Position these guys the plus version making a puppy teams and 5 MB 2004. average file system 2009 could be compressed by 22% by whole file deduplication, i.e. reduced to 78% of original size by replacing identical copies of files with links. Compression could be increased to 28% by dividing files into 64 KB blocks and removing duplicate blocks, or to 31% using 8 KB blocks. If alignment restrictions are removed, then compression increases to 39% by replacing duplicate segments larger than 64 KB, or 42% using 8 KB segments. If links are allowed to point to duplicates on other file systems, then compression improves by about 3 percentage points for each doubling of the number of computers. For the entire 857 file systems, whole file deduplication compressed by 50%, and 8K segments by 69%. The obvious application of deduplication is incremental backups. It is only necessary to back up data that has changed. the 2009 study, the median time since last modification was 100 days. 7% of files of files were modified within one week, and 75% within one year. A code is assignment of bit strings to symbols such that the strings can be decoded unambiguously to recover the original data. The optimal code for a symbol with probability p have a length of log 2 p bits. Several efficient coding algorithms are known. Huffman developed algorithm that calculates optimal assignment over alphabet of n symbols O time. deflate and bzip2 use Huffman codes. However, Huffman codes are inefficient practice because code lengths must be rounded to a whole number of bits. If a symbol probability is not a power of 1, then the code assignment is less than optimal. This coding inefficiency can be reduced by assigning probabilities to longer groups of symbols but only at the cost of exponential increase alphabet size, and thus run time. The algorithm is as follows. We are given alphabet and a probability for each symbol. We construct a binary tree by starting with each symbol its own tree and joining the two trees that have the two smallest probabilities until we have one tree. Then the number of bits each Huffman code is the depth of that symbol the tree, and its code is a description of its path from the root For example, suppose that we are given the alphabet with each symbol having probability 0. We start with each symbol a one-node tree: .1 .1 .1 .1 .1 0 2 4 6 8 Because each small tree has the same probability, we pick any two and combine them: .2 .1 .1 .1 .1 .1 1 3 5 7 9 Continuing, .2 .2 .1 .1 .1 .1 .1 0 2 4 6 8 At this point, 8 and 9 have the two lowest probabilities we have to choose those: .2 .2 .2 .1 .1 .1 .1 .1 1 3 5 7 9 all of the trees have probability .2 we choose any pair of them: .4 .2 .2 .1 .1 .1 .1 .1 0 2 4 6 8 We choose any two of the three remaining trees with probability .2 .4 .2 .2 .1 .1 .1 .1 .1 0 2 4 6 8 the two smallest probabilities are .2 and one of the .4 .4 .2 .2 .2 .1 .1 .1 .1 .1 1 3 5 7 9 the two smallest are .4 and .6. After this step, the tree is finished. We can label the branches 0 for left and 1 for right, although the choice is arbitrary. 1 0 .6 1 .4 1 1 .2 .2 .1 .1 .1 .1 .1 0 2 4 6 8 From this tree we construct the code: Symbol Code 0 1 2 3 4 5 6 7 8 9 A code be static or dynamic. A static code is computed by the compressor and transmitted to the decompresser as part of the compressed data. A dynamic code