Simultaneous localization and mapping is a fundamental problem in mobile robotics: while a robot navigates in an unknown environment, it must incrementally build a map of its surroundings and localize itself within that map. Traditional approaches to the problem are based upon Kalman filters, but suffer from complexity issues: first, the belief state grows quadratically in the size of the map; and second, the filtering operation can take time quadratic in the size of the map. I have developed a linear-space filter that maintains a tractable approximation of the filtered belief state as a thin junction tree. The junction tree grows under measurement and motion updates and is periodically "thinned" to remain tractable. The time complexity of the filter operation is linear in the size of the map, and further approximations permit constant-time approximate filtering.