When we traverse the tree post order to assign coordinates to each of the internal nodes. Do this for each of the internal nodes.

Create 4 buckets 1 for each quadrant on a circle.

Also create four flags 1 for each quadrant. A true value of a flag for a quadrant indicates the presence of atleast one node in that quadrant.

Step 1:

for each child node, child of parent node.

Putting the node in the appropriate bucket depending on its quadrant.

Mark the flag true for that quadrant.

Step 2:

Sort each bucket so that the minimum values are towards the top or the beginning in the bucket. Use the sorting algorithm discussed in the previous post. If you have to sort negative numbers sort mod of them and then reverse the order.

It is to be noted that since you are sorting only within a bucket all elements within the same bucket have the same sign. This takes O(d) time where d is the maximum degree of any node in the phylogeny.

Step 3:

def distance (bucket a, bucket b):

the maximum distance between any two nodes in these buckets.

maxDistance = distance(bucket 1, bucket 3)

maxDistance = max { maxDistance, distance(bucket 2, bucket 4) }

maxDistance = max { maxDistance, distance(bucket 1, bucket 2)}

maxDistance = max { maxDistance, distance(bucket 2, bucket 3)}

maxDistance = max { maxDistance, distance(bucket 3, bucket 4)}

maxDistance = max { maxDistance, distance(bucket 4, bucket 1)}

I was trying to find out a general formula interms of variables that saves all this but my attempts in that direction didn't yield any feasible results. So now I believe that no one universal formula exists which can capture the centroid location for all the nodes on the globe so therefore these elaborations.

Once we find the 2 nodes between which we have the maximum distance we can transform the coordinates so as to make the coordinate of the one of these nodes zero.

Create 4 buckets 1 for each quadrant on a circle.

Also create four flags 1 for each quadrant. A true value of a flag for a quadrant indicates the presence of atleast one node in that quadrant.

Step 1:

for each child node, child of parent node.

Putting the node in the appropriate bucket depending on its quadrant.

Mark the flag true for that quadrant.

Step 2:

Sort each bucket so that the minimum values are towards the top or the beginning in the bucket. Use the sorting algorithm discussed in the previous post. If you have to sort negative numbers sort mod of them and then reverse the order.

It is to be noted that since you are sorting only within a bucket all elements within the same bucket have the same sign. This takes O(d) time where d is the maximum degree of any node in the phylogeny.

Step 3:

def distance (bucket a, bucket b):

the maximum distance between any two nodes in these buckets.

maxDistance = distance(bucket 1, bucket 3)

maxDistance = max { maxDistance, distance(bucket 2, bucket 4) }

maxDistance = max { maxDistance, distance(bucket 1, bucket 2)}

maxDistance = max { maxDistance, distance(bucket 2, bucket 3)}

maxDistance = max { maxDistance, distance(bucket 3, bucket 4)}

maxDistance = max { maxDistance, distance(bucket 4, bucket 1)}

I was trying to find out a general formula interms of variables that saves all this but my attempts in that direction didn't yield any feasible results. So now I believe that no one universal formula exists which can capture the centroid location for all the nodes on the globe so therefore these elaborations.

Once we find the 2 nodes between which we have the maximum distance we can transform the coordinates so as to make the coordinate of the one of these nodes zero.

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