✏️ Explanatory Question
The efficiency of an algorithm is n3. If each step takes 1 nanosecond (10−9 seconds), how long will the algorithm take to process an input of size 1000?
The efficiency of an algorithm is n3. If each step takes 1 nanosecond (10−9 seconds), how long will the algorithm take to process an input of size 1000?
Ans. For n = 1000:
(1000)3 × 10−9 = 109 × 10−9 = 1 second
💡 Explanation:
First, substitute n = 1000 into the efficiency expression:
n3 = (1000)3 = 1,000,000,000 = 109 steps
Since each step takes 1 nanosecond = 10−9 seconds, the total execution time is:
109 × 10−9 seconds = 100 seconds = 1 second
Notice how much slower n3 is than 5n2. For the same input size, n = 1000, the cubic algorithm takes 1 second, whereas the quadratic algorithm in Q9 takes only 5 milliseconds. This illustrates the significant cost of a higher-order growth rate.