Are there any interesting algorithms using both a stack and queue (deque) ADT?

The Melkman algorithm (for computing the convex hull of a simple polygonal chain in linear time) uses a double-ended queue (a.k.a deque) to store an incremental hull for the vertices already processed.

Input: a simple polyline W with n vertices V[i]

    Put first 3 vertices onto deque D so that:
    a) 3rd vertex V[2] is at bottom and top of D
    b) on D they form a counterclockwise (ccw) triangle

    While there are more polyline vertices of W to process
    Get the next vertex V[i]
    {
        Note that:
        a) D is the convex hull of already processed vertices
        b) D[bot] = D[top] = the last vertex added to D

        // Test if V[i] is inside D (as a polygon)
        If V[i] is left of D[bot]D[bot+1] and D[top-1]D[top]
            Skip V[i] and Continue with the next vertex

        // Get the tangent to the bottom
        While V[i] is right of D[bot]D[bot+1]
            Remove D[bot] from the bottom of D
        Insert V[i] at the bottom of D

        // Get the tangent to the top
        While V[i] is right of D[top-1]D[top]
            Pop D[top] from the top of D
        Push V[i] onto the top of D
    }

    Output: D = the ccw convex hull of W.

Source: http://softsurfer.com/Archive/algorithm_0203/algorithm_0203.htm

Joe Mitchell: Melkman’s Convex Hull Algorithm (PDF)

This structure is called Deque, that is a queue where elements can be added to or removed from the head or tail. See more at 1.