[英]Time Complexity of Nested for Loops with if Statements
How does the if-statement of this code affect the time complexity of this code?此代码的 if 语句如何影响此代码的时间复杂度?
Based off of this question: Runtime analysis , the for loop in the if statement would run n*n times.基于这个问题:运行时分析,if 语句中的 for 循环将运行 n*n 次。 But in this code, j outpaces i so that once the second loop is run
j = i^2
.但是在这段代码中, j超过了i ,因此一旦运行第二个循环
j = i^2
。 What does this make the time complexity of the third for loop then?那么这使第三个 for 循环的时间复杂度如何呢? I understand that the first for loop runs n times, the second runs
n^2
times, and the third runs n^2
times for a certain amount of times when triggered.我知道第一个 for 循环运行 n 次,第二个运行
n^2
次,第三个在触发时运行n^2
次。 So the complexity would be given by n*n^2(xn^2)
for which n
is the number of times the if statement is true.因此,复杂性将由
n*n^2(xn^2)
给出,其中n
是 if 语句为真的次数。 The complexity is not simply O(n^6)
because the if-statement is not true n times right?复杂性不仅仅是
O(n^6)
,因为 if 语句不是真的 n 次,对吧?
int n;
int sum;
for (int i = 1; i < n; i++)
{
for (int j = 0; j < i*i; j++)
{
if (j % i == 0)
{
for (int k = 0; k < j; k++)
{
sum++;
}
}
}
}
The if
condition will be true when j
is a multiple of i
;当
j
是i
的倍数时, if
条件为真; this happens i
times as j
goes from 0 to i * i
, so the third for
loop runs only i
times.当
j
从 0 变为i * i
时,这会发生i
次,因此第三个for
循环仅运行i
次。 The overall complexity is O(n^4).总体复杂度为 O(n^4)。
for (int i = 1; i < n; i++)
{
for (int j = 0; j < i*i; j++) // Runs O(n) times
{
if (j % i == 0) // Runs O(n) × O(n^2) = O(n^3) times
{
for (int k = 0; k < j; k++) // Runs O(n) × O(n) = O(n^2) times
{
sum++; // Runs O(n^2) × O(n^2) = O(n^4) times
}
}
}
}
The complexity is not simply O(n^6) because the if-statement is not true n times right?
复杂性不仅仅是 O(n^6),因为 if 语句不是真的 n 次,对吧?
No, it is not.不它不是。
At worst, it is going to be O(n^5)
.在最坏的情况下,它将是
O(n^5)
。 It is less than that since j % i
is equal to 0
only i
times.它小于这个值,因为
j % i
仅i
次等于0
。
The first loop is run n
times.第一个循环运行
n
次。
The second loop is run O(n^2)
times.第二个循环运行
O(n^2)
次。
The third loop is run at most O(n)
times.第三个循环最多运行
O(n)
次。
The worst combined complexity of the loop is going to be O(n) x O(n^2) x O(n)
, which is O(n^4)
.循环的最差组合复杂度将是
O(n) x O(n^2) x O(n)
,即O(n^4)
。
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