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We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O (( k +1)log 2 n ) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x -monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O (log 2 n ) time; for a query point x 0 ∈ ℝ, it can report, in O (log n ) time, the point on the lower envelope with x -coordinate x 0 ; and for a query point q ∈ ℝ 2 , it can return all k pseudo-lines lying below q in time O (log n + k log 2 n ). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D , all input arcs intersecting D can be reported in O ( n 1/2+ɛ + k ) time, where k is the output size and ɛ > 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O (log 2 n ) time. 

Rearranging objects on a planar surface arises in a variety of robotic applications, such as product packaging. Using two arms can improve efficiency but introduces new computa- tional challenges. This paper studies the problem structure of object rearrangement using two arms in synchronous, monotone tabletop setups and develops an optimal mixed integer model. It then describes an efficient and scalable algorithm, which first minimizes the cost of object transfers and then of moves between objects. This is motivated by the fact that, asymptotically, object transfers dominate the cost of solutions. Moreover, a lazy strategy minimizes the number of motion planning calls and results in significant speedups. Theoretical arguments support the benefits of using two arms and indicate that synchronous execution, in which the two arms perform together either transfers or moves, introduces only a small overhead. Experiments support these claims and show that the scalable method can quickly compute solutions close to the optimal for the considered setup.