I'm trying to figure out the time complexity for this nested loop code snippet. Here's what it looks like:
```java
for (int i = 0; i < n; ++i) {
for (int j = 0; j < i * i; ++j) {
for (int k = 0; k < j; ++k) {
++sum;
}
}
}
```
Any insights would be greatly appreciated! Thanks!
1 Answer
The innermost loop runs for 'j' iterations, and the middle loop runs for 'i^2' iterations since 'j' ranges from 0 to 'i^2 - 1'. If we add it all up, the total number of operations for a single 'i' is essentially the sum of 'j' from 0 to 'i^2 - 1', which gives us the formula (i² - 1)(i²) / 2. When you take into account the outer loop running 'n' times, we can conclude that the overall complexity sums to O(n^5). It's typically what you'd expect in time complexity analyses.
Absolutely! That's a solid breakdown. You can simplify the last part a bit more to get 1/2 * (sum of i^2 (i^2 - 1)), which shows that the sum of i^2 becomes less significant compared to the sum of i^4. There are different ways to prove this is indeed in O(n^5), such as using integrals or asymptotic analysis.