2022-408DS first question (finding time complexity)

1 title description

The time complexity of the following program segment is

int sum = 0;
for(int i = 1; i < n; i *= 2)
    for(int j = 0; j < i; j++)
        sum++;

2 ideas

Observing the program, you can find thatthe number of executions of the inner layer is consistent with the number of executions of the outer layer, and the inner layer and the outer layer are "bundled" Relationship, in short, the outer layer is executed once and the inner layer is executed i times. So the final result is the time complexity of the inner layer.
The overall idea is to first find the number of executions of the outer layer, and then substitute it into the inner layer to calculate the final time complexity.

3 answers

Fayi

First recall the geometric sequence summation formula:
$$S=a_1\frac{1-q^n}{1-q}$$Among thema1Representative head, qRepresentative common ratio,nNumber of numbers.

Back to this question, first observe the outermost for cycle, the changing trend of i is (wherek Expressed as the kth execution):
$$1,2,4,8,...,2^{k-1}$$Roughly obtained outer layer order:
$$2^{k-1}=n$$$$k-1=lo{g}_{2}n$$$$k=lo{g}_{2}n+1$$$$\approx lo{g}_{2}n$$
Next, find the time complexity of the inner layer. Since the number of executions of the inner layer is the same as the number of executions of the outer loop variablei, then The changing trend of the execution times of the inner loop can be drawn:$$1,2,4,8,...,lo{g}_{2}n$$Chinese neck head1，Kobi 为2，Ichigo$lo{g}_{2}For\; the\; n$ term, substitute it into the geometric sequence summation formula to get:$$1\cdot \frac{1-2^{lo{g}_{2}n}}{1-2}$$$$=\frac{1-n}{-1}$$$$\to O(n)$$

Houji

Refer to method 1. The maximum value of j in the inner layer is equal to the maximum value of i, that is, $2^{k-1}$, the most trips to the outer layer$lo{g}_{2}n$ times [k is the number of executions], write the formula:
$$\sum _{k=0}^{lo{g}_{2}n}\sum _{j=0}^{2^{k-1}}1=\sum _{k=0}^{lo{g}_{2}n}{2}^{k+1}=\frac{1-2^{lo{g}_2(n+1)}}{1-2}=n+1-1$$$$=O\left(n\right)$$

Summarize

To put it simply, this question can also be thought of like this:

i=1, executed once,
i=2, executed 2 times,
i=4, executed 4 times, The result is obtained by summing the geometric sequence, which is the final time complexity i=2 to the kth power, indicating that a total of 2 to the kth power is executed ; …
I=8, executed 8 times,

The content is for reference only, please point out if there are any errors, thank you!