Top-bottom routing around a rectangle is as easy as computing prefix minima

A new parallel algorithm for the prefix minima problem is presented for inputs drawn from the range of integers [l ...s]. For an input of size n, it runs in O (log log log s) time and O(n) work (which is optimal). A faster algorithm is presented for the special case s = n; it runs in O(log n) time with optimal work. Both algorithms are for the Priority concurrent-read concurrent-write parallel random access machine (CRCW PRAM). A possibly surprising outcome of this work is that, whenever the range of the input is restricted, the prefix minima problem can be solved significantly faster than the Ω (log log n) time lower bound in a decision model of parallel computation, as described by Valiant [SIAM J. Comput., 4 (1975), pp. 348-355]. The top-bottom routing problem, which is an important subproblem of routing wires around a rectangle in two layers, is also considered. It is established that, for parallel (and hence for serial) computation, the problem of top-bottom routing is no harder than the prefix minima problem with s = n, thus giving an O(log n) time optimal parallel algorithm for the top-bottom routing problem. This is one of the first nontrivial problems to be given an optimal parallel algorithm that runs in sublogarithmic time.