timsort: step by step implementation
Understanding timsort features by implementing step by step manner.
Overview
All imprementations are written by JavaScript. It can be written as compact code, and has primitive functionoalities: e.g. shift ops, for loop. Notice that the codes are not faster than native Array.sort really.
Implementation files has numbering file name. Each implementation is improvement of former implementation.
At first, prepare the basic knowledge for implementing timsort.
- 00binarysort.js
- 01mergesort.js
Timsort is improvement of merge sort for two aspects: split and merge.
- split improvement has 1X numbered file name
- merge improvement has 2X numbered file name
At last, merged both improvements as
- 30timsort.js
Every implementation files are executable. runner.js executes the sort in the file with several examples:
node runner.js 01mergesort.js
I checked by node-v0.6.0
00binarysort.js
Binary sort is improvement of insertion sort.
In usual insertion sort, comparing and swap are mixed in the inner loop. To divide them as searching inserting point and inserting a element, the former can apply binary search, and the latter is just a right cyclic shift. It is named "binary sort".
We use sort function interface as
sort(array, first, last, lessThan)
For representing array range, usually use [first,last). Notice that the "last" is not last index. it is last index + 1. The range representaion usually makes the code simpler than [first,lastIndex] representation.
- whole array: [0,length).
Our codes use customizable comparator for array elements. Sorts use the relation "less than"(a < b) as comparator, not use three state comparator(-1,0,1). "a <= b" is made as "!(b < a)".
bool lessThan(a, b)
The key is
- change to apply binary search to sort impl
- change to apply bulk copy (not swap) to sort impl
- range replesentation of array and algorithm
- relationship with relation lessThan and stable insertion point when included same values.
The binary sort is also one of the part of timsort.
01mergesort.js
The merge sort is the base of improvements to timsort.
The key of merge sort is
- merge only "neighbor" two chunks.
Split improvements keeps the chunk neighborness. Merge improvements also depends on the neighborness.
11loop_mergesort.js
The improvement changes recursive splitting to looped split and merge. Looped one uses chunk stack. Merge strategy is designed merge count as O(Nlog(N)).
- e.g. length in chunk stack: [1]=>[1,1]=>[2]=>[2,1]=>[2,1,1]=>[2,2]=>[4]=>...
The key is
- split each 1 element chunks
- merge strategy is based on chunk size of last 2 chunks in the stacks.
12runs_timsort.js
Timsort uses monotonic part(asc or desc) as basic splitted chunk. It is named to "run".
- e.g. runs on pre-sorted: [a,c,c,|b,a,|b,c,d,|f]
The ascendant run are already sorted. The descendant run shoud be reversed, so becomes also sorted part.
- desc part reversed: [a,c,c,|a,b,|b,c,d,|f]
Entire array parted sorted chunks. it is similar as mid state of merge sort execution. Merging strategy is extended one of the loop merge sort and method of merge neighbor is same as the merge sort.
- merges: [a,a,b,c,c,|b,c,d,|f] => [a,a,b,c,c,|b,c,d,f] => [a,a,b,b,c,c,c,d,f]
The key is
- Timsort can be applyed same improvement of merge of mergesort.
13minrun_timsort.js
Handling shorter runs is not efficient. It introduce minimun run size (as 7).
If forcely run exprode minimun run size, the run(minrun) is not orderd elements. So, the minrun is sorted by the binary sort.
- e.g. [a,b,c,d,c,b,e,f,g,...]
- cut as run: => [a,b,c,d,|c,b,e,f,g,...]
- if run length < minrun size then extend it to minrun: => [a,b,c,d,c,b,e,|f,g,...]
- sort minrun: =>[a,b,b,c,c,d,e,|f,g,...]
14improve_minrun_timsort.js
Whole minrun is not orderd. But former side of minrun(part of before extended run) ensures orderd.
Binary sort can start at after the extended part.
21onebuffer_mergesort.js
NOTICE: Apply merge improvement to the basic merge sort (not timsort). One reason is timsort to smaller array is not enter the merge process (perform minrun binary sort only).
First improvement is reduce merging escaped buffer to one. It escapes only shorter side chunk.
Merge process is varied two. It depends on which side is shorter: left side or right side. The difference is direction of iterating and compare relation.
- e.g. [a,b,|a,c,d,e] => [a,b,|a,c,d,e] and [a,b] : merge them from leftmost
- e.g. [a,b,c,d,|d,e] => [a,b,c.d,|c,e] and [c,e] : merge them from rightmost
22onebuffer_with_searchmergestart_mergesort.js
Escaping shorter side chunk does not need to escape entire chunk. To seek a merge point of first of the larger side, The point is used as escape range start.
- e.g. [a,b,|a,c,d,e] => [a,b,|a,c,d,e] and [b]
- e.g. [a,b,c,d,|d,e] => [a,b,c.d,|c,e] and [c]
23refactored_mergesort.js
This improvement is refactored to packing state to a struct. It is for passing functions. It makes functions dividable.
24galloping_mergesort.js
Usual merge is comparing heads of both side and copy back one of heads. It named "one-pair mode".
the other is "galloping mode". Galloping mode is copy back as block manner. End of blocks to copy is found with binary sort.
The improvement mixes one-pair mode and galloping mode. Mode switching is based on counts of copy back of eachcountinuous side.
25improve_gallop_search_mergesort.js
The improvement is using special binary sort.
At merge's left side is shorter, for each n=1,2,3,..., searching just in [2^(n-1), 2^n) range which includes the result.
- ranges: [0,1),[1,3),[3,7),[7,15),....
When right side is shorter, the ranges begins last side.
- ranges(length is l): [l-1,l-0),[l-3,l-1),[l-7,l-3),[l-15,l-7),....
26varying_galloping_mergesort.js
This improvement makes continuous count of mode switching variable.
30timsort.js
It is mixed merge improvements until 26varying_galloping_mergesort.js to timsort until 14improve_minrun_timsort.js.
License
All of the improvement step code is written with non-C language from scratch to reconstruct my understanding of the timsort mechanisms described at the articles of understanding timsort.
Every code of the steps is published to the Public Domain.