Parallel photon mapping computations to enable fast approximate solutions to the art gallery and watchman route problems

The art gallery and watchman route problems (AGP and WRP) are NP-hard constrained optimization problems concerned with providing static and dynamic sensing, respectively, to environments such that the maximum amount of information is sensed at a minimal cost. What being an NP-hard problem means, practically, is that when an AGP or WRP solution is calculated for a particular time step t, any small change in the environment requires that an entirely new solution must be computed. Extending 3D AGP- and WRP-solving computations into 4D (i.e. considering time´s effects on the solutions generated) means that a large number of computational resources would be consumed if the updates to the AGP and WRP solutions are performed serially - since each time step´s solution would be computed sequentially. Our particular AGP- and WRP-solving algorithms are built upon the photon mapping algorithm in order to model the information obtainable in the sensed environment. The photon mapping algorithm models the propagation of multispectral photons through an environment and stores the result of the photons´ interaction with their environment in a k-d tree data structure called a photon map. Since each virtual photon can operate independently of every other virtual photon, a photon map generated at a particular time step t can be generated independently of every other photon map populated at every other time step using a graphics processing unit (GPU). Thus given an n-sized time sequence, a photon map can be populated by each member of an n-core GPU. Once the photon map is updated, our AGP/WRP-solving algorithms can be executed in parallel over the time sequence using the particular core assigned to a photon map´s population. We present the results of our computations and compare both serial- and GPU-based performance.