Peter Todd wrote:
My solution was to simply state that vertexes that happened to cause the
tree to be unbalanced would be discarded, and set the depth of inbalance
such that this would be extremely unlikely to happen by accident. I'd
rather see someone come up with something better though.
Here is a simpler solution. (most of this message repeats the content
of my reply to the forum)
Suppose we were talking about a binary search tree, rather than a
Merkle tree. It's important to balance a binary search tree, so that
the worst-case maximum length from the root to a leaf is bounded by
O(log N). AVL trees were the original algorithm to do this, Red-Black
trees are also popular, and there are many similar methods. All
involve storing some form of 'balancing metadata' at each node. In a
RedBlack tree, this is a single bit (red or black). Every operation on
these trees, including search, inserting, deleting, and rebalancing,
requires a worst-case effort of O(log N).
Any (acyclic) recursive data structure can be Merkle-ized, simply by
adding a hash of the child node alongside each link/pointer. This way,
you can verify the data for each node very naturally, as you traverse
the structure.
In fact, as long as a lite-client knows the O(1) root hash, the rest
of the storage burden can be delegated to an untrusted helper server.
Suppose a lite-client wants to insert and rebalance its tree. This
requires accessing at most O(log N) nodes. The client can request only
the data relevant to these nodes, and it knows the hash for each chunk
of data in advance of accessing it. After computing the updated root
hash, the client can even discard the data it processed.
This technique has been well discussed in the academic literature,
e.g. [1,2], although since I am not aware of any existing
implementation, I made my own, intended as an explanatory aid:
https://github.com/amiller/redblackmerkle/blob/master/redblack.py
[1] Certificate Revocation and Update
Naor and Nissim. 1998
http://static.usenix.org/publications/library/proceedings/sec98/full_papers/nissim/nissim.pdf
[2] A General Model for Authenticated Data Structures
Martel, Nuckolls, Devanbu, Michael Gertz, Kwong, Stubblebine. 2004
http://truthsayer.cs.ucdavis.edu/algorithmica.pdf
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