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The beacon model is a recent paradigm for guiding the trajectory of messagesor small robotic agents in complex environments. A beacon is a fixed point withan attraction pull that can move points within a given polygon. Points movegreedily towards a beacon: if unobstructed, they move along a straight line tothe beacon, and otherwise they slide on the edges of the polygon. The Euclideandistance from a moving point to a beacon is monotonically decreasing. A givenbeacon attracts a point if the point eventually reaches the beacon.  The problem of attracting all points within a polygon with a set of beaconscan be viewed as a variation of the art gallery problem. Unlike mostvariations, the beacon attraction has the intriguing property of beingasymmetric, leading to separate definitions of attraction region and inverseattraction region. The attraction region of a beacon is the set of points thatit attracts. It is connected and can be computed in linear time for simplepolygons. By contrast, it is known that the inverse attraction region of apoint---the set of beacon positions that attract it---could have $\Omega(n)$disjoint connected components.  In this paper, we prove that, in spite of this, the total complexity of theinverse attraction region of a point in a simple polygon is linear, and presenta $O(n \log n)$ time algorithm to construct it. This improves upon the bestprevious algorithm which required $O(n^3)$ time and $O(n^2)$ space. Furthermorewe prove a matching $\Omega(n\log n)$ lower bound for this task in thealgebraic computation tree model of computation, even if the polygon ismonotone.

An Optimal Algorithm to Compute the Inverse Beacon Attraction Region