HRW rendezvous hashing in log time?

Question

The Wikipedia page for Rendezvous hashing (Highest Random Weight "HRW") makes the following claim:

While it might first appear that the HRW algorithm runs in O(n) time, this is not the case. The sites can be organized hierarchically, and HRW applied at each level as one descends the hierarchy, leading to O(log n) running time, as in.[7]

I got a copy of the referenced paper, "Hash-Based Virtual Hierarchies for Scalable Location Service in Mobile Ad-hoc Networks." However the hierarchy referenced in their paper seems to be very specific to their application domain. As far as I can discern, there is no clear indication of how to generalize the method. The Wikipedia remark makes it seem like log is the general case.

I looked at a few general HRW implementations, and none of them seemed to support anything better than linear time. I gave it some thought, but I don't see any way to organize sites hierarchically without causing parent nodes to cause inefficient remapping when they drop out, significantly defeating the main advantage of HRW.

Does anybody know how to do this? Alternatively, is Wikipedia incorrect about there being a general way to implement this in log time?

Edit: Investigating mcdowella's approach:

OK, I think I see how this could work. But you need a little more than you've specified.

If you just do what you've described, you get in a situation where each leaf probably just has either zero or one nodes in it, and there's significant variance in how many nodes are in the leaf-most subtrees. If you swap using HRW at each level with just making the whole thing a regular search tree, you get exactly the same effect. Essentially, you've got an implementation of consistent hashing, along with its flaw of having unequal loading between buckets. Computing the combined weights, the defining implementation of HRW, adds nothing; you're better off just doing a search at each level, since it saves doing the hashes, and can be implemented without looping over each radix value

It's fixable though: you just need to be using HRW to choose from many alternatives at the final level. That is, you need all of the leaf nodes to be in large buckets, comparable to the number of replicas you'd have in consistent hashing. These large buckets should be approximately equally-loaded compared to each other, and then you're using HRW to choose the specific site. Since the bucket sizes are fixed, this is an O(n) algorithm, and we get all of the key HRW properties.

Honestly though, I think this is pretty questionable. It isn't so much an implementation of HRW, as it is just combining HRW with consistent hashing. I guess there's nothing wrong with that, and it might even be better than the usual technique of using replicas, in some cases. But I think it's misleading to state that HRW is log(n), if this is actually what the author meant.

Additionally, the original description is also questionable. You don't need to apply HRW at each level, and you shouldn't, as there is no advantage in doing so; you should do something fast (such as indexing), and just use HRW for the final choice.

Is this really the best we can do, or is there some other way to make HRW O(log(n))?

Answer 1

If you give each site a sufficiently long random id expressed in radix k (perhaps by hashing a non-random id) then you can associate the sites with leaves of a tree which has at most k descendants at each node. There is no need to associate any site with an internal node of the tree.

To work out where to store an item, use HRW to work out from the root of the tree down which way to branch at tree nodes, stopping when you reach a leaf, which is associated with a site. You can do this without having to communicate with any site until you work out which site you want to store the item at - all you need to know is the hashed ids of the sites to construct a tree.

Because sites are associated only with leaves there is no way an internal node of the tree can drop out, except if all of the sites associated with leaves under it drop out, at which point it will become irrelevant.