## Sunday, June 10, 2018

### graph equivalence part 2: comparing the graph signatures

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In the part 1 I've told how to build the signatures but haven't told how to compare them.

So, a signature of the graph is essentially the list of its nodes ordered in a specific way, by the algorithm from the part 1.  The basic idea of comparing the signatures is to compare the first node, if they are equal then the next node, and so on. But how do we compare the nodes?

Obviously, first we compare the tags of the nodes. Then we've got to compare the links. But we've got to compare the links. We've got to check that the links with the same tags lead to the nodes at the same indexes in the list. The easy way to do this is to order the links by their tags in advance. And this happens to be the same order that was used during the signature building, so it can be just kept from there. But there still is the problem of what to do when the links have the same tags. In this case we've got to check that these links lead to the sets of the same nodes in both graphs. The easy way to do it is to sort these links by the indexes of the nodes they point to, which can also be done in advance.

So after these pre-computations the comparisons can be done pretty fast, maybe even as a simple byte-string comparison. And the computed node hashes can be checked up front to make the comparison faster. And these hashes can be further combined into one value to produce a hash for the hash map.

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