Efficient way to insert a number into a sorted array of numbers

Question

I have a sorted JavaScript array  and want to insert one more item into the array such the resulting array remains sorted   I could certainly implement a simple quicksort-style insertion function   var array    1 2 3 4 5 6 7 8 9   var element   3 5  function insert element  array      array splice locationOf element  array    1  0  element     return array     function locationOf element  array  start  end      start   start    0    end   end    array length    var pivot   parseInt start    end - start    2  10     if  end-start  lt   1    array pivot      element  return pivot    if  array pivot   lt  element        return locationOf element  array  pivot  end       else       return locationOf element  array  start  pivot          console log insert element  array       WARNING  this code has a bug when trying to insert to the beginning of the array  e g  insert 2   3  7  9   produces incorrect   3  2  7  9     However  I noticed that implementations of the Array sort function might potentially do this for me  and natively   var array    1 2 3 4 5 6 7 8 9   var element   3 5  function insert element  array      array push element     array sort function a  b        return a - b          return array     console log insert element  array      Is there a good reason to choose the first implementation over the second   Edit  Note that for the general case  an O log n   insertion  as implemented in the first example  will be faster than a generic sorting algorithm  however this is not necessarily the case for JavaScript in particular  Note that    Best case for several insertion algorithms is O n   which is still significantly different from O log n    but not quite as bad as O n log n   as mentioned below   It would come down to the particular sorting algorithm used  see Javascript Array sort implementation   The sort method in JavaScript is a native function  so potentially realizing huge benefits -- O log n   with a huge coefficient can still be much worse than O n  for reasonably sized data sets

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Had your first code been bug free  my best guess is  it would have been how you do this job in JS  I mean   Make a binary search to find the index of insertion Use splice to perform your insertion   This is almost always 2x faster than a top down or bottom up linear search and insert as mentioned in domoarigato s answer which i liked very much and took it as a basis to my benchmark and finally push and sort  Of course under many cases you are probably doing this job on some objects in real life and here i have generated a benchmark test for these three cases for an array of size 100000 holding some objects  Feel free to play with it

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Don t re-sort after every item  its overkill    If there is only one item to insert  you can find the location to insert using binary search  Then use memcpy or similar to bulk copy the remaining items to make space for the inserted one  The binary search is O log n   and the copy is O n   giving O n   log n  total  Using the methods above  you are doing a re-sort after every insertion  which is O n log n    Does it matter  Lets say you are randomly inserting k elements  where k   1000  The sorted list is 5000 items     Binary search   Move   k  n   log n    1000  5000   12    5 000 012    5 million ops Re-sort on each   k  n log n     60 million ops   If the k items to insert arrive whenever  then you must do search move  However  if you are given a list of k items to insert into a sorted array - ahead of time - then you can do even better  Sort the k items  separately from the already sorted n array  Then do a scan sort  in which you move down both sorted arrays simultaneously  merging one into the other    - One-step Merge sort   k log k   n   9965   5000    15 000 ops  Update  Regarding your question  First method   binary search move   O n   log n   Second method   re-sort   O n log n  Exactly explains the timings you re getting

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Here are a few thoughts  Firstly  if you re genuinely concerned about the runtime of your code  be sure to know what happens when you call the built-in functions  I don t know up from down in javascript  but a quick google of the splice function returned this  which seems to indicate that you re creating a whole new array each call  I don t know if it actually matters  but it is certainly related to efficiency  I see that Breton  in the comments  has already pointed this out  but it certainly holds for whatever array-manipulating function you choose   Anyways  onto actually solving the problem   When I read that you wanted to sort  my first thought is to use insertion sort   It is handy because it runs in linear time on sorted  or nearly-sorted lists  As your arrays will have only 1 element out of order  that counts as nearly-sorted  except for  well  arrays of size 2 or 3 or whatever  but at that point  c mon   Now  implementing the sort isn t too too bad  but it is a hassle you may not want to deal with  and again  I don t know a thing about javascript and if it will be easy or hard or whatnot  This removes the need for your lookup function  and you just push  as Breton suggested    Secondly  your  quicksort-esque  lookup function seems to be a binary search algorithm  It is a very nice algorithm  intuitive and fast  but with one catch  it is notoriously difficult to implement correctly  I won t dare say if yours is correct or not  I hope it is  of course       but be wary if you want to use it   Anyways  summary  using  push  with insertion sort will work in linear time  assuming the rest of the array is sorted   and avoid any messy binary search algorithm requirements  I don t know if this is the best way  underlying implementation of arrays  maybe a crazy built-in function does it better  who knows   but it seems reasonable to me        - Agor

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The best data structure I can think of is an indexed skip list which maintains the insertion properties of linked lists with a hierarchy structure that enables log time operations  On average  search  insertion  and random access lookups can be done in O log n  time    An order statistic tree enables log time indexing with a rank function    If you do not need random access but you need O log n  insertion and searching for keys  you can ditch the array structure and use any kind of binary search tree    None of the answers that use array splice   are efficient at all since that is on average O n  time  What  39 s the time complexity of array splice   in Google Chrome

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Just as a single data point  for kicks I tested this out inserting 1000 random elements into an array of 100 000 pre-sorted numbers using the two methods using Chrome on Windows 7   First Method   54 milliseconds Second Method   57 seconds   So  at least on this setup  the native method doesn t make up for it   This is true even for small data sets  inserting 100 elements into an array of 1000   First Method  1 milliseconds Second Method  34 milliseconds

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TypeScript version with custom compare method  const   compare     new Intl Collator undefined      numeric  true    sensitivity   quot base quot       const insert    items  string    item  string    gt        let low   0      let high   items length       while  low  lt  high            const mid    low   high   gt  gt  1          compare items mid   item   gt  0                high   mid                 low   mid   1              items splice low  0  item       Use  const items        insert items   quot item 12 quot    insert items   quot item 1 quot    insert items   quot item 2 quot    insert items   quot item 22 quot     console log items         quot item 1 quot    quot item 2 quot    quot item 12 quot    quot item 22 quot

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Here s a comparison of four different algorithms for accomplishing this  https   jsperf com sorted-array-insert-comparison 1  Algorithms   Naive  just push and sort   afterwards Linear  iterate over array and insert where appropriate Binary Search  taken from https   stackoverflow com a 20352387 154329  Quick Sort Like   the refined solution from syntheticzero  https   stackoverflow com a 18341744 154329    Naive is always horrible  It seems for small array sizes  the other three dont differ too much  but for larger arrays  the last 2 outperform the simple linear approach

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Here s a version that uses lodash   const     require  lodash    sortedArr splice   sortedIndex sortedArr valueToInsert   0 valueToInsert     note  sortedIndex does a binary search

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Your insertion function assumes that the given array is sorted  it searches directly for the location where the new element can be inserted  usually by just looking at a few of the elements in the array   The general sort function of an array can t take these shortcuts  Obviously it at least has to inspect all elements in the array to see if they are already correctly ordered  This fact alone makes the general sort slower than the insertion function   A generic sort algorithm is usually on average O n  sdot  log n   and depending on the implementation it might actually be the worst case if the array is already sorted  leading to complexities of O n2   Directly searching for the insertion position instead has just a complexity of O log n    so it will always be much faster

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There s a bug in your code  It should read   function locationOf element  array  start  end      start   start    0    end   end    array length    var pivot   parseInt start    end - start    2  10     if  array pivot      element  return pivot    if  end - start  lt   1      return array pivot   gt  element   pivot - 1   pivot    if  array pivot   lt  element        return locationOf element  array  pivot  end       else       return locationOf element  array  start  pivot           Without this fix the code will never be able to insert an element at the beginning of the array

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Very good and remarkable question with a very interesting discussion  I also was using the Array sort   function after pushing a single element in an array with some thousands of objects   I had to extend your locationOf function for my purpose because of having complex objects and therefore the need for a compare function like in Array sort     function locationOf element  array  comparer  start  end        if  array length     0          return -1       start   start    0      end   end    array length      var pivot    start   end   gt  gt  1      should be faster than dividing by 2      var c   comparer element  array pivot        if  end - start  lt   1  return c    -1   pivot - 1   pivot       switch  c            case -1  return locationOf element  array  comparer  start  pivot           case 0  return pivot          case 1  return locationOf element  array  comparer  pivot  end                 sample for objects like  lastName   Miller        var patientCompare   function  a  b        if  a lastName  lt  b lastName  return -1      if  a lastName  gt  b lastName  return 1      return 0

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For a small number of items  the difference is pretty trivial  However  if you re inserting a lot of items  or working with a very large array  calling  sort   after each insertion will cause a tremendous amount of overhead   I ended up writing a pretty slick binary search insert function for this exact purpose  so I thought I d share it  Since it uses a while loop instead of recursion  there is no overheard for extra function calls  so I think the performance will be even better than either of the originally posted methods  And it emulates the default Array sort   comparator by default  but accepts a custom comparator function if desired   function insertSorted arr  item  comparator        if  comparator    null               emulate the default Array sort   comparator         comparator   function a  b                if  typeof a      string   a   String a               if  typeof b      string   b   String b               return  a  gt  b   1    a  lt  b   -1   0                             get the index we need to insert the item at     var min   0      var max   arr length      var index   Math floor  min   max    2       while  max  gt  min            if  comparator item  arr index    lt  0                max   index            else               min   index   1                    index   Math floor  min   max    2                 insert the item     arr splice index  0  item        If you re open to using other libraries  lodash provides sortedIndex and sortedLastIndex functions  which could be used in place of the while loop  The two potential downsides are 1  performance isn t as good as my method  thought I m not sure how much worse it is  and 2  it does not accept a custom comparator function  only a method for getting the value to compare  using the default comparator  I assume

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I know this is an old question that has an answer already  and there are a number of other decent answers   I see some answers that propose that you can solve this problem by looking up the correct insertion index in O log n  - you can  but you can t insert in that time  because the array needs to be partially copied out to make space  Bottom line   If you really need O log n  inserts and deletes into a sorted array  you need a different data structure - not an array   You should use a B-Tree  The performance gains you will get from using a B-Tree for a large data set  will dwarf any of the improvements offered here  If you must use an array  I offer the following code  based on insertion sort  which works  if and only if the array is already sorted  This is useful for the case when you need to resort after every insert  function addAndSort arr  val        arr push val       for  i   arr length - 1  i  gt  0  amp  amp  arr i   lt  arr i-1   i--            var tmp   arr i           arr i    arr i-1           arr i-1    tmp            return arr     It should operate in O n   which I think is the best you can do  Would be nicer if js supported multiple assignment  here s an example to play with  Update  this might be faster  function addAndSort2 arr  val        arr push val       i   arr length - 1      item   arr i       while  i  gt  0  amp  amp  item  lt  arr i-1             arr i    arr i-1           i -  1            arr i    item      return arr     Updated JS Bin link

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Here is my function  uses binary search to find item and then inserts appropriately    x000D   x000D  function binaryInsert val  arr   x000D      let mid   x000D      len arr length  x000D      start 0  x000D      end len-1  x000D      while start  lt   end   x000D          mid   Math floor  end   start  2   x000D          if val  lt   arr mid    x000D              if val  gt   arr mid-1    x000D                  arr splice mid 0 val   x000D                  break  x000D                x000D              end   mid-1  x000D           else  x000D              if val  lt   arr mid 1    x000D                  arr splice mid 1 0 val   x000D                  break  x000D                x000D              start   mid 1  x000D            x000D        x000D      return arr  x000D    x000D   x000D  console log binaryInsert 16    x000D      5    6   14   19  23  44  x000D     35   51   86   68  63  71  x000D     87  117 x000D        x000D   x000D   x000D

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function insertOrdered array  elem        let  array   array      let i   0      while   i  lt  array length  amp  amp  array i   lt  elem    i           array splice i  0  elem       return  array

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Simple  Demo    function sortedIndex array  value        var low   0          high   array length       while  low  lt  high            var mid    low   high   gt  gt  gt  1          if  array mid   lt  value  low   mid   1          else high   mid            return low

[javascript] Efficient way to insert a number into a sorted array of numbers?

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