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Fibonacci Heaps. History. [Fredman and Tarjan, 1986] Ingenious data structure and analysis. Original motivation: improve Dijkstra's shortest path algorithm from O(E log V ) to O(E + V log V ). V insert, V delete-min, E decrease-key. Basic idea. Similar to binomial heaps, but less rigid structure.
5.2 Fibonacci Heaps Frank StajanoThomas Sauerwald Lent 2016 Structure of Fibonacci Heaps Forest of MIN-HEAPs Nodes can be marked (roots are always unmarked) Tree roots are stored in a circular, doubly-linked list Min-Pointer pointing to the smallest element Fibonacci Heap 58 30 10 43 41 41 33 70
Structure of a Fibonacci Heap Fibonacci heaps, like binomial heaps, are made up of trees maintaining the heap property. But these trees need not be binomial trees!
A Fibonacci heap is a collection of trees satisfying the minimum-heap property, that is, the key of a child is always greater than or equal to the key of the parent. This implies that the minimum key is always at the root of one of the trees. Compared with binomial heaps, the structure of a Fibonacci heap is more flexible.
The Fibonacci heap has these runtimes: enqueue: O(1) meld: O(1) find-min: O(1) extract-min: O(log n), amortized. decrease-key: O(1), amortized. Cost of Prim's or Dijkstra's algorithm: = O(n T enq + n T ext + m T dec) = O(n + n log n + m) = O(m + n log n) This is theoretically optimal for a comparison-based
19 Fibonacci Heaps. The Fibonacci heap data structure serves a dual purpose. First, it supports a set of operations that constitutes what is known as a “mergeable heap.” Second, several Fibonacci-heap operations run in constant amortized time, which makes this data structure well suited for applications that invoke these operations frequently.
Definition 13.1: A Fibonacci heap H is a collection of heap-ordered trees that have the following properties: The roots of these trees are kept in a doubly-linked list (the root list of. H), The root of each tree contains the minimum element in that tree (this follows from being a heap-ordered tree),
