An m × n Young tableau is an m × n matrix such that the entries of each row are in sorted order from left to right and the entries of each column are in sorted order from top to bottom. Some of the entries of a Young tableau may be ∞, which we treat as nonexistent elements. Thus, a Young tableau can be used to hold r ≤ mn finite numbers.

a. Draw a 4 × 4 Young tableau containing the elements {9, 16, 3, 2, 4, 8, 5, 14, 12}.

b. Argue that an m × n Young tableau Y is empty if Y [1, 1] = ∞. Argue that Y is full (contains mn elements) if Y [m, n] < ∞.

c. Give an algorithm to implement EXTRACT-MIN on a nonempty m × n Young tableau that runs in O(m + n) time. Your algorithm should use a recursive subroutine that solves an m × n problem by recursively solving either an (m – 1) × n or an m × (n – 1) sub problem. Think about MAX-HEAPIFY. Define T(p),  where p = m + n, to be the maximum running time of EXTRACT-MIN on any m × n Young tableau. Give and solve a recurrence for T(p) that yields the O(m + n) time bound.

d. Show how to insert a new element into a non full m × n Young tableau in O(m + n) time.

e. Using no other sorting method as a subroutine, show how to use an n × n Young tableau to sort n² numbers in O(n³) time.

f. Give an O(m + n)-time algorithm to determine whether a given number is stored in a given m × n Young tableau.