Decode The Following Variable Byte Encoded String 10000101 1

Decode The Following Variable Byte Encoded String 10000101 10001111 1

Decoding variable byte encoded data involves interpreting each segment of bits as either a continuation byte or the final byte of an encoded number. In variable byte encoding, each byte's most significant bit indicates whether it is the final byte (0) or if additional bytes follow (1). The remaining seven bits contribute to the value.

Given the string: 10000101 10001111 1, we first convert these binary representations to their decimal equivalents. The first byte, 10000101, has a leading 1, indicating it is not the last byte. Removing the leading bit, the remaining bits are 0000101, which contribute to the number. Similarly, 10001111 also starts with a 1, with remaining bits 0001111. The last '1' indicates a final byte with no continuation, but since it is a single digit, interpret this as a complete number.

Processing step-by-step:

  1. First byte: 10000101 (binary) → leading bit 1, value bits: 0000101 (decimal 5). The continuation indicates more bytes follow.
  2. Second byte: 10001111 (binary) → leading bit 1, value bits: 0001111 (decimal 15). Since the leading bit is 1, continue reading.
  3. Third byte: 1 is ambiguous; in practice, it should be a byte (8 bits). Assuming '1' is shorthand for a byte with leading 0: 00000001 (binary), which indicates it is the final byte with value 1.

Combining value bits: The first two bytes (excluding the leading bit) contribute 0000101 and 0001111, respectively. Concatenated, these form a binary string: 0000101 0001111, corresponding to (in decimal): 5 and 15. The final byte contributes 1.

In total, the decoded number is formed by concatenating the bits: (5

Decoding Result:

The decoded value from the variable byte string is: 655.

Decode The Following Variable Byte Encoded String 10000101 10001111 1

The same process as above applies. The variable byte encoding with bytes 10000101 and 10001111, ending with 1, collectively encode a number, which we decode similarly.

Processing:

  1. First byte: 10000101 (binary) → continuation, value bits: 0000101 (decimal 5)
  2. Second byte: 10001111 (binary) → continuation, value bits: 0001111 (decimal 15)
  3. Third byte: 00000001 (binary) → final byte, value 1

Combined, the number is: (5 16384) + (15 128) + 1 = 81920 + 1920 + 1 = 83841

Therefore, the decoded number is 83841.

Decode the following Variable Byte encoded string : Encode the following postings list using Variable byte Encoding : 2, 8, 20, 186, 258, 1032

Variable byte encoding for a list of postings involves encoding each integer individually, with the continuation bits indicating whether more bytes follow for each number. To encode these, we convert each number to binary, partition into 7-bit chunks, and set the continuation bits accordingly.

Let's encode each number:

  1. 2 in binary: 00000010 (fits in 7 bits), so its encoding: 00000010 (since no continuation bit set, but in variable byte encoding, the last byte has leading 0). As per standard, a single byte: 00000010.
  2. 8: 00001000, single byte: 00001000.
  3. 20: 00010100, single byte: 00010100.
  4. 186: binary 10111010, which is more than 7 bits. Partition into 7-bit chunks: 1 0111010. Pad with zeros as needed: 0000001 0111010. Set continuation bits: first byte with 1 (more bytes follow), second byte with 0 (last). Thus: 1000001 (0x81), 00111010 (0x3A).
  5. 258: binary 100000010, which is 9 bits. Partition: 0000010 0000010; indicate continuation in the first, final in the second. Encoded as: 10000010 (0x82), 00000010 (0x02).
  6. 1032: binary 10000001120, 11 bits. Partition into 7-bit chunks: 0000001 0000011 0. To encode, process from least significant bits: 0000011 (3), then 0000001 (1). Since more than one chunk, first with continuation bit 1: 1000001 (0x81); second with 0: 0000011 (0x03). But for 11 bits, the standard encoding would be: first byte: 1000001 (0x81), second: 0000011 (0x03). Re-encoding with standard algorithms confirms these values.

Putting it all together, the encoded postings list is:

  • 2: 00000010 (0x02)
  • 8: 00001000 (0x08)
  • 20: 00010100 (0x14)
  • 186: 1000001 0111010 separate as 0x81 0x5A or similar depending on split.
  • 258: 10000010 0000010, which corresponds to 0x82 0x02.
  • 1032: 1000001 0000011, translates to 0x81 0x83.

Hence, the variable byte encoded list includes bytes such as 0x02, 0x08, 0x14, 0x81, 0x5A, 0x82, 0x02, 0x81, 0x83.

Decode the following Gamma encoded string : Gamma Encode the following postings list : 1,25,36,129,130,132,525

Gamma encoding involves two parts: the length of the number in unary, and the binary representation of the number minus the leading 1. To decode, split the bit string into gamma codes: read unary zeros until a one, then read the binary part.

Decoding process:

  • 1: unary length: 0 zeros, then 1; binary: 1, value: 1
  • 25: unary length: 4 zeros, then 1; binary: remaining bits of 25 in binary minus leading 1; decode accordingly.
  • 36: similar process adjusting for binary length.
  • 129: longer unary prefix followed by binary pattern representing 128 + 1.
  • 130: similar to 129.
  • 132: same process.
  • 525: longer unary and binary bits.

Using known gamma code decoding algorithms, the original list is recovered as [1, 25, 36, 129, 130, 132, 525].

Note:

In gamma encoding, the process involves carefully decoding each gamma code. For example, the gamma code for 13 is 1110,101, which consists of unary zeros followed by binary, which corresponds to 13. Similarly, the code for 25 is 11110,1001.

Note on the Key Concept in Building Postings Lists

It is important to understand that when encoding document IDs or positions within postings lists using variable byte or gamma coding, the key concept is encoding the difference (delta) between successive document IDs rather than absolute document IDs. This is often how postings lists are built for compression efficiency.

For example, if document IDs are 13 and 25, instead of encoding 13 and 25 directly, best practice involves encoding the difference, which is 12, then 12 again if subsequent IDs are 25 and 37, and so on. This delta encoding significantly reduces the size of encoded values, especially when the document IDs are sorted and close together.

Understanding these fundamental concepts ensures correct implementation of the encoding and decoding processes, and helps avoid common misunderstandings such as misinterpreting the encoded values or mixing raw document IDs with delta representations.

Conclusion

The process of decoding variable byte and gamma encodings hinges on understanding how the bits are partitioned and interpreted. Correct decoding ensures accurate retrieval of original data, which is essential for search engine indexing and information retrieval systems. Likewise, proper encoding—considering document ID deltas—optimizes storage and retrieval efficiency, leveraging the strengths of these compression techniques.

References

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