# Binary tree questions geeksforgeeks

In the field of data compressionShannon—Fano codingnamed after Claude Shannon and Robert Fanois a technique for constructing a prefix code based on a set of symbols and their probabilities estimated or measured. It is suboptimal in the sense that it does not achieve the lowest possible expected code word length like Huffman coding.

The technique was proposed in Shannon's " A Mathematical Theory of Communication ", his article introducing the field of information theory. The method was attributed to Fano, who later published it as a technical report. In Shannon—Fano coding, the symbols are arranged in order from most probable to least probable, and then divided into two sets whose total probabilities are as close as possible to being equal.

Binary tree questions geeksforgeeks symbols then have the first digits of their codes assigned; symbols in the first set receive "0" and symbols in the second set receive "1". As long as any sets with more than one member remain, the same process is repeated on those sets, to determine successive digits of their codes.

When a set has been reduced to one symbol this means the symbol's code is binary tree questions geeksforgeeks and will not form the prefix of any other symbol's code.

The algorithm produces fairly efficient variable-length encodings; when the two smaller sets produced by a partitioning are in fact of equal probability, the one bit of information used to distinguish them is used most efficiently. For this reason, Shannon—Fano is almost never used; Huffman coding is almost as computationally simple and produces prefix codes that always achieve the lowest expected code word length, under the constraints that each symbol is represented by a code formed of an integral number of bits.

This is a constraint binary tree questions geeksforgeeks is often unneeded, since the codes will be packed end-to-end in long sequences. If we consider groups of codes at a time, symbol-by-symbol Huffman coding is only optimal if the probabilities of the symbols are independent and are some power of a half, i.

In most situations, arithmetic coding can produce greater overall compression than either Huffman or Shannon—Fano, since it can encode in fractional numbers of bits which more closely approximate the actual information content of the symbol.

However, arithmetic coding has binary tree questions geeksforgeeks superseded Huffman the way that Huffman supersedes Shannon—Fano, both because arithmetic coding is more computationally expensive and because it is covered by multiple patents. A Shannon—Fano tree is built according to a specification designed to define an effective code table.

The actual algorithm is simple:. The example shows the construction of the Shannon code for a small alphabet. The five symbols which can be coded have the following frequency:. All symbols are sorted by frequency, from left to right shown in Figure a. Putting the dividing line between symbols B and C results in a total of 22 in the left group and binary tree questions geeksforgeeks total of 17 in the right group.

This minimizes the difference in totals between the two groups. With this division, A and B will each have a code that starts binary tree questions geeksforgeeks a 0 bit, and the C, D, and E codes will all start with a 1, as shown in Figure b.

Subsequently, the left half of the tree gets a new division between A and B, which puts A on a leaf with code 00 and B on a leaf with code After four division procedures, a tree of codes results. In the final tree, the three symbols with the highest frequencies have all been assigned 2-bit codes, and two symbols with lower counts have 3-bit codes as shown table below:.

The Shannon—Fano algorithm doesn't always generate an optimal code. InDavid A. Huffman gave a different algorithm that always produces an optimal tree for any given symbol weights probabilities.

While the Shannon—Fano tree is created from the root to the leaves, the Huffman algorithm works in the opposite direction, from the leaves to the root. The lowest pair now are B and C so they're allocated 0 and 1 and grouped together with a combined probability of 0. This leaves BC and DE now with the lowest probabilities so 0 and 1 are prepended to their codes and binary tree questions geeksforgeeks are combined. This leaves us with a single node and our algorithm is complete.

The code lengths for the different characters this time are 1 bit for A and 3 bits for all other characters. From Wikipedia, the free encyclopedia. The Imploding algorithm is actually a combination of two distinct algorithms. The first algorithm compresses repeated byte sequences using a sliding dictionary. The second algorithm is used to compress the encoding of the sliding dictionary output, using multiple Shannon—Fano trees. Compression formats Compression software codecs. Retrieved from " https: Lossless compression algorithms Claude Shannon.

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