Given a tree $T$, I want to find a subset $N$ of $n$ leaves that are farthest apart. I.e., I want to find $N$ that maximizes function:

$$f(N)=\sum\limits_{x_1,x_2 \in N, x_1 \neq x_2}{dist(x_1,x_2)}$$

where $dist(x_1, x_2)$ is the number of edges in a path between two vertices/nodes $x_1$ and $x_2$.

What algorithm can I use?


If you have a binary tree, you should be able to solve this using dynamic programming.

Let $A[v,j,k]$ denote the maximum possible value of the objective function

$$f'(N) = \sum_{x_1,x_2 \in N} \text{dist}(x_1,x_2) + k \sum_{x \in N} \text{dist}(v,x)$$

where $N$ ranges over all subsets of exactly $j$ leaves from among those that are descendants of $v$ (i.e., leaves of the subtree rooted at $v$).

If $v$ is a leaf, it is easy to compute $A[v,j,k]$ for all $j,k$.

If $v$ has one child $v'$, it is easy to compute $A[v,j,k]$ for all $j,k$ from $A[v',\cdot,\cdot]$.

So now suppose $v$ has two children $v',v''$. Then we can work out a recursive equation for $A[v,j,k]$ in terms of values $A[w,\cdot,\cdot]$ where the $w$'s are descendants of $v$:

$$A[v,j,k] = \max A[v',j',k+j''] + A[v'',j'',k+j'] + 2 j' j'' + jk$$

where $j',j''$ range over all values such that $j'+j'' = j$, $0 \le j',j'' \le j$. The intended meaning is that $j'$ counts the number of leaves in $N$ that are descendants of $v'$ and $j''$ counts the number of leaves in $N$ that are descendants of $v''$. In other words, we split $N=N' \cup N''$ where $N'$ contains $j'$ leaves from the descendants of $v'$, and $N''$ contains $j''$ leaves from the descendants of $v''$; then (loosely speaking) we compute the maximum value of $f'(N)$ in terms of the maximum values of $f'(N')$ and $f'(N'')$. The $2jj'$ term counts distances of the form $\text{dist}(x_1,x_2)$ where $x_1 \in N'$ and $x_2 \in N''$. The $jk$ term accounts for the fact that the $j$ root-to-leaf paths all need to be extended by one edge (thank you @j_random_hacker).

If you have an arbitrary tree (not necessarily a binary tree), you can convert it to a binary tree as follows: whenever you have a node $v$ with $m$ children, create a little binary tree with $m$ leaves, then put $v$ at the root of that binary tree and paste the children of $v'$ in at its $m$ leaves. Put a distance of 0 on all the edges of that new little binary tree except the top level. Repeat recursively.

In this way you obtain a binary tree where every edge has a distance of either 0 or 1 on it, and the distance between two vertices $v,w$ is the sum of the distances on the edges in the path between $v$ and $w$. Now you should be able to generalize the above algorithm to that case. I'll let you handle the details.

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    $\begingroup$ @dzieciou, Thank you, those are excellent points, my answer had some errors in it. See revised answer. Please review carefully; it might well have other errors as well. $\endgroup$ – D.W. Jan 17 at 20:39
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    $\begingroup$ Nice approach! I think that there needs to be an additional $jk$ term in the recursion formula though, to account for the fact that the $j$ leaf-to-$v$ paths that the 2 subproblem solutions refer to all need to be extended by 1 weight-$k$ edge. $\endgroup$ – j_random_hacker Jan 18 at 12:36
  • $\begingroup$ Thanks for revising the answer. One more note as I'm not sure if I understood your approach for converting arbitrary trees to binary ones. I've drawn an example. Here children from the original tree are always encoded as a left child in binary tree and edges to left childs have weight 1. Remaining edges (those to right children) have 0 weight. I thought about using Wikipedia procedure for encoding m-trees as binary trees, but it changes some leaves into internal nodes. $\endgroup$ – dzieciou Jan 18 at 18:25
  • $\begingroup$ @j_random_hacker, ooh, good point! Fixed. $\endgroup$ – D.W. Jan 18 at 20:03
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    $\begingroup$ @dzieciou, yup, that looks like it should work! (I was imagining something slightly different but yours should work just as well.) $\endgroup$ – D.W. Jan 18 at 20:04

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