O（N + m）与O（NM）之间的复杂度计算差异

Question

In the algorithm below I cannot understand why the complexity is O(N+M),as calculated by codility.com examination tool, instead of be O(NM). I suppose it is O(NM) because of the two iteration one nested to the other one.

function solution($N, $A) {
// write your code in PHP7.0
$counter=[];
$max=0;

foreach($A as $value){
    if($value >= 1 && $value <= $N){
        isset($counter[$value-1]) ? $counter[$value-1]++ : $counter[$value-1] = 1;
        if($counter[$value-1] > $max){
            $max = $counter[$value-1];

            }
        }
    if($value == $N+1){
        for($i = 0; $i<$N; $i++){
            $counter[$i]=$max;
            }
        }
    }
return $counter;
}

Since I have not really understood the differences between the two complexities, can anybody give me two simple algorithms showing the two complexities?

Answer 1

如果A数组中的所有值都等于N+1 ，则counter数组将被填充M=Length(A)次，因此最坏的情况是O(N*M) ，如果A内容没有特殊限制。