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This article follows the insertion sort (concept article) and presents the implementation steps and implementation code of the binary insertion sort

Binary Insertion Sort Algorithm Steps

1. Treat the first element of the first sequence to be sorted as an ordered sequence, and the second element to the last element as an unsorted sequence.
2. Scan the unsorted sequence from beginning to end, and insert each scanned element into the appropriate position of the ordered sequence. Binary insertion sort uses the binary search method to find the appropriate position in the ordered sequence to insert the unsorted elements. (One thing to note here is that if there is an element in the ordered sequence that is equal to the element to be inserted, the element to be inserted will be found after this element. This way of insertion sorting is stable. If it is inserted in front of this element, then this way of insertion sorting is unstable)

Implementation code (PHP code)

$arr1 = array(15,77,23,43,90,87,68,32,11,22,33,99,88,66,44,113,224,765
,980,159,456,7,998,451,96,0,673,82,91,100);
for($i=1;$i<count($arr);$i++){
    if($arr[$i]<$arr[$i-1]){
        //使用二分查找法查找适当的位置
        $low = 0;
        $high = $i-1;
        $pos = floor(($low+$high)/2);
        $key = $arr[$i];
        while($low<$high){
            if($arr[$pos]>$key){
                $high = $pos-1;
            }elseif($arr[$pos]<=$key){
                $low = $pos+1;
            }
            $pos = floor(($low+$high)/2);
        }
        //二分查找法结束
        if($arr[$pos]>$arr[$i]){
            $pos = $pos-1;
        }
        for($j=$i-1;$j>$pos;$j--){
            $arr[$j+1]=$arr[$j];
        }
        $arr[$j+1] = $key;
    }
}

The above is the implementation code of Binary Insertion Sort. From the code, we can see that the only difference between Binary Insertion Sort and Direct Insertion Sort is the code for finding the position. The rest are the same. Binary Insertion Sort needs to move the same elements as Direct Insertion Sort. Therefore, its time complexity is also O(n²).

But further analysis shows that even though the time complexity of the two is the same, binary insertion sort is faster than direct insertion sort overall. The following is a test I did (the data required here cannot be the data in the above code. Here I constructed an array of 100 data for testing)

$arr = array(15,77,23,43,90,87,68,32,11,22,33,99,88,66,44,113,224,765,980,
159,456,7,998,451,96,0,673,82,91,100,36,57,1334,5567,4432,11178,9997,88851,
4449,33332,9992,76853,67434,1239,98716,5349,90718,3589,213450,6754,24,
562,77,16,127,361,572,13343,55674,44325,1117,99979,88850,44491,3333,99923,
7685,6743,12395,9871,53497,9071,35899,21345,67541,24,56,774,16,127,111,112,
113,114,115,116,117,118,119,110,101,102,103,104,105,106,107,108,109,1000);

The direct insertion sort code uses the following paragraph

for($i=1;$i<count($arr);$i++){
    $p = $arr[$i];
    for($j=$i-1;$j>=0;$j--){
        if($arr[$j]>$p){
            $arr[$j+1] = $arr[$j];
        }else{
            break;
        }
    }
    $arr[$j+1] = $p;
}

Based on the same set of data, we run these two codes multiple times and record the time taken for each. The following is the time comparison for each time. The one above is binary insertion sort and the one below is direct insertion sort.

From the above comparisons, we can find that the time required for binary insertion is shorter than that required for direct insertion. The larger the amount of data, the more obvious the time difference between the two methods. Therefore, in actual projects, if we really need to use the insertion sort algorithm, I personally recommend using binary insertion sort. Binary insertion may not be better than direct insertion in terms of space usage, because binary insertion requires more other variables than direct insertion. So when we only have dozens of data, run the above two codes again, you will be surprised to find that the time of the former (binary insertion) is actually longer than the latter (direct insertion). This is because so many variables need to allocate memory for them, and the time spent accounts for a large proportion of the entire running process, so the time of the former is longer than the latter. However, as the amount of data increases, the proportion of time allocated for variables gradually becomes smaller and smaller, which can be almost ignored. Therefore, when the amount of data is large, the time of the former is shorter than the latter. Moreover, with the current hardware level, in my opinion, the advantage of time can make up for the relative lack of space. If in actual situations our data volume really reaches the point where the advantage of time can no longer make up for the space, then there are other sorting algorithms for us to choose.