 Given a set of points P in the plane, the classic Unit Covering (UC) problem asks to compute the minimum number of unit disks (possibly intersecting) required to cover the points in P, along with a placement of the disks . In this paper, we study an online version of UC in L_2, which we call as Online Unit Covering (OUC)in L_2, or, simply Online Unit Covering (OUC). In OUC, the points arrive online sequentially. Once a point p arrives, a decision needs to be taken - either assign p to a previously placed disk (if possible)or place a new disk to cover p since none of the previously placed disks can cover p. In this setup, once a disk is placed, its placement cannot be altered in future. Also, the points remain at their places forever. We show that  competitive ratio of an optimal deterministic online algorithm for OUC is at least 3 and at most 5. Furthermore, we show that for the input point sequences which are subsets of the infinite square or hexagonal lattices, / \log{\log{d}}), due to Charikar et al. (2004). (III) We obtain lower bounds of 4 and 5 for the } competitive ratio of an optimal deterministic online algorithm for OUC is at least 2.5 and at most^2 and respectively \mathbb{R}^3; the previous best lower bounds were both } 3. The lower bounds derived in this paper are independent of computational complexity.
We revisit the online Unit Covering problem in higher dimensions: Given a set of n points in \mathbb{R . In this paper, we work in \mathbb{R We show that the competitive ratio of every deterministic online algorithm (with an adaptive deterministic adversary) for Unit Covering in \mathbb{R is at least d+1 for every d \geq 1. This greatly improves upon the previous best lower bound, \Omega(\log{d / \log{\log{d}}), due to Charikar et al. (2004). (III) We obtain lower bounds of 4 and 5 for the } competitive ratio of any deterministic algorithm for online Unit Covering in \mathbb{R^2 and respectively \mathbb{R}^3; the previous best lower bounds were both } 3. (IV) When the input points are taken from the square or hexagonal lattices in \mathbb{R
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We revisit the online Unit Covering problem in higher dimensions:
R
points P in the plane, the classic Unit Covering (UC) problem asks to compute the minimum number of unit disks (possibly intersecting) required to cover the points in P, along with a placement of the disks
n points in \mathbb{R
R
study an online version of UC in L_2, which we call as Online Unit Covering (OUC)in L_2, or, simply Online Unit Covering (OUC). In OUC, the points arrive online sequentially. Once a point p arrives, a decision needs to be taken - either assign p to a previously placed disk (if possible)or place a new disk to cover p since none of the previously placed disks can cover p. In this setup, once a disk is placed, its placement cannot be altered in future. Also, the points remain at their places forever.
work in \mathbb{R
A
the
R
an optimal
every
R
for OUC
(with an adaptive deterministic adversary) for Unit Covering in \mathbb{R
R
3 and at most 5. Furthermore, we show that for the input point sequences which are subsets of the infinite square or hexagonal lattices,
d+1 for every d \geq 1. This greatly improves upon the previous best lower bound, \Omega(\log{d
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an optimal deterministic online algorithm for OUC is at least 2.5 and at most
any deterministic algorithm for online Unit Covering in \mathbb{R
R
The lower bounds derived in this paper are independent of computational complexity.
(IV) When the input points are taken from the square or hexagonal lattices in \mathbb{R