The skew binary number system is a non-standard positional numeral system in which the nth digit contributes a value of
times as they do in binary.
Each digit has a value of 0, 1, or 2.
A number can have many skew binary representations.
For example, a decimal number 15 can be written as 1000, 201 and 122.
Each number can be written uniquely in skew binary canonical form where there is only at most one instance of the digit 2, which must be the least significant nonzero digit.
In this case 15 is written canonically as 1000.
Canonical skew binary representations of the numbers from 0 to 15 are shown in following table:[1] The advantage of skew binary is that each increment operation can be done with at most one carry operation.
Incrementing a skew binary number is done by setting the only two to a zero and incrementing the next digit from zero to one or one to two.
When numbers are represented using a form of run-length encoding as linked lists of the non-zero digits, incrementation and decrementation can be performed in constant time.
Other arithmetic operations may be performed by switching between the skew binary representation and the binary representation.
[2] To convert from decimal to skew binary number, one can use following formula:[3] Base case:
Induction case:
To convert from skew binary number to decimal, one can use definition of skew binary number:
, st. only least significant bit (lsb)
Given a skew binary number, its value can be computed by a loop, computing the successive values of
A more efficient method is now given, with only bit representation and one subtraction.
The skew binary number of the form
1s is equal to the binary number
represents the digit
The skew binary number of the form
1s is equal to the binary number
Similarly to the preceding section, the binary number
1s equals the skew binary number
Note that since addition is not defined, adding
corresponds to incrementing the number
and incrementation takes constant time.
Hence transforming a binary number into a skew binary number runs in time linear in the length of the number.
The skew binary numbers were developed by Eugene Myers in 1983 for a purely functional data structure that allows the operations of the stack abstract data type and also allows efficient indexing into the sequence of stack elements.
[4] They were later applied to skew binomial heaps, a variant of binomial heaps that support constant-time worst-case insertion operations.