Most digital computers internally represent all of their data as electronic representations of binary numbers, so processing the digits of integer representations by groups of binary digit representations is most convenient. Two classifications of radix sorts are least significant digit (LSD) radix sorts and
Most digital computers internally represent all of their data as electronic representations of binary numbers, so processing the digits of integer representations by groups of binary digit representations is most convenient. Two classifications of radix sorts are least significant digit (LSD) radix sorts and most significant digit (MSD) radix sorts. LSD radix sorts process the integer representations starting from the least significant digit and move towards the most significant digit. MSD radix sorts work the other way around.
The integer representations that are processed by sorting algorithms are often called "keys," which can exist all by themselves or be associated with other data. LSD radix sorts typically use the following sorting order: short keys come before longer keys, and keys of the same length are sorted lexicographically. This coincides with the normal order of integer representations, such as the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. MSD radix sorts use lexicographic order, which is suitable for sorting strings, such as words, or fixed-length integer representations. A sequence such as "b, c, d, e, f, g, h, i, j, ba" would be lexicographically sorted as "b, ba, c, d, e, f, g, h, i, j". If lexicographic ordering is used to sort variable-length integer representations, then the representations of the numbers from 1 to 10 would be output as 1, 10, 2, 3, 4, 5, 6, 7, 8, 9, as if the shorter keys were left-justified and padded on the right with blank characters to make the shorter keys as long as the longest key for the purpose of determining sorted order.
Radix Sort is adapted to fixed-length keys, which means that much practical data would need to be adjusted. A crude implementation would "temp = key, index(key) = key, and then key = temp, then repeat for number of keys". A bug in that implementation is that it only works on data where no key is already in place. It's written here only to illustrate that implementations benefit from using long fields, sixteen bits (64k) being a good compromise between the size and the number of programs that typically run at one time on a computer. That method will also fail if it transposes values. floating point numbers would be sorted first by their mantissa, coalesced, and then sorted by their exponent. Variable length keys are padded with terminal zero digits to make all of the keys a fixed length, which requires processing time proportional to the number of keys.
A least significant digit (LSD) radix sort is a fast stable sorting algorithm which can be used to sort keys in lexicographic order. Keys may be a string of characters, or numerical digits in a given 'radix'. The processing of the keys begins at the least significant digit (i.e., the rightmost digit), and proceeds to the most significant digit (i.e., the leftmost digit). The sequence in which digits are processed by a least significant digit (LSD) radix sort is the opposite of the sequence in which digits are processed by a most significant digit (MSD) radix sort.
A least significant digit (LSD) radix sort operates in O(nk) time, where n is the number of keys, and k is the average key length. You can get this performance for variable-length keys by grouping all of the keys that have the same length together and separately performing an LSD radix sort on each group of keys for each length, from shortest to longest, in order to avoid processing the whole list of keys on every sorting pass.
A radix sorting algorithm was originally used to sort punched cards in several passes. A computer algorithm was invented for radix sort in 1954 at MIT by Harold H. Seward. In many large applications needing speed, the computer radix sort is an improvement on (slower) comparison sorts.
LSD radix sorts have resurfaced as an alternative to other high performance comparison-based sorting algorithms (like heapsort and mergesort) that require Ω(n · log n) comparisons, where n is the number of items to be sorted. Comparison sorts can do no better than Ω(n · log n) execution time but offer the flexibility of being able to sort with respect to more complicated orderings than a lexicographic one; however, this ability is of little importance in many practical applications.
Each key is first figuratively dropped into one level of buckets corresponding to the value of the rightmost digit of each key. Each bucket preserves the original order of the keys as the keys are dropped into each bucket. There is a one-to-one correspondence between the number of buckets and the number of values that can be represented by a digit. Then, the process repeats with the next neighboring digit until there are no more digits to process. In other words:
170, 45, 75, 90, 2, 24, 802, 66
170, 90, 2, 802, 24, 45, 75, 66
Notice that we keep 2 before 802, because 2 occurred before 802 in the original list, and similarly for 170 and 90.
2, 802, 24, 45, 66, 170, 75, 90
2, 24, 45, 66, 75, 90, 170, 802
It is important to realize that each of the above steps requires just a single pass over the data, since each item can be placed in its correct bucket without having to be compared with other items.
Some LSD radix sort implementations allocate space for buckets by first counting the number of keys that belong in each bucket before moving keys into those buckets. The number of times that each digit occurs is stored in an array. Consider the previous list of keys viewed in a different way:
The first counting pass starts on the least significant digit of