# javascript - Efficient way to insert a number into a sorted array of numbers?

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 <= 1 || array[pivot] === element) return pivot;if (array[pivot] < element) {return locationOf(element, array, pivot, end);} else {return locationOf(element, array, start, pivot);}}console.log(insert(element, array));``

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.

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
``````

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) >> 1;  // should be faster than the above calculation

var c = comparer(element, array[pivot]);
if (end - start <= 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 < b.lastName) return -1;
if (a.lastName > b.lastName) return 1;
return 0;
};
``````

Simple (Demo):

``````function sortedIndex(array, value) {
var low = 0,
high = array.length;

while (low < high) {
var mid = (low + high) >>> 1;
if (array[mid] < value) low = mid + 1;
else high = mid;
}
return low;
}
``````

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 <= 1)
return array[pivot] > element ? pivot - 1 : pivot;
if (array[pivot] < 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.

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 ⋅ 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.

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 > b ? 1 : (a < 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 > min) {
if (comparator(item, arr[index]) < 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).

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.

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 > 0 && arr[i] < 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 > 0 && item < arr[i-1]) {
arr[i] = arr[i-1];
i -= 1;
}
arr[i] = item;
return arr;
}
``````

Updated JS Bin link

You should also take into consideration that depending on javascript implementation built in functions might be implemented in native code, and thus be a lot faster then your own JS code. And although your algorithm has lower complexity, sort + push might be faster.

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.