In this paper, we explore some basic issues concerning the impact of uncertainty on the complexity of map learning: assimilating sensor data to support spatial queries. The primary computational task involves consolidating a set of measurements to support queries concerning the relative position of identifiable locations. We are concerned with the complexity of providing the best possible answers to such queries. We provide a partial categorization of map learning problems according to the sorts of sensors available and the probability distributions that govern their accuracy. The objective of this exercise is to precisely define a set of computational problems and investigate the properties of algorithms that provide solutions to these problems. We show that hard problems involving uncertain measurements arise even in a single-dimensional setting. We also show that there are interesting two-dimensional map learning problems that can be solved in polynomial time. By considering basic time/space tradeoffs, we provide some insight into exactly what makes map learning hard, and how various sensors might be employed to circumvent combinatorial problems.
"On The Complexity Of Integrating Spatial Measurements", Proc. SPIE 1003, Sensor Fusion: Spatial Reasoning and Scene Interpretation, (5 January 1989); doi: 10.1117/12.948950; https://doi.org/10.1117/12.948950