Fast deterministic processor allocation

Interval allocation has been suggested as a possible formalization for the PRAM of the (vaguely defined) processor allocation problem, which is of fundamental importance in parallel computing. The interval allocation problem is, given $n$ nonnegative integers $x_1,\ldots,x_n$, to allocate $n$ nonoverlapping subarrays of sizes $x_1,\ldots,x_n$ from within a base array of $O(\sum_{j=1}^n x_j)$ cells. We show that interval allocation problems of size $n$ can be solved in $O((\log\log n)^3)$ time with optimal speedup on a deterministic CRCW PRAM. In addition to a general solution to the processor allocation problem, this implies an improved deterministic algorithm for the problem of approximate summation. For both interval allocation and approximate summation, the fastest previous deterministic algorithms have running times of $\Theta({{\log n}/{\log\log n}})$. We also describe an application to the problem of computing the connected components of an undirected graph