Refined Buneman Trees

So how do we find σ ′ We start by counting T bottom-up with respect to
U and V ,wehavethatforeverynodew ∈ T which hangs from some edge e
that u w is equal the the value |U ′ ∩ U|. U ′ is the set of leaves which lie in the
subtree of w, sou w is both the number of elements from U in w’s subtree and
the number of elements that U and U ′ have in common. Similarly, we also have
that v w = |U ′ ∩ V | by the same argument.
Now we apply basic set theory to find the values |V ′ ∩ U| = |U|−|U ′ ∩ U|
and |V ′ ∩ V | = |V |−|V ′ ∩ V |. We have all four quantities, and we can check if
any one of them is zero, and answer the question whether w is the node we are
looking for (consequently whether e is the edge we are looking for).
The time spent searching for an incompatible node is linear. We use linear
time decorating T with U/V counts, and we use linear time checking each one for
incompatibility. Given u w , v w , |U| and |V | we can in constant time determine
incompatibility by calculating the quantities described above.
We are not guaranteed to find an incompatible split, but if we do, we can report
it in linear time. To do this, we allocate a bitvector b of length n and search
through one of the subtrees induced by e. Wemarkanentryinb corresponding
to the index of every leaf we find.
6.3 Testing the tree data structure
The tree data structure has not been tested formally. Of course, all tree operations
have been unit-tested, and have been found to be correct. Also, performance
tests have been run to some extent, but this is not easy: if we want to
test the performance of the tree operations, we would need to be able to create
highly resolved trees by inserting a large number of compatible splits created
at random, which could then be a basis for inserting, deleting or searching for
splits. Doing the operations on an unresolved tree would not give a realistic
picture of the performance. It is quite easy to generate a larvae-tree by starting
out with a bitvector in the form 11000..., and then generating splits of
the form 111000..., 1111000..., 11111000... andsoon. Thiswasusedfor
informal performance testing, but clearly this kind of tree is biased, and the
author has chosen not to present a formal test on this basis. Still, the author
claims that the tree data structure does indeed run as specified, referring to
the performance test for the whole algorithm — if the tree data structure did
not support linear time insertion, deletion and searching, the refined Buneman
algorithm would not be able to perform as specified.
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