Why is this LINQ so slow? - linq

Question

Can anyone please explain why the third query below is orders of magnitude slower than the others when it oughtn't to take any longer than doing the first two in sequence?
var data = Enumerable.Range(0, 10000).Select(x => new { Index = x, Value = x + " is the magic number"}).ToList();
var test1 = data.Select(x => new { Original = x, Match = data.Single(y => y.Value == x.Value) }).Take(1).Dump();
var test2 = data.Select(x => new { Original = x, Match = data.Single(z => z.Index == x.Index) }).Take(1).Dump();
var test3 = data.Select(x => new { Original = x, Match = data.Single(z => z.Index == data.Single(y => y.Value == x.Value).Index) }).Take(1).Dump();
EDIT: I've added a .ToList() to the original data generation because I don't want any repeated generation of the data clouding the issue.
I'm just trying to understand why this code is so slow by the way, not looking for faster alternative, unless it sheds some light on the matter. I would have thought that if Linq is lazily evaluated and I'm only looking for the first item (Take(1)) then test3's:
data.Select(x => new { Original = x, Match = data.Single(z => z.Index == data.Single(y => y.Value == x.Value).Index) }).Take(1);
could reduce to:
data.Select(x => new { Original = x, Match = data.Single(z => z.Index == 1) }).Take(1)
in O(N) as the first item in data is successfully matched after one full scan of the data by the inner Single(), leaving one more sweep of the data by the remaining Single(). So still all O(N).
It's evidently being processed in a more long winded way but I don't really understand how or why.
Test3 takes a couple of seconds to run by the way, so I think we can safely assume that if your answer features the number 10^16 you've made a mistake somewhere along the line.

Answer 1

The first two "tests" are identical, and both slow. The third adds another entire level of slowness.
The first two LINQ statements here are quadratic in nature. Since your "Match" element potentially requires iterating through the entire "data" sequence in order to find the match, as you progress through the range, the length of time for that element will get progressively longer. The 10000th element, for example, will force the engine to iterate through all 10000 elements of the original sequence to find the match, making this an O(N^2) operation.
The "test3" operation takes this to an entirely new level of pain, since it's "squaring" the O(N^2) operation in the second single - forcing it to do another quadratic operation on top of the first one - which is going to be a huge number of operations.
Each time you do data.Single(...) with the match, you're doing an O(N^2) operation - the third test basically becomes O(N^4), which will be orders of magnitude slower.

Answer 2

Fixed.
var data = Enumerable.Range(0, 10000)
.Select(x => new { Index = x, Value = x + " is the magic number"})
.ToList();
var forward = data.ToLookup(x => x.Index);
var backward = data.ToLookup(x => x.Value);
var test1 = data.Select(x => new { Original = x,
Match = backward[x.Value].Single()
} ).Take(1).Dump();
var test2 = data.Select(x => new { Original = x,
Match = forward[x.Index].Single()
} ).Take(1).Dump();
var test3 = data.Select(x => new { Original = x,
Match = forward[backward[x.Value].Single().Index].Single()
} ).Take(1).Dump();
In the original code,
data.ToList() generates 10,000 instances (10^4).
data.Select( data.Single() ).ToList() generates 100,000,000 instances (10^8).
data.Select( data.Single( data.Single() ) ).ToList() generates 100,000,000,000,000,000 instances (10^16).
Single and First are different. Single throws if multiple instances are encountered. Single must fully enumerate its source to check for multiple instances.