Consider an equal number of mobile robotic agents and distinct target locations dispersed in an environment. Each agent has a limited communication range and knowledge of every target's position. We study the following target assignment problem: design a distributed algorithm with which the agents divide the targets among themselves and, simultaneously, move to their unique target. This paper focuses on "dense" environments, where the sum of the communication footprints is larger than the area of the environment. We introduce the class of monotonic algorithms, whose worst-case completion time is lower bounded by the area of the environment. We propose a monotonic algorithm called Grid Assgmt and characterize its asymptotic performance: First, the algorithm is an asymptotically optimal monotonic algorithm for worst-case initial conditions. Second, for uniformly randomly distributed agents and targets the completion time is upper bounded by the diameter of the environment with high probability. Third, if the number of agents exceeds the number of targets by a logarithmic factor, then the completion time is constant with high probability. Our algorithm also solves a sensor based target assignment problem where agents have no initial knowledge of target positions, but acquire them via limited range sensing.