Brief Summary
This video explains Fibonacci heaps, a data structure that supports efficient merging and key decrease operations. It begins by reviewing binary heaps and introducing binomial heaps as a stepping stone. The discussion covers binomial tree construction, heap properties, and the implementation of operations like adding elements and merging heaps. Finally, the video transitions to Fibonacci heaps, detailing their structure, operations, and amortized analysis to achieve constant-time merging and key decrease, crucial for graph algorithms like Dijkstra's and Prim's.
- Fibonacci heaps support efficient merging and key decrease operations.
- Binomial heaps serve as a foundation for understanding Fibonacci heaps.
- Amortized analysis is used to demonstrate the performance of Fibonacci heap operations.
Intro to Fibonacci Heaps
The lecture introduces Fibonacci heaps as a data structure that, like binary heaps, supports adding elements and removing minimal elements. However, Fibonacci heaps also support merging two heaps into one and decreasing the key of an element already in the heap. Decreasing the key involves taking an element x within the heap and changing its key to a smaller value y, where y < x.
Binary Heaps Recap
Binary heaps allow adding elements, removing the minimum element, and decreasing a key, all in logarithmic time. Merging two binary heaps isn't straightforward, but it can be achieved in O(log² n) time by inserting elements from the smaller heap into the larger one. The presenter mentions that while Fibonacci heaps can be implemented relatively easily, they are often not used in practice due to a large constant factor that makes them less efficient than binary heaps for most problems.
Binomial Heaps
Binomial heaps are built upon binomial trees, contrasting with simple heaps based on binary trees. Binomial trees have specific ranks: a tree of rank 0 consists of a single vertex, while a tree of rank 1 consists of two vertices. A binomial tree of rank k has a root with k children, where the i-th child is a binomial tree of rank i-1. Alternatively, a binomial tree of rank k can be constructed by taking a binomial tree of rank k-1 and adding another child that is also a binomial tree of rank k-1.
Binomial Heap Properties
A heap is built on top of binomial trees by placing elements in each node while maintaining the heap invariant: each element is less than or equal to its children. A binomial tree of rank k contains exactly 2^k nodes. To represent n elements where n is not a power of two, multiple trees of different ranks are used. For example, 11 elements can be represented using trees of ranks 3 (8 elements), 1 (2 elements), and 0 (1 element). The number of trees is minimized to ensure efficiency.
Binomial Heap Operations: Adding Elements
All operations in a binomial heap work in logarithmic time. To add an element, it is initially considered a binomial tree of rank 0. If the binomial heap does not already contain a tree of rank 0, the new element is simply added as a new tree. If a tree of rank 0 exists, the two trees are merged into a tree of rank 1. This merging process continues, combining trees of equal rank until a unique tree is formed. Merging two trees of rank k into a tree of rank k+1 can be done in constant time by linking the root with the larger key as a child of the root with the smaller key.
Binomial Heap Operations: Merging Heaps
Merging two binomial heaps is similar to merging two sorted arrays. Each binomial heap is maintained as a list of trees sorted by rank. The merging process involves iterating through both lists, combining trees of the same rank when encountered. If three trees of the same rank exist temporarily (one from each heap and one resulting from a merge), two of them are merged into a tree of the next higher rank. The time complexity of merging is logarithmic, proportional to the number of trees in both heaps.
Binomial Heap Operations: Removing Minimum Element
To remove the minimum element from a binomial heap, first, the minimum element is located, which is always the root of one of the binomial trees. After removing this root, its children, which are also binomial trees of different ranks, must be reincorporated into the heap. This is done by merging the heap formed by the children with the original heap (minus the removed root). The next minimum element is then found by iterating through the roots of the remaining trees.
Fibonacci Heaps: Introduction and Goals
Fibonacci heaps aim to improve upon binomial heaps by achieving constant time for add and merge operations, as well as decrease key operations. The motivation for optimizing the decrease key operation stems from its importance in graph algorithms like Dijkstra's and Prim's, where decreasing the key is frequently needed when updating the minimum element in a set. The presenter notes that the constant-time complexities are amortized, meaning they are achieved over a series of operations.
Fibonacci Heaps: Basic Operations
In Fibonacci heaps, adding an element is done by creating a new tree of rank zero and adding it to the heap. Merging two Fibonacci heaps involves simply concatenating the lists of trees from both heaps, achieving constant-time merging. A pointer to the minimum element is maintained and updated during add and merge operations.
Fibonacci Heaps: Removing Minimum Element
Removing the minimum element in a Fibonacci heap involves taking the minimal element and removing it from the heap. All its children are added as single trees to the root list of the Fibonacci heap. To maintain balance and facilitate finding the next minimum element, a cleanup procedure is initiated. This procedure involves iterating through all roots, merging trees of the same rank until all trees have different ranks. This cleanup ensures that the number of trees is logarithmic.
Fibonacci Heaps: Amortized Analysis of Remove Minimum
The amortized cost of the cleanup process during the remove minimum operation is analyzed using a potential function. The potential function is defined as the number of trees in the Fibonacci heap. The real time of the cleanup operation is m, the number of trees before the cleanup. After the cleanup, there are at most log n trees. The change in potential is log n - m, resulting in an amortized cost of log n.
Fibonacci Heaps: Decreasing Key Operation
To achieve constant-time decrease key operations, Fibonacci heaps use a different type of tree structure compared to binomial trees. These trees allow a node to lose at most one child. If a key is decreased such that it violates the heap property, the node is detached from its parent and moved to the root list. If the parent already has a child removed (indicated by a mark), it is also detached and moved to the root list. This process continues up the tree until a node without a mark is reached, at which point a mark is added.
Fibonacci Heaps: Tree Properties and Ranks
In Fibonacci heaps, if a tree has rank k, its children have ranks of at least 0, 1, 2, ..., k-1. A tree can also have a mark indicating that it has lost a child. The rank of the root will be no more than c multiplied by log n, for some constant c.
Fibonacci Heaps: Amortized Analysis of Decrease Key
The amortized time complexity of the decrease key operation is analyzed using a potential function defined as the number of trees plus twice the number of marked nodes. When a key is decreased and nodes are moved to the root list, the number of trees increases, but the number of marked nodes decreases. The change in potential is such that the amortized cost of the decrease key operation is constant.
