Skew binary number system
The skew binary number system is a non-standard positional numeral system in which the nth digit contributes a value of times the digit (digits are indexed from 0) instead of times as they do in binary. Each digit has a value of 0, 1, or 2. Notice that 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 first non-zero least significant digit. In this case 15 is written canonically as 1000.
Skew binary representations of the numbers from 0 to 15 are shown in following table:
The advantage of skew binary is that each increment operation can be done with at most one carry operation. This exploits the fact that . 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, a form called the sparse representations of numbers by Okasaki, incrementation and decrementation can be performed in constant time.
Other arithmetic operations are performed by switching between the skew binary representation and the binary representation.
From skew binary representation to binary representation
Given a skew binary number, its value can be computed by a loop, computing the successive values of and adding it once or twice for each such that the th digit is 1 or 2 respectively. A more efficient method is no given, with only bit representation and one substraction.
The skew binary number of the form without 2 and with 1s is equal to the binary number minus . Let represents the digit repeated times. The skew binary number of the form with 1s is equal to the binary number minus .
From binary representation to skew binary representation
Similarly to the preceding section, the binary number of the form with 1s equals the skew binary number plus . Note that since addition is not defined, adding corresponds to incrementing the number times. However, is bounded by the logarithm of 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.
Skew binary numbers find applications in skew binomial heaps, a variant of binomial heaps that support worst-case O(1) insertion, and in skew binary random access lists, a purely functional data structure. They also find use in bootstrapped skew binomial heaps, which have excellent asymptotic guarantees.
If smoothsort is implemented using perfect binary trees (rather than the more common Leonardo trees), the heap is divided into trees which correspond to the digits of the skew binary representation of the heap size.
- skew binary numbers
- Okasaki, Chris. Purely Functional Data Structures.
- Elmasry, Amr; Jensen, Claus; Katajainen, Jyrki (2012). "Two Skew-Binary Numeral Systems and One Application" (PDF). Theory of Computing Systems. 50: 185–211. doi:10.1007/s00224-011-9357-0.