Simulating a stack using queues

It is well known that a queue can be simulated by two stacks using a constant number of stack operations per queue operation. In this paper we consider the forgotten converse problem of simulating a stack using several queues. We consider several variants of this problem. For the offline variant, we obtain a tight upper and lower bounds for the worst-case number of queue operations needed to simulate a sequence of n stack operations using k queues. For the online variant, when the number of queues k is constant, and n is the maximum number of items in the stack at any given time, we obtain tight Θ(n1/k) upper and lower bounds on the worst-case and amortized number of queue operations needed to simulate one stack operation. When k is allowed to grow with n, we prove an upper bound of O(n1/k + logk n) and a lower bound of on the amortized number of queue operations per stack operation. We also prove an upper bound of O(kn1/k) and a lower bound of Ω(n1/k + logk n) on the worst-case number of queue operations per stack operation. We also show that the specific but interesting sequence of n pushes followed by n pops can be implemented much faster using a total number of only Θ(n logk n) queue operations, for every k ≥ 2, an amortized number of Θ(logk n) queue operations per stack operation, and this bound is tight. On the other hand, we show that the same sequence requires at least Ω(n1/k) queue operations per stack operation in the worst case.