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Half-plane point retrieval queries with independent and dependent geometric uncertainties

This paper addresses a family of geometric half-plane retrieval queries of points in the plane in the presence of geometric uncertainty. The problems include exact and uncertain point sets and half-plane queries defined by an exact or uncertain line whose location uncertainties are independent or dependent and are defined by k real-valued parameters. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We present an efficient O(k2) time and space algorithm for computing the envelope of the LPGUM line that defines the half-plane query. For an exact line and an LPGUM n points set, we present an O(log⁡nk+mk) time query and O(nk) space algorithm, where m is the number of LPGUM points on or above the half-plane line. For a LPGUM line and an exact points set, we present a O(k2+(klog⁡nlog⁡log⁡n)ε+m) time and O(n2log⁡n+kε) space approximation algorithm, where 0<ε≤1 is the desired approximation error. For a LPGUM line and an LPGUM points set, we present two O(k2+(klog⁡nklog⁡log⁡nk)ε+mk) and O(mk2+(klog⁡nklog⁡log⁡nk)ε) time query and O((nk)2log⁡nk+kε) space approximation algorithms for the independent and dependent case, respectively.

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Multi-robot motion planning for unit discs with revolving areas

We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R′ that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm. On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(log⁡nlog⁡log⁡n)-approximation algorithm for this problem.

Open Access