A Survey of Decoding Algorithms in Insertion and Deletion Channels

In this survey, we are going to talk about some algorithms and codes that can make communications on Insertion/Deletion Channel effective. First of all,what is the Insertion/Deletion Channel? Generally speaking, it is a kind of channel that can easily occur errors which may result in loss of synchronization. Insertion errors and deletion errors are typical ones of these errors. In the following step, we are going to show the definition of them. The definition of insertion channel is that: each time we transimit a symbol on the channel,1 bit will be inserted into the transmition sequence with a fixed probability p. The definition of deletion channel is that: each bit we transmit on this channel might be independently deleted with fixed probability p. Here comes a example to make these definition more intuitively: On a channel with both insertion and deletion error, if a sequence of symbols of 11001100 have been transmitted, by inserting a ”1” into the second digit and deleting a ”0” from the third digit, the original sequence we send turn outs to be 11101100. Here comes a question, can we communicate reliablly and effectively on such kind of channels? The answer is yes. By using some specific encoding and decoding algorithms, this goal can be reached. The purpose of this survey is to briefly show the theory of the encoding and decoding algorithms on insertion/deletion channel, and the algorithms(or we can rather call them coding strategy) will be in a increasinig order of the capability of correcting insertion and deletion errors. In section 2, we will first discuss some elements of coding on the insertion/deletion channel, that is the Maximum likelihood decoder, Hamming distance. And by finding out that Hamming distance can only satisfy the requirement of decoding on the channel occurs substitution errors, in the section 3, we introduce a stronger algorithm that fits to decode on the insertion and deletion channel, that is Levenshtein distance. While we again find out that simply use Levenshtein distance may not help much in decoding on the insertion and deletion channels, Greedy algorithm, which is much stronger, will be introduced in section 4, and again we will find out that even Greedy algorithm is a good decoding algorithm, it can not deal with the channels with serious deletion (insertion), then the Jigsaw Puzzles algorithm will be introduced in section 5. At the end of this survey, we will make a conclusion and leave some open questions. This survey takes some older works on codes used in insertion and deletion channels as basic knowledges, such as [2], [5], [15], [16], [19], and [1], [9], [8], [11], [18], [13], [4], [3], [12], [17], [6], [10], [14], [7] are read as backgroud knowledge of the new algorithms such as Greedy algorithm and Jigsaw algorithm, etc. The survey gave by Michael Mitzenmacher [10] really helps in the train of thought of this survey. After all, thanks to the Wikipedia!