What s the fastest algorithm for sorting a linked list

Question

I m curious if O n log n  is the best a linked list can do

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Mergesort is the best you can do here

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Here s an implementation that traverses the list just once  collecting runs  then schedules the merges in the same way that mergesort does   Complexity is O n log m  where n is the number of items and m is the number of runs  Best case is O n   if the data is already sorted  and worst case is O n log n  as expected   It requires O log m  temporary memory  the sort is done in-place on the lists    updated below  commenter one makes a good point that I should describe it here   The gist of the algorithm is       while list not empty         accumulate a run from the start of the list         merge the run with a stack of merges that simulate mergesort s recursion     merge all remaining items on the stack   Accumulating runs doesn t require much explanation  but it s good to take the opportunity to accumulate both ascending runs and descending runs  reversed   Here it prepends items smaller than the head of the run and appends items greater than or equal to the end of the run   Note that prepending should use strict less-than to preserve sort stability    It s easiest to just paste the merging code here       int i   0      for     i  lt  stack size      i            if   stack i               break          run   merge run  stack i   comp           stack i    nullptr            if  i  lt  stack size              stack i    run        else           stack push back run           Consider sorting the list  d a g i b e c f j h   ignoring runs   The stack states proceed as follows                  d              a d           g    a d                 a d g i           b      a d g i              b e   a d g i           c   b e   a d g i                     a b c d e f g i           j         a b c d e f g i              h j      a b c d e f g i      Then  finally  merge all these lists   Note that the number of items  runs  at stack i  is either zero or 2 i and the stack size is bounded by 1 log2 nruns   Each element is merged once per stack level  hence O n log m  comparisons  There s a passing similarity to Timsort here  though Timsort maintains its stack using something like a Fibonacci sequence where this uses powers of two   Accumulating runs takes advantage of any already sorted data so that best case complexity is O n  for an already sorted list  one run   Since we re accumulating both ascending and descending runs  runs will always be at least length 2   This reduces the maximum stack depth by at least one  paying for the cost of finding the runs in the first place   Worst case complexity is O n log n   as expected  for data that is highly randomized    Um    Second update    Or just see wikipedia on bottom-up mergesort

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The question is LeetCode  148  and there are plenty of solutions offered in all major languages  Mine is as follows  but I m wondering about the time complexity  In order to find the middle element  we traverse the complete list each time  First time n elements are iterated over  second time 2   n 2 elements are iterated over  so on and so forth  It seems to be O n 2  time  def sort linked list  LinkedList int   - gt  LinkedList int         Return n    2 element     def middle head  LinkedList int   - gt  LinkedList int           if not head or not head next              return head         slow   head         fast   head next          while fast and fast next              slow   slow next             fast   fast next next          return slow      def merge head1  LinkedList int   head2  LinkedList int   - gt  LinkedList int           p1   head1         p2   head2         prev   head   None          while p1 and p2              smaller   p1 if p1 val  lt  p2 val else p2             if not head                  head   smaller             if prev                  prev next   smaller             prev   smaller              if smaller    p1                  p1   p1 next             else                  p2   p2 next          if prev              prev next   p1 or p2         else              head   p1 or p2          return head      def merge sort head  LinkedList int   - gt  LinkedList int           if head and head next              mid   middle head              mid next   mid next               Makes it easier to stop             mid next   None              return merge merge sort head   merge sort mid next           else              return head      return merge sort linked list

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It is reasonable to expect that you cannot do any better than O N log N  in running time     However  the interesting part is to investigate whether you can sort it in-place  stably  its worst-case behavior and so on   Simon Tatham  of Putty fame  explains how to sort a linked list with merge sort   He concludes with the following comments      Like any self-respecting sort algorithm  this has running time O N log N   Because this is Mergesort  the worst-case running time is still O N log N   there are no pathological cases       Auxiliary storage requirement is small and constant  i e  a few variables within the sorting routine   Thanks to the inherently different behaviour of linked lists from arrays  this Mergesort implementation avoids the O N  auxiliary storage cost normally associated with the algorithm    There is also an example implementation in C that work for both singly and doubly linked lists   As  J  rgen Fogh mentions below  big-O notation may hide some constant factors that can cause one algorithm to perform better because of memory locality  because of a low number of items  etc

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Depending on a number of factors  it may actually be faster to copy the list to an array and then use a Quicksort  The reason this might be faster is that an array has much better cache performance than a linked list  If the nodes in the list are dispersed in memory  you may be generating cache misses all over the place  Then again  if the array is large you will get cache misses anyway  Mergesort parallelises better  so it may be a better choice if that is what you want  It is also much faster if you perform it directly on the linked list  Since both algorithms run in O n   log n   making an informed decision would involve profiling them both on the machine you would like to run them on  --- EDIT I decided to test my hypothesis and wrote a C-program which measured the time  using clock    taken to sort a linked list of ints  I tried with a linked list where each node was allocated with malloc   and a linked list where the nodes were laid out linearly in an array  so the cache performance would be better  I compared these with the built-in qsort  which included copying everything from a fragmented list to an array and copying the result back again  Each algorithm was run on the same 10 data sets and the results were averaged  These are the results  N   1000   Fragmented list with merge sort  0 000000 seconds Array with qsort  0 000000 seconds Packed list with merge sort  0 000000 seconds  N   100000   Fragmented list with merge sort  0 039000 seconds Array with qsort  0 025000 seconds Packed list with merge sort  0 009000 seconds  N   1000000   Fragmented list with merge sort  1 162000 seconds Array with qsort  0 420000 seconds Packed list with merge sort  0 112000 seconds  N   100000000   Fragmented list with merge sort  364 797000 seconds Array with qsort  61 166000 seconds Packed list with merge sort  16 525000 seconds  Conclusion  At least on my machine  copying into an array is well worth it to improve the cache performance  since you rarely have a completely packed linked list in real life  It should be noted that my machine has a 2 8GHz Phenom II  but only 0 6GHz RAM  so the cache is very important

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Merge sort doesn t require O 1  access and is O   n ln n    No known algorithms for sorting  general data are better than O   n ln n     The special data algorithms such as radix sort   limits size of data   or histogram sort   counts discrete data   could sort a linked list with a lower growth function  as long as you use a different structure with O 1  access as temporary storage    Another class of special data is a comparison sort of an almost sorted list with k elements out of order  This can be sorted in O   kn   operations   Copying the list to an array and back would be O N   so any sorting algorithm can be used if space is not an issue   For example  given a linked list containing uint 8  this code will sort it in O N  time using a histogram sort    include  lt stdio h gt   include  lt stdint h gt   include  lt malloc h gt   typedef struct  list list t  struct  list       uint8 t value      list t   next       list t  sort list   list t  list         list t  heads 257     0       list t  tails 257     0           O N  loop     for   list t  it   list  it    0  it   it - gt  next             list t  next   it - gt  next           if   heads  it - gt  value      0                 heads  it - gt  value     it            else               tails  it - gt  value   - gt  next   it                     tails  it - gt  value     it             list t  result   0          constant time loop     for   size t i   255  i--  gt  0              if   tails i                  tails i  - gt  next   result              result   heads i                        return result     list t  make list   char  string         list t head       for   list t  it    amp head   string  it   it - gt  next    string             it - gt  next   malloc   sizeof   list t              it - gt  next - gt  value     uint8 t     string          it - gt  next - gt  next   0             return head next     void free list   list t  list         for   list t  it   list  it    0              list t  next   it - gt  next          free   it            it   next           void print list   list t  list         printf                if   list             printf     c   list - gt  value             for   list t  it   list - gt  next  it    0  it   it - gt  next               printf       c   it - gt  value               printf       n         int main   int nargs  char   args         list t  list   make list   nargs  gt  1   args 1     wibble           print list   list         list t  sorted   sort list   list          print list   sorted         free list   list

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As I know  the best sorting algorithm is O n log n   whatever the container - it s been proved that sorting in the broad sense of the word  mergesort quicksort etc style  can t go lower  Using a linked list will not give you a better run time    The only one algorithm which runs in O n  is a  hack  algorithm which relies on counting values rather than actually sorting

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Not a direct answer to your question  but if you use a Skip List  it is already sorted and has O log N  search time

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You can copy it into an array and then sort it     Copying into array O n   sorting O nlgn   if you use a fast algorithm like merge sort    copying back to linked list O n  if necessary    so it is gonna be O nlgn    note that if you do not know the number of elements in the linked list you won t know the size of array  If you are coding in java you can use an Arraylist for example

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A Radix sort is particularly suited to a linked list  since it s easy to make a table of head pointers corresponding to each possible value of a digit

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As stated many times  the lower bound on comparison based sorting for general data is going to be O n log n    To briefly resummarize these arguments  there are n  different ways a list can be sorted   Any sort of comparison tree that has n   which is in O n n   possible final sorts is going to need at least log n   as its height  this gives you a O log n n   lower bound  which is O n log n     So  for general data on a linked list  the best possible sort that will work on any data that can compare two objects is going to be O n log n    However  if you have a more limited domain of things to work in  you can improve the time it takes  at least proportional to n    For instance  if you are working with integers no larger than some value  you could use Counting Sort or Radix Sort  as these use the specific objects you re sorting to reduce the complexity with proportion to n   Be careful  though  these add some other things to the complexity that you may not consider  for instance  Counting Sort and Radix sort both add in factors that are based on the size of the numbers you re sorting  O n k  where k is the size of largest number for Counting Sort  for instance    Also  if you happen to have objects that have a perfect hash  or at least a hash that maps all values differently   you could try using a counting or radix sort on their hash functions

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Comparison sorts  i e  ones based on comparing elements  cannot possibly be faster than n log n  It doesn t matter what the underlying data structure is  See Wikipedia   Other kinds of sort that take advantage of there being lots of identical elements in the list   such as the counting sort   or some expected distribution of elements in the list  are faster  though I can t think of any that work particularly well on a linked list

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This is a nice little paper on this topic   His empirical conclusion is that Treesort is best  followed by Quicksort and Mergesort   Sediment sort  bubble sort  selection sort perform very badly   A COMPARATIVE STUDY OF LINKED LIST SORTING ALGORITHMS by Ching-Kuang Shene  http   citeseerx ist psu edu viewdoc summary doi 10 1 1 31 9981

[algorithm] What's the fastest algorithm for sorting a linked list?

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