing Articles

### OpenURL

### Abstract

We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k-point subsets, matching point sets under translation, computing rectilinear p-centers and discrete 1centers, and solving linear programs with k violations. 1 Introduction Consider the classic randomized algorithm for finding the minimum of r numbers minfA[1]; : : : ; A[r]g: Algorithm rand-min 1. randomly pick a permutation hi 1 ; : : : ; i r i of h1; : : : ; ri 2. t /1 3. for k = 1; : : : ; r do 4. if A[i k ] ! t then 5. t / A[i k ] 6. return t By a well-known fact [27, 44], the expected number of times that step 5 is execut...