Minimum Popping

You are given a double-ended queue $Q$. Initially, it contains elements $Q_1, Q_2, \ldots, Q_M$ in this order. Each of these elements is an integer between $1$ and $N$ (inclusive) and each integer between $1$ and $N$ (inclusive) occurs in the queue at least once. We want to pop some (possibly zero) elements from the front of the queue and some (not necessarily the same number, possibly zero) elem