Consider a complete binary tree where the left and the right sub-trees of the root are max-heaps. The lower bound for the number of operations to convert the tree to a heap is

A binary tree $$T$$ has $$20$$ leaves. The number of nodes in $$T$$ having two children is _______________.

Consider the following transaction involving two bank accounts x and y.

read (x) ; x: = x - 50; write (x); read (y); y: = y + 50;   write (y)

The constraint that the sum of the accounts x and y should remain constant is that of

Consider a simple checkpointing protocol and the following set of operations in the log.

(start, $$T4$$); (write, $$T4, y, 2, 3$$); (start, $$T1$$); (commit, $$T4$$); (write, $$T1, z, 5, 7$$);
(checkpoint);
(start, $$T2$$); (write, $$T2, x, 1, 9$$); (commit, $$T2$$); (start, $$T3$$), (write, $$T3, z, 7, 2$$);

If a crash happens now and the system tries to recovver using both undo and redo operations, what are the contents of the undo list and the redo list?