If algorithm A and another algorithm B take $$log_{2} (n)$$​ and $$\sqrt{n}$$​  microseconds, respectively to solve a problem, then the largest size n of a problem these algorithms can solve, respectively, in one second are _____________ and _________.

Correct option is B

Let's break down and solve the question step by step:
Problem Given:
• Algorithm A takes $$\log_2(n)$$​ microseconds.
• Algorithm B takes $$\sqrt{n}$$​  microseconds.
• You need to find the largest size n that these algorithms can solve in one second.
Step 1: Convert time into microseconds
• We know that 1 second = $$10^6$$​ microseconds.
Step 2: Solve for the maximum n for each algorithm
For Algorithm A:
• The time taken by Algorithm A is $$\log_2(n)$$​ microseconds.
• We are asked to find the maximum size n such that it can be solved within 1 second, i.e., within $$10^6$$​ microseconds.
​$$\log_2(n) = 10^6$$​​
• Solve for n:
​$$n = 2^{10^6}$$​​
• So, $$n = 2^{10^6}$$​​
For Algorithm B:
• The time taken by Algorithm B is $$\sqrt{n}$$​  microseconds.
• We are asked to find the maximum size n such that it can be solved within 1 second, i.e., within $$10^6$$​ microseconds.
​$$\sqrt{n} = 10^6$$​​
• Solve for n:
​$$n = (10^6)^2 = 10^{12}$$​​
So, $$n = 10^{12}$$​.
Final Answer:
The largest sizes n that each algorithm can solve in 1 second are:
• For Algorithm A: $$2^{10^6}$$​​
• For Algorithm B: $$10^{12}$$​​