On the efficient simulation of networks by hypercube machines
The mapping problem arises whenever a parallel algorithm is implemented on an array of processors. The natural or preferred interconnection network implied by the communication needs of the parallel algorithm must be simulated by some actually available network whose interconnections are different from those of the preferred one. This requires the communication graph of the preferred network be mapped, or embedded into that of the available one. The embedding should keep certain simulation costs small, the most significant being dilation and expansion. Parallel algorithms suitable for hypercube execution are given to embed any mesh in its optimal hypercube, thereby minimizing expansion. It is argued that hypercube embeddings should, if possible, be computed by parallel algorithms, underscoring the significance of the algorithms presented, since existing algorithms appear to be inherently sequential. Except for 3-D meshes, the dilation achieved here is no more than for the sequential algorithms. For 2-D meshes dilation is at most two, which is optimal. For 3-D meshes, dilation, is at most eight, slightly worse than the best sequential algorithms guaranteeing dilation at most seven. For n-D meshes dilation is at most 4n minus 1, marginally improving on the best sequential algorithm guaranteeing dilation at most 4n plus 1. Another class of source networks is also considered: those whose communication graphs are known only to be planar and of vertex degree at most five. It is shown that it is NP-complete to decide if any given graph in this class has a dilation one embedding in its optimal hypercube. This complements a recent result showing the problem is NP-complete when the source is a tree (hence planar) of unbounded degree and the target is a hypercube whose size is given, but not optimal.
- Research Organization:
- Northwestern Univ., Evanston, IL (United States)
- OSTI ID:
- 5919052
- Country of Publication:
- United States
- Language:
- English
Similar Records
Embedding of tree networks into hypercubes
Generalized gray codes and embedding in hypercubes