Single Step Undirected Reconfigurable Networks

Constant time algorithms for the transitive closure and some related graph problems on processor arrays with reconngurable bus systems. In this work we have shown a general construction so that a T steps rn-mesh of size n n can be simulated by a single step rn-mesh, where the corresponding mesh size increase by a factor of n O(T). The construction itself is simpler and smaller as compared to a similar result previously obtained for the directed model 5]. A previous work that gave self-simulation of large meshes on smaller ones may be seen as the complement of the current paper 2]. Combining these two results yield a complete scalability framework in terms of a Time{Size Tradeoo for computing with reconngurable meshes. Namely, if more hardware is available so that the mesh is large then computation can be accelerated. In the opposite direction, one may always scale his algorithm to a smaller mesh at the expense of slowing down the algorithm. Interestingly, one consequence of this result is that whenever the deenition of a complexity class for the reconngurable mesh involves polynomial size meshes (as, for example, is the case in 3]), it is insensitive to the precise number of steps required to solve a problem. For speciic problems it makes sense to give tradeoos of the form time c1 size c2 f(n) (where n is the size of the input). In fact, it was previously observed that the rn-mesh follows the Thompson VLSI model, and thus AT 2-type lowerbounds in the Thompson model apply to it 16]. More work is required in this direction, e.g. when the power of broadcasts is restricted to a certain \length" of the bus. Finally, this work leaves a lot of space for further research. One may derive bounds on the minimal number (as a constant) of steps that speciic problems require when the size of the mesh remains xed. Further eeort may nd ways to improve our general construction in terms of the blowup in the size of the mesh. One may want to give explicit single step constructions for known reconngurable algorithms which use reduced size than that required by the general construction. self simulation algorithms for reconng-urable arrays. 16 node v 0 such that the value of v 0 in G is one ii the value of v in G is zero. The size of G is polynomial in the size of …