DNA Compression (Encoded using Huffman Encoding Method)
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The document discusses DNA compression algorithms. It describes the common components of most DNA compression algorithms, which include finding repeat segments in a DNA sequence, considering approximate repeats allowing for operations like substitutions, and selecting the best set of compatible repeats. It then provides an example demonstrating how these steps may be applied to a sample DNA sequence to identify repeat segments and encode them for compression.
DNA Compression (Encoded using Huffman Encoding Method)
- 1. (Encoded using Huffman Coding Method) Marwa K. Al-Rikaby University of Babylon/ College of Information Technology
- 2. DNA One of the building blocks in the organisms bodies. Consists of four chemical bases: Adenine (A). Thymine (T). Cytosine (C). Guanine (G). DNA bases pair up with each other, A with T and C with G, to form units called base pairs. DNA in humans contains around 3 billion bases and these are similar in two persons for about 99% of the total bases.
- 3. Goal: analyzing, saving space and time. The DNA sequences constructed from the alphabet {A, T, C, G}, and those sequences have various repeats usually approximate. Only lossless algorithms are valid. DNA compression model is preferred to be: Based on a biological knowledge. Give compression. Simple, few parameters. Can give per symbol information content. Efficient algorithm.
- 4. Since DNA sequences only contain the four bases {a, c, g, t} they can be stored using two bits per input symbol. The standard compression tools, such as gzip and bzip, usually fail to achieve any compression since they use more than two bits per symbol. When compressing 229354 bases (57338 bytes), we get: HEHCMVCG: 57338 bytes (without compression). gzip: 66741 bytes (negative compression). bzip2: 62169 bytes (negative compression).
- 5. In the case of multiple genomes from the same species, associated with ‘resequencing’ technologies, the flat text file approach is clearly wasteful since for the most part the sequences are identical. A simple approach is to store a reference sequence, and then for each other sequence, encode only the differences (or ‘deltas’) with respect to the original sequence. Consider the sequences AACGACTAGTAATTTG and CACGTCTAGTAATGTG which are identical, except for a substitution in position 1 (A→C), 5 (A→T) and 14 (T→G). Each SNP can be encoded by a pair (i, X), where i is an integer encoding the position and X represents the value of the substitution relative to the reference.
- 6. Although the basic idea is easy to understand, and not new, a precise implementation requires addressing a number of important technical issues: One can use local relative addresses, i.e. intervals, rather than absolute addresses. Using intervals, the above example ‘1C5T14G’becomes ‘0C4T9G’. With intervals the dynamic range of the integers to be encoded may be considerably smaller than with absolute addresses. The relatively modest price to pay is that intervals must be added to recover absolute coordinates. If the positions at which variations occur in the population are fixed and form a relatively small subset of all possible positions, then additional savings may result by focusing only on those positions. The choice of the reference sequence.
- 7. All applications of the basic ideas hinge on a fundamental technical problem: how to encode integers, representing for instance absolute or relative genomic addresses or read lengths, into binary strings? we are interested in binary encoding schemes for sequences of integers that can be parsed automatically and that, consistently with information theory, are entropy efficient, in the sense that fewer bits are used to encode more frequent events.
- 8. Common components of most of DNA compression algorithms: Finding the candidate repeat segments. Considering approximate repeats. Selecting the best subset of compatible repeats. Encoding of the repeat segments. Encoding of the non-repeat segments.
- 9. Suppose we have the following DNA sequence: TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTG ATAGATGATAGGGATATCACGTAGTCCCTAGCTCTTGGCG CTGGATGGGGCGGACGGTAAGGGAAATCGACCGTTGATA GTCCAAATTCGGTCGTATGATAGAAATTTCGAATGGAAAT TCTGATACATAGGTGATAGTAGATGTAAGATGATAGATGAT AGATAGATAGATGATAGACAGATTGATAGATGATAGAGAG A
- 10. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 11. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 12. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 13. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 14. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 15. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 16. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 17. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 18. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 19. 1. Finding the candidate repeat segments: Let “TGATAG” be a candidate segment, so we’ll find its repetitions in the example. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 20. The total number of “TGATAG” occurrences is 14. All segments repetitions should be indicated in this way. The counted numbers are kept for using in the encoding.
- 21. 2. Considering approximate repeats: Scanning the sequence to find out any similarity between the segments, i.e. segments can be identical after applying any operation from the four basic operations: Insertion: “AAATTCG”==“AAATTCTG” after Ins(T,6). Deletion: “AAATTCG”==“AAATTG” after Del(5,1). Replacement: “AAATTCG”==“AATTTCG” after Rep(2,T). Reverse: “AAATTCG”==“GCTTAAA” after Rev().
- 22. 2. Considering approximate repeats: let “ATATGA” be a reference segment, then “ATATCA” is identical to it if we replace “G” by “C” TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 23. 2. Considering approximate repeats: “ATAGA” is identical to “ATATGA” when deleting “T” at position 3. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 24. 2. Considering approximate repeats: “ATATGA” is identical to “ATAGA” when deleting “T” at position 3. “GGCGC” is identical to “GGCGG” when replacing “C” by “G” at position 4. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 25. 2. Considering approximate repeats: “AATGG” is identical to “GGTAA” when reversing it. TGATAGGTGATAGATATGATTGATAGATGATAGAAGATTGATAGATGAT AGGGATATCACGTAGTCCCTAGCTCTTGGCGCTGGATGGGGCGGACG GTAAGGGAAATCGACCGTTGATAGTCCAAATTCGGTCGTATGATAGAA ATTTCGAATGGAAATTCTGATACATAGGTGATAGTAGATGTAAGATGAT AGATGATAGATAGATAGATGATAGACAGATTGATAGATGATAGAGAGA
- 26. 3. Selecting the best subset of compatible repeats: The choosing of the reference segment is a major and a very sensitive process since the design of the reference sequence impacts not only the variants to be recorded, but also the intervals, and therefore it must also take into consideration any constraints a particular implementation may place on the intervals and their encodings. In our example, The segments that we have detected should have integer numbers pointing to its indexes in the reference table.
- 27. Segment Index A 0 T 1 C 2 G 3 TGATAG 4 ATATGA 5 AAATTCG 6 GGTAA 7 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 The reference table contains: • The four basic symbols {A, T, G, C}. • The candidates segments. • The basic operations, each one with the available parameters applied on the sequence.
- 28. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 ATATGA 5 AAATTCG 6 GGTAA 7 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 29. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 ATATGA 5 AAATTCG 6 GGTAA 7 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 30. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 AAATTCG 6 GGTAA 7 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 31. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 AAATTCG 6 GGTAA 7 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 32. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 AAATTCG 6 1 GGTAA 7 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 33. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 AAATTCG 6 1 GGTAA 7 1 GGCGC 8 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 34. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 AAATTCG 6 1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 35. 4. Encoding of the Repeat segment: initially the repetitions of each candidate segment must be counted in the same way shown in step 1. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 AAATTCG 6 1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 36. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 AAATTCG 6 1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 37. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 +1 AAATTCG 6 1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 InsT 11 Rev 12 RepG 13 1 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 38. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 1 +1 +1 AAATTCG 6 1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 1 InsT 11 Rev 12 RepG 13 1 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 39. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 1 +1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 1 +1 InsT 11 Rev 12 RepG 13 1 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 40. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 1 +1 +1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 2 InsT 11 1 Rev 12 RepG 13 1 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 41. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 1 +1 +1 +1 GGTAA 7 1 GGCGC 8 1 RepC 9 Del 10 2 InsT 11 1 +1 Rev 12 RepG 13 1 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 42. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 1 +1 GGCGC 8 1 RepC 9 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 43. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 1 +1 +1 GGCGC 8 1 RepC 9 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 44. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 3 GGCGC 8 1 +1 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 45. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 2 GGCGC 8 2 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 46. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 C 2 G 3 25 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 2 GGCGC 8 2 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 47. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 T 1 19 C 2 G 3 25 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 2 GGCGC 8 2 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 48. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 28 T 1 19 C 2 G 3 25 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 2 GGCGC 8 2 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 49. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 28 T 1 19 C 2 14 G 3 25 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 2 GGCGC 8 2 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 50. 5. Encoding of the Non-Repeat segment: The approximate segment must be processed in the same way shown in step 2. Segment Index repetitions A 0 28 T 1 19 C 2 14 G 3 25 TGATAG 4 14 ATATGA 5 3 AAATTCG 6 4 GGTAA 7 2 GGCGC 8 2 RepC 9 1 Del 10 2 InsT 11 2 Rev 12 1 RepG 13 1 RepT 14 1 TGATAGGTGATAGATATGATTGATAGAT GATAGAAGATTGATAGATGATAGGGAT ATCACGTAGTCCCTAGCTCTTGGCGCT GGATGGGGCGGACGGTAAGGGAAATC GACCGTTGATAGTCCAAATTCGGTCGT ATGATAGAAATTTCGAATGGAAATTCT GATACATAGGTGATAGTAGATGTAAGA TGATAGATGATAGATAGATAGATGATAG ACAGATTGATAGATGATAGAGAGA
- 51. First, find each segment probability: Segment Index repetitions probability A 0 28 28/119 T 1 19 19/119 C 2 14 14/119 G 3 25 25/119 TGATAG 4 14 14/119 ATATGA 5 3 3/119 AAATTCG 6 4 4/119 GGTAA 7 2 2/119 GGCGC 8 2 2/119 RepC 9 1 1/119 Del 10 2 2/119 InsT 11 2 2/119 Rev 12 1 1/119 RepG 13 1 1/119 RepT 14 1 1/119 No. of segments = 119
- 52. Arrange the segments in non-decreasing order according to its probability. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119
- 69. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1).
- 70. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0
- 71. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0
- 72. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0 1 0
- 73. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0 1 0 1 0
- 74. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0 1 0 1 0 1 0
- 75. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0 1 0 1 0 1 0 1 0
- 76. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0 1 0 1 0 1 0 1 0 0 1
- 77. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0
- 78. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Next, beginning from the root and backing to the leaves, give each branch with the small value the value (0) and the large the value (1). 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
- 79. 14 13 12 9 11 10 8 7 5 6 4 2 1 3 0 1/119 1/119 1/119 1/119 2/119 2/119 2/119 2/119 3/119 4/119 14/119 14/119 19/119 25/119 28/119 2/119 2/119 4/119 4/119 4/119 7/119 8/119 11/119 19/119 28/119 38/119 53/119 66/119 119/119 Finally, encode the segments via reading its code from the root to its leaf. 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
- 83. Segment Index Repetitions Probability Code A 0 28 28/119 1 0 T 1 19 19/119 1 1 1 C 2 14 14/119 0 1 1 G 3 25 25/119 0 0 TGATAG 4 14 14/119 0 1 0 ATATGA 5 3 3/119 1 1 0 1 1 0 AAATTCG 6 4 4/119 1 1 0 1 1 1 GGTAA 7 2 2/119 1 1 0 1 0 1 GGCGC 8 2 2/119 1 1 0 1 0 0 RepC 9 1 1/119 1 1 0 0 1 1 1 Del 10 2 2/119 1 1 0 0 0 1 InsT 11 2 2/119 1 1 0 0 0 0 Rev 12 1 1/119 1 1 0 0 1 1 0 RepG 13 1 1/119 1 1 0 0 1 0 1 RepT 14 1 1/119 1 1 0 0 1 0 0 The final reference table is: Keep in mind that only the segments and the codes are important for the decoder.
- 84. The previous coding satisfy both prefix property and the information theory in that : • There is no code given for a segment is a prefix in an other segment code. •The shortest codes given to segments that are more frequent while long ones assigned to those which are less frequent.