FibonacciCascadingProof

Today we are going to introduce a new concept called amortized time.

This is about the process of retrieving time consumption and flexible amorizized function positioning.

Here it's time to prove 2 operation bounds:

First comes some basic rules:

1, X为斐波那契堆任一节点，ci is the newing i-th child node, then ci has more than i-2 rank.

Brief verify: the moment ci is attached to X, X previously have the child node c1,c2,...ci-1.Thus ci has more than i-1 child nodes. As it might lose more than one, it has at least i-2 nodes;

2, F0 = 1, F1 = 1, and Fk = Fk-1 + Fk-2, than with any R >=1 node it has more than Fk+1 nodes.

Brief verify:

3, FINALLY COMES THE: any random node has the rank O(logN).

现在到了关键时刻，时间有限，只给出deleteMin操作的界限。

所有树中，具有最小元素的树的rank为R，T是操作前树的棵数，为了执行规定操作，分离所有节点，再进行合并，得到开销

cost = T+R+log(N).

最多可以generate log(N)棵树，在操作过程中，标记节点不会损失，那么位势变化量最多为log(N)-T, 二者相加amortized bound为log(N).

这个关键在于一个探索过程，也就是如何定义amortized function。准确的定义，可以使低功耗操作增加，而使高功耗操作减少。