Minced trees, with applications to fault-tolerant VLSI processor arrays

Abstract

We derive here a lower bound on the number of edgesf(c, d) that one must remove from a depth-d complete binary tree in order to partition the tree intoc equal size pieces (to within rounding). We show that for the sequence of integersc _l = _def 3× 2^l

$$f\left( {c_l ,d} \right) \geqslant \frac{{c_l }}{{16}}\left( {d - 2l - {{19} \mathord{\left/ {\vphantom {{19} 6}} \right. \kern-\nulldelimiterspace} 6}} \right)$$

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We then apply this bound to a graph-embedding problem related to the design of fault-tolerant VLSI processor arrays. An earlier study has exhibited a fault-tolerant implementation of arbitrary binary trees, using a particular design strategy. We show here that that implementation is optimal in area consumption (to within constant factors) among designs using that strategy, even when the array to be simulated must have the structure of acomplete binary tree.