Building a Scalable Network Search

WARNING: This whole discussion assumes that every node is well-behaved, and not malicious. Let's take this stuff one step at a time, ok? 
ALSO: Sorry this is so long. Lot to say.

Not sure how to preamble this so... I'll just dive in. Picture of a kitten.

Plain ol' search

 

Since we already know (more or less) how think about tree algorithms, let's model our networks as them, the same way I show in my earlier post. 

This makes common-sense search pretty easy - let the node send a search request to all its subtrees, and make them pass it on. If a node has the desired information, it can call back straight to the origin, and celebrate a job well done with a pint of orange juice.
 
Here's what it looks like: If our tree has a branching factor of two (the network then has a branching factor of three, since we include the pseudo-parent node), each node checks its own records and passes the request to its two pseudo-children:

        ---C
       /
   ---B
  /    \
 /      ---D
A
 \      ---F
  \    /
   ---E
       \
        ---G

(fyi I labelled the children in pre-order)

Let's call the time taken for the request to reach a child delta_t. We can assume that actually firing a request might take negligible time, since the parent need not wait for a response to send the next request.  It's like how you can send five letters from the post office at the same time, instead of sending one, waiting for a response, then the second, and so on.

Taking time steps through the process looks like this:

t = 0:  A fires a request to B and then to E.
t = delta_t:  B fires a request to C then F, and E asks F and G
t = 2 * delta_t:  C, D, F, and G each send requests to their children, and so on.

This tells us that we can reach twice as many peers with each increment delta_t, and therefore search takes logarithmic time - doubling the number of peers only results in a constant-time incremental increase! Changing the branching factor or the magnitude of delta_t will just affect the magnitude of that increment. 

But here's the catch: this type of search will hit every peer, both in the best and the worst case. Even if the first child node B has the desired information, the remaining children will not only propagate down their own trees, but also down B's subtree! Even though B does not propagate, if C, D, E, etc.. are connected to any of B's children, those children will receive the request! sadface.gif!

This sucks. It means that although search scales well, the amount of work every node will do as part of other nodes' searches is linear with N. Because of this, large networks will quickly choke on all the bandwidth that searches require. Suppose one node runs, on average, one search per minute. The number of searches it must process is N per minute. Imagine doing this with a million nodes, with the majority of results turning up negative - it's a complete waste.

Slightly better (maybe) search

So obviously we need to modify our search algorithm. We'll certainly need to sacrifice some things, but let's talk about what we want to maintain:

1)  Full searchability: If one node in N has the data we're looking for, we want to find it eventually.
2)  (More-or-less) Constant time node work: A node ought not to do much more work in a network with large N (say a few million nodes) than with small N. We could of course *let* a node do more work if it wants to take it on...
3)  Time of data retrieval is related to abundance of data: Looking for something common like cat pictures should take less or equal time to something rare like videos of me doing something stupid at that party one time.

We can add to this list, but this is a good place to start. Here's my solution for now: Limiting the number of hops a request can make. For each hop, increment the 'hop counter' on the request, and if it reaches a sustainable limit J, hold the request. If any node in J has replied positive with the information, it should broadcast that the search can be called off. Ideally, information has been found within that radius and propagation may cease. Such a search results in node work proportional to J, and keeps a nice search time.

But what if nobody in the radius returns positive? I'd suggest letting one node in the outside circle propagate the request after waiting some time T for confirmation that the item has not been found. It may be helpful for each failure to broadcast their unique ID; this way, propagating the request will ignore nodes that have already been hit. To be honest, I'm not sure what the work complexity is of this kind of search, because there's a lot of sub-algorithms going on.

Every time the bounds have been exceeded, the search will experience a deficit of T extra time. Furthermore, the more 'common' the search (small number of extra T's) the less work every node does.  To this end, it is helpful for networks to be clustered into groups with similar interests (if I want cat pictures, I should hang with people that have a lot of cats). But I think this post is long enough : )

At this point I fell asleep, so...

...That's all for now. If you got this far, wow what the heck! We should hang out!

Or just leave a comment, that would be nice : )