Neighbour Joining

1. Neighbour Joining By Sivan Yogev

2. Neighbour Joining • Algorithm presented by Saitou and Nei in 1987 • An approximation algorithm • If the given matrix is additive, the algorithm returns the correct tree

3. Description • Input: A set of n objects, with a distance between every two objects • Output: unrooted tree with the given objects as leaves • We will have a graph with the original objects as nodes and add nodes to this geaph. We keep a distance matrix between each two “active” nodes

4. Bear Raccoon Weasel Seal Dog Initialization • Each object in a separate node, distance by input • We will use an example of 5 mammal species

5. Iterations • We iterate until there is only one tree • At each iteration we perform: • Find the two nodes x and y with minimal value of D(x,y). • Add a new node z, and connect x and y to this node. • Calculate the distance from x and y to z. • Compute the distance between z and the other remaining nodes, and remove x and y from the set.

6. Example • First iteration:

7. BD Bear Dog Raccoon Weasel Seal Example (cont.) • Select minimal D(i,j) • We shall select Bear and Dog

8. BD 6 26 Bear Dog Raccoon Weasel Seal Update • Calculate distance from new node to its “sons”

9. Update (cont.) • Update computation – for each t x,y we set:

10. BDR 1.5 19.5 BD 6 26 Bear Dog Raccoon Weasel Seal Example (cont.) • Second iteration:

11. BDR 1.5 19.5 BD WS 6 26 22.25 21.75 Bear Dog Raccoon Weasel Seal Example (cont.) • Third iteration:

12. BDR 1.5 1.5 19.5 WS BD 6 26 22.25 21.75 Bear Dog Raccoon Weasel Seal Example (cont.) • Fourth (and last) iteration: