Fibonacci Heap - Implementation of Operations

Implementation of Operations

To allow fast deletion and concatenation, the roots of all trees are linked using a circular, doubly linked list. The children of each node are also linked using such a list. For each node, we maintain its number of children and whether the node is marked. Moreover we maintain a pointer to the root containing the minimum key.

Operation find minimum is now trivial because we keep the pointer to the node containing it. It does not change the potential of the heap, therefore both actual and amortized cost is constant. As mentioned above, merge is implemented simply by concatenating the lists of tree roots of the two heaps. This can be done in constant time and the potential does not change, leading again to constant amortized time.

Operation insert works by creating a new heap with one element and doing merge. This takes constant time, and the potential increases by one, because the number of trees increases. The amortized cost is thus still constant.

Operation extract minimum (same as delete minimum) operates in three phases. First we take the root containing the minimum element and remove it. Its children will become roots of new trees. If the number of children was d, it takes time O(d) to process all new roots and the potential increases by d−1. Therefore the amortized running time of this phase is O(d) = O(log n).

However to complete the extract minimum operation, we need to update the pointer to the root with minimum key. Unfortunately there may be up to n roots we need to check. In the second phase we therefore decrease the number of roots by successively linking together roots of the same degree. When two roots u and v have the same degree, we make one of them a child of the other so that the one with the smaller key remains the root. Its degree will increase by one. This is repeated until every root has a different degree. To find trees of the same degree efficiently we use an array of length O(log n) in which we keep a pointer to one root of each degree. When a second root is found of the same degree, the two are linked and the array is updated. The actual running time is O(log n + m) where m is the number of roots at the beginning of the second phase. At the end we will have at most O(log n) roots (because each has a different degree). Therefore the difference in the potential function from before this phase to after it is: O(log n) − m, and the amortized running time is then at most O(log n + m) + O(log n) − m = O(log n). Since we can scale up the units of potential stored at insertion in each node by the constant factor in the O(m) part of the actual cost for this phase.

In the third phase we check each of the remaining roots and find the minimum. This takes O(log n) time and the potential does not change. The overall amortized running time of extract minimum is therefore O(log n).

Operation decrease key will take the node, decrease the key and if the heap property becomes violated (the new key is smaller than the key of the parent), the node is cut from its parent. If the parent is not a root, it is marked. If it has been marked already, it is cut as well and its parent is marked. We continue upwards until we reach either the root or an unmarked node. In the process we create some number, say k, of new trees. Each of these new trees except possibly the first one was marked originally but as a root it will become unmarked. One node can become marked. Therefore the potential decreases by at least k − 2. The actual time to perform the cutting was O(k), therefore the amortized running time is constant.

Finally, operation delete can be implemented simply by decreasing the key of the element to be deleted to minus infinity, thus turning it into the minimum of the whole heap. Then we call extract minimum to remove it. The amortized running time of this operation is O(log n).