We explore two ways of incorporating parallelism into priority queues. The first is t o speed up the execution of individual priority operations so that they can be performed one operation per time step, unlike sequential implementations which require O(1og N ) time steps per operation for an N element heap. We give an optimal parallel implementation that uses a linear array of O(1og N ) processors.Second, we consider parallel operations on the priority queue. We show that using a d-dimensional array (constant d ) of P processors we can insert or delete the smallest P elements from a heap in time O(P1ldlog1-'ld P ) , where the number of elements in the heap is assumed to be polynomial in P . We also show a matching lower bound, based on communication complexity arguments, for a range of deterministic implementations. Finally, using randomization, we show that the tame can be reduced to the optimal O ( P 1 l d ) time with high probability.
For an integrable dynamical system with one degree of freedom, "painting" the integral over the phase space proves to be very effective for uncovering the global flow down to minute details. Applied to the main problem in artificial satellite theory, for instance, the technique reveals an intricate configuration of equilibria and bifurcations when the polar component of the angular momentum approaches zero.
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