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Recursive indexing

## Summary

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When number (generally large number) is represented in a finite alphabet set, and it cannot be represented by just one member of the set, recursive indexing is used.

Recursive indexing itself is a method to write the successive differences of the number after extracting the maximum value of the alphabet set from the number, and continuing recursively till the difference falls in the range of the set.

Recursive indexing with a 2-letter alphabet is called unary code.

## Encoding

To encode a number N, keep reducing the maximum element of this set (Smax) from N and output Smax for each such difference, stopping when the number lies in the half closed half open range [0 – Smax).

Example:

Let S = [0 1 2 3 4 … 10], be an 11-element set, and we have to recursively index the value N=49.

According to this method, we need to keep removing 10 from 49, and keep proceeding till we reach a number in the 0–10 range.

So the values are 10 (N = 49 – 10 = 39), 10 (N = 39 – 10 = 29), 10 (N = 29 – 10 = 19), 10 (N = 19 – 10 = 9), 9. Hence the recursively indexed sequence for N = 49 with set S, is 10, 10, 10, 10, 9.

## Decoding

Keep adding all the elements of the index, stopping when the index value is between (inclusive of ends) the least and penultimate elements of the set S.

Example:

Continuing from above example we have 10 + 10 + 10 + 10 + 9 = 49.

## Uses

This technique is most commonly used in run-length encoding systems to encode longer runs than the alphabet sizes permit.