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CompressionandHuffmanCoding
Dr. CPSC Marina Gavrilova
Computer Science
Unive rs ity o fC a lg a ry C a na d a
Lecture Overview
Huffman Coding Non-determinism of the algorithm Implementations:
Singly-linked
List Doubly-linked list Recursive top-down Using heap
Adaptive Huffman coding
Huffman Coding
Algorithm is used to assign a codework to each character in the text according to their frequencies. The codework is usually represented as a bitstring. Algorithm starts with the set of individual trees, consisting of a single node, sorted in the order of increasing character probabilities. Then two trees with the smallest probabilities are selected and processed so that they become the left and the right sub-tree of the parent node, combining their probabilities. In the end, 0 are assigned to all left branches of the tree, 1 to all right branches, and the codework for all leaves (characters) of the tree is generated.
Non-determinism of the Huffman Coding
Non-determinism of the Huffman Coding
Huffman Algorithm Implementation Linked List
Implementation depends on the ways to represent the priority queue, which requires removing two smallest probabilities and inserting the new probability in the proper positions. The first way to implement the priority queue is the singly linked list of references to trees, which resembles the algorithm presented in the previous slides. The tree with the smallest probability is replaced by the newly created tree. From the trees with the same probability, the first trees encountered are chosen.
Doubly Linked List
All probability nodes are first ordered, the first two trees are always removed. The new tree is inserted at the end of the list in the sorted A order. doubly-linked list of references to trees with immediate access to the beginning and to the end of this list is used.
Doubly Linked-List implementation
Recursive Implementation
Top-down approach for building a tree starting from the highest probability. The root probability is known if lower probabilities, in the roots children, have been determined, the latter are known if the lower probabilities have been computed etc. Thus, the recursive algorithm can be used.
Implementation using Heap
The min-heap of probabilities is built. The highest probability is put in the root. Next, the heap property is restored The smallest probability is removed and the root probability is set to the sum of two smallest probabilities. The processing is complete when there is only one node in the heap left.
Huffman implementation with a heap
Huffman Coding for pairs of characters
Adaptive Huffman Coding
Devised by Robert Gallager and improved by Donald Knuth. Algorithm is based on the sibling property: if each node...