Finding the Largest and Second Largest Numbers in n + log₂(n) - 2 Comparisons - 常规

To find the largest and second largest numbers among n given numbers, we can use a modified version of the tournament method.

Algorithm:

Divide the given n numbers into pairs and compare each pair. Keep track of the larger number in each pair.
Repeat step 1 until we have only two numbers left.
- For each pair, compare the larger numbers obtained in the previous step and keep track of the largest number overall.
Now, we have the largest number among the n numbers.
To find the second largest number, we need to compare the remaining n-1 numbers (excluding the largest number found in step 3) with the largest number found in step 3.
- Compare the largest number found in step 3 with each of the remaining n-1 numbers and keep track of the second largest number.
Return the largest and second largest numbers.

Correctness:

In each pair comparison, we keep track of the larger number. So, after comparing all pairs, we have the largest number among the given n numbers.
In step 2, we compare the larger numbers obtained from the previous step until we have only two numbers left. So, we can find the largest number overall.
In step 4, we compare the remaining n-1 numbers with the largest number found in step 3. By doing this, we can identify the second largest number among the given n numbers.

Number of comparisons:

In each pair comparison (step 1), we compare two numbers, which requires 1 comparison. Since we divide the n numbers into pairs, the number of comparisons in step 1 is n/2.
In step 2, we need to compare the larger numbers obtained from the previous step. In each comparison, we compare two numbers, which requires 1 comparison. Since we repeat step 2 log₂(n) times, the number of comparisons in step 2 is log₂(n).
In step 4, we compare the remaining n-1 numbers with the largest number found in step 3. This requires n-1 comparisons.
Therefore, the total number of comparisons is n/2 + log₂(n) + n-1 = n + log₂(n) - 2.

Hence, the algorithm finds the largest and second largest numbers in n + log₂(n) - 2 comparisons.