Question:
Published on: 1 July, 2022

What is a minimum spanning tree ? Describe Huffman’s Algorithm.

A spanning tree of a graph is just a subgraph that contains all the vertices and is a tree. A graph may have many spanning trees; for instance the complete graph on four vertices

Huffman Algorithm

Huffman’s algorithm is a method for building an extended binary tree with a minimum weighted path length from a set of given weights.

Huffman coding is based on the frequency of occurance of a data item (pixel in images). The principle is to use a lower number of bits to encode the data that occurs more frequently. Codes are stored in a Code Book which may be constructed for each image or a set of images. In all cases the code book plus encoded data must be transmitted to enable decoding.

The Huffman algorithm is now briefly summarised:

·       A bottom-up approach

1. Initialization: Put all nodes in an OPEN list, keep it sorted at all times (e.g., ABCDE).

2. Repeat until the OPEN list has only one node left:

(a) From OPEN pick two nodes having the lowest frequencies/probabilities, create a parent node of them.

(b) Assign the sum of the children's frequencies/probabilities to the parent node and insert it into OPEN.

(c) Assign code 0, 1 to the two branches of the tree, and delete the children from OPEN.

Symbol        Count           Code        Subtotal (# of bits)

----------          -----             ---------     --------------------

A                  11                 0                   11

B                   5               100                  15

C                   6            101                    18

D                   5            110                    15

E                   6            111                    18

TOTAL (# of bits): 77

The following points are worth noting about the above algorithm:

Decoding for the above two algorithms is trivial as long as the coding table (the statistics) is sent before the data. (There is a bit overhead for sending this, negligible if the data file is big.)

Unique Prefix Property: no code is a prefix to any other code (all symbols are at the leaf nodes) -> great for decoder, unambiguous.

If prior statistics are available and accurate, then Huffman coding is very good.

In the above example:

Number of bits needed for Huffman Coding is: 77 / 33 = 2.3

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