# UberProf performance improvements, or: when O(N^3 + N) is not fast enough

While working on improving the performance of the profiler, I got *really* annoyed. The UI times for updating sessions with large amount of statements was simply unacceptable. We already virtualized the list UI, so it couldn’t be that. I started tracking it down, and it finally came down to the following piece of code:

protected override void Update(IEnumerable<IStatementSnapshot> snapshots) { foreach (var statementSnapshot in snapshots) { var found = (from model in unfilteredStatements where model.Id == statementSnapshot.StatementId select model).FirstOrDefault(); if (found == null) { unfilteredStatements.Add( new StatementModel(applicationModel, statementSnapshot) {CanJumpToSession = false}); } else { found.UpdateWith(statementSnapshot); } } if (snapshots.Any()) { hasPolledForStatements = true; StatementsChanged(this, new StatementsChangedEventArgs {HasPolledForStatments = hasPolledForStatements}); } }

This looks pretty innocent, right? At a guess, what is the performance characteristics of this code?

It isn’t O(N), that is for sure, look at the linq query, it will perform a linear search in the unfilteredStatements, which is an O(N) operation. *

At first glance, it looks like O(N*N), right? Wrong!

unfilteredStatements is an ObservableCollection, and guess what is going on in the CollectionChanged event? A statistics function that does yet *another* O(N) operation for *every single added statement*.

Finally, we have the StatementsChanged handler, which will also perform an O(N) operation, but only once.

protected override void Update(IEnumerable<IStatementSnapshot> snapshots) { var hasStatements = false; foreach (var statementSnapshot in snapshots) { IStatementModel found; statementsById.TryGetValue(statementSnapshot.StatementId, out found); hasStatements = true; if (found == null) { unfilteredStatements.Add(new StatementModel(applicationModel, statementSnapshot) {CanJumpToSession = false}); statementsById.Add(statementSnapshot.StatementId, found); } else { found.UpdateWith(statementSnapshot); } } if (hasStatements == false) return; HandleStatementsOrFiltersChanging(true); }

The changes here are subtle, first, we killed off the O(N) linq query in favor of an O(1) lookup in a hashtable. Second, unfilteredStatements is now a simple List<T>, and HandleStatementsOrFilterChanging is the one responsible for notifications. Just these two changes were sufficent from the O(N^3+N) that we had before to a simple O(N+N) (because we still run some statistical stuff at the end) which collapses so a simple O(N).

Once that is done, showing sessions with 50,000 statements in them became instantaneous.

* Yes, it is supposed to be O(M) because the unfilteredStatements.Count is different than snapshots.Count, but I am simplifying here to make things easier.

## Comments

Can't wait to see these performance improvements in action myself. Keep up the good work!

At second glance, it still does to me.

Your loop body is a sequence of two O(n) operations, summing to O(n). So we have a O(n²) loop followed by a O(n) operation, that's O(n²+n) = O(n²).

...whatever - another great example of real world performance improvements!

Sebastian,

Note the next statement, unflitedStaetments is firing a CollectionChanged event that has a subscriber that does another O(N) operation

I think Sebantian is right, this is O(N^2).

Unless the CollectionChangedEvent is performing an O(N) operation and for each value of N the subscriber is performing, everything in unfilteredStatements is O(N), not O(N^2).

It is O(N), the for loop, operating on N+N+N, the three O(N) statements, which comes out to O(N^2).

Mike,

As I said in the post, the collection changed event is making an O(N) operation every time it is invoked.

Just my fifty cents.

Anyway, why haven't you simply written O(N^3) instead of O(N^3 + N) and O(N) instead of O(N + N)?

For me it's confusing to see odd O(N^3 + N) instead of O(N^3). And even more confusing to see this in the post title...

I also think it's O(N^2). The LINQ query takes O(N) and the unfilteredStatements.Add statements takes O(N). This sums up to O(N + N) and not O(N * N) within the foreach loop.

arent people reading your post ayende? three comments about n^2 already...

Yep, it's not O(N^3) but O(N^2) like Manu said. If it were O(N^3) the program would probably take 50 years to process 50 000 statements.

Vadim,

O(N^3 + N) express what is going on better than just N^3. Note the comment in the end of the post.

We have..

1.a. Linq query - O(N)

1.b OrderedCollection.Add - O(N)

Now the Linq query and the OrderCollection.Add are in the scope of the outer loop, so we have O(N*2N+N) which is O(N^2)

I get your point. Though I still don't agree.

Anyway, you use big o notation to discuss the complexity of the algorithm. N in the O(N^3 + N) barely adds anything to the overall time complexity.