Covering a Set of Points with a Minimum Number of Lines

We consider the minimum line covering problem: given a set S of n points in the plane, we want to find the smallest number l of straight lines needed to cover all n points in S. We show that this problem can be solved in O(n log l) time if l O(log1n), and that this is optimal in the algebraic computation tree model (we show that the (n log l) lower bound holds for all values of l up to O( n)). Furthermore, a O(log l)-factor approximation can be found within the same O(n log l) time bound if l O( 4 n). For the case when l (log n) we suggest how to improve the time complexity of the exact algorithm by a factor exponential in l.

• Type: Conference
• Year: 2006
• Where: CIAC
• Authors: Magdalene Grantson, Christos Levcopoulos
• Keywords: Algorithms, CIAC 2006, Line Covering Problem, Log L, N Log L