The 1-center clustering with outliers problem asks about identifying a prototypical robust statistic that approximates the location of a cluster of points. Given some constant 0<α<1 and n points such that αn of them are in some (unknown) ball of radius r, the goal is to compute a ball of radius
Nearby in the stack
O(r)
that also contains
αn
points. This problem can be formulated with the points in a normed vector space such as
Rd
or in a general metric space. The problem has a simple randomized solution: a randomly selected point is a correct solution with constant probability, and its correctness can be verified in linear time. However, the deterministic complexity of this problem was not known. In this paper, for any
ℓp
vector space, we show an
O(nd)
-time solution with a ball of radius
O(r)
for a fixed
α>21,
and for any normed vector space, we show an
O(nd)
-time solution with a ball of radius
O(r)
when
α>21
as well as an
O(ndlog(k)(n))
-time solution with a ball of radius
O(r)
for all
α>0,k∈N,
where
log(k)(n)
represents the
k
th iterated logarithm, assuming distance computation and vector space operations take
O(d)
time. For an arbitrary metric space, we show for any
C∈N
an
O(n1+1/C)
-time solution that finds a ball of radius
2Cr,
assuming distance computation between any pair of points takes
O(1)
-time. Moreover, this algorithm is optimal for general metric spaces, as we show that for any fixed
α,C,
there is no
o(n1+1/C)
-query and thus no
o(n1+1/C)
-time solution that deterministically finds a ball of radius