ROMM is a class of Randomized, Oblivious, Multi{phase, Minimal routing algorithms. ROMM routing o ers a potential for improved performance compared to both fully randomized algorithms and deterministic oblivious algorithms, under both light and heavy loads. ROMM routing also offers close to best case performance for many common routing problems. In previous work, these claims were supported by extensive simulations on binary cube networks 30, 31]. Here we present analytical and empirical results for ROMM routing on wormhole routed mesh and torus networks. Our simulations show that ROMM algorithms can perform several representative routing tasks 1.5 to 3 times faster than fully randomized algorithms, for medium{sized networks. Furthermore, ROMM algorithms are always competitive with deterministic, oblivious routing, and in some cases, up to 2 times faster.1 Introduction E cient data motion has been critical in high performance computing as long as computers have been in existence. Massively parallel computers use a sparse interconnection network between processing nodes with local memories. Minimizing the potential for high congestion of communication links is an important goal in the design of routing algorithms and networks. Moreover, the packaging technology introduces constraints on the data motion bandwidth at module boundaries, such as chips and boards.These constraints suggest that preserving locality of reference with careful data allocation and minimizing network load by using minimal algorithms are desirable objectives. However, it is important to note that minimal algorithms and locality preserving data mappings do not always guarantee the best routing times. In fact, for certain problems, non{minimal routing algorithms and data allocations which do not preserve locality can utilize more of the available network bandwidth and cause less communication link congestion, without the corresponding increase in demand for network bandwidth.In this paper, we report on a class of minimal algorithms designed to keep the risk of severe congestion low