Problems 20-2: More Fibonacci-heap operations

We wish to augment a Fibonacci heap H to support two new operations
without changing the amortized running time of any other Fibonacci-
heap operations.

(a)The operation FIB-HEAP-CHANGE-KEY(H, x, k) changes the key of node
x to the value k. Give an efficient implementation of FIB-HEAP-CHANGE-
KEY, and analyze the amortized running time of your implementation for
the cases in which k is greater than, less than, or equal to key[x].

(b)Give an efficient implementation of FIB-HEAP-PRUNE(H, r), which
deletes min(r, n[H]) nodes from H. Which nodes are deleted should be
arbitrary. Analyze the amortized running time of your implementation.
(Hint: You may need to modify the data structure and potential
function.)

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