Algorithm to find diameter of a tree using BFS/DFS. Why does it work?

Question

This link provides an algorithm for finding the diameter of an undirected tree using BFS/DFS. Summarizing:

Run BFS on any node s in the graph, remembering the node u discovered last. Run BFS from u remembering the node v discovered last. d(u,v) is the diameter of the tree.

Why does it work ?

Page 2 of this provides a reasoning, but it is confusing. I am quoting the initial portion of the proof:

Run BFS on any node s in the graph, remembering the node u discovered last. Run BFS from u remembering the node v discovered last. d(u,v) is the diameter of the tree.

Correctness: Let a and b be any two nodes such that d(a,b) is the diameter of the tree. There is a unique path from a to b. Let t be the first node on that path discovered by BFS. If the paths $p_1$ from s to u and $p_2$ from a to b do not share edges, then the path from t to u includes s. So

$d(t,u) \ge d(s,u)$

$d(t,u) \ge d(s,a)$

....(more inequalities follow ..)

The inequalities do not make sense to me.

Answer 1

All parts of proving the claim hinge on 2 crucial properties of trees with undirected edges:

• 1-connectedness (ie. between any 2 nodes in a tree there is exactly one path)
• any node can serve as the root of the tree.

Choose an arbitrary tree node $s$. Assume $u, v \in V(G)$ are nodes with $d(u,v) = diam(G)$. Assume further that the algorithm finds a node $x$ starting at $s$ first, some node $y$ starting at $x$ next. wlog $d(s,u) \geq d(s,v)$. note that $d(s,x) \geq d(s,y)$ must hold, unless the algorithm's first stage wouldn't end up at $x$. We will see that $d(x,y) = d(u,v)$.

The most general configuration of all nodes involved can be seen in the following pseudo-graphics ( possibly $s = z_{uv}$ or $s = z_{xy}$ or both ):

(u)                                            (x)
  \                                            /
   \                                          /
    \                                        /
     ( z_uv )---------( s )----------( z_xy )
    /                                        \
   /                                          \
  /                                            \
(v)                                            (y)

we know that:

1. $d(z_{uv},y) \leq d(z_{uv},v)$. otherwise $d(u,v) < diam(G)$ contradicting the assumption.
2. $d(z_{uv},x) \leq d(z_{uv},u)$. otherwise $d(u,v) < diam(G)$ contradicting the assumption.
3. $d(s,z_{xy}) + d(z_{xy},x) \geq d(s,z_{uv}) + d(z_{uv},u)$, otherwise stage 1 of the algorithm wouldn't have stopped at $x$.
4. $d(z_{xy},y) \geq d(v,z_{uv}) + d(z_{uv},z_{xy})$, otherwise stage 2 of the algorithm wouldn't have stopped at $y$.

1) and 2) imply $\, \\ d(u,v) = d(z_{uv},v) + d(z_{uv},u) \\ \qquad\geq d(z_{uv},x) + d(z_{uv},y) = d(x,y) + 2\, d(z_{uv}, z_{xy}) \\ \qquad\qquad\geq d(x,y)$.

3) and 4) imply $\, \\ d(z_{xy},y) + d(s,z_{xy}) + d(z_{xy},x) \\ \qquad\geq d(s,z_{uv}) + d(z_{uv},u) + d(v,z_{uv}) + d(z_{uv},z_{xy}) \qquad\qquad\qquad\qquad \\ \, $ equivalent to $\, \\ d(x,y) = d(z_{xy},y) + d(z_{xy},x) \\ \qquad\geq 2*\,d(s,z_{uv}) + d(v,z_{uv}) + d(u,z_{uv}) \\ \qquad\qquad\geq d(u,v)$.

therefore $d(u,v) = d(x,y)$.

analogue proofs hold for the alternative configurations

                 (u)                          (x)
                   \                          /
                    \                        /
                     \                      /
     ( s )---------( z_uv )----------( z_xy )
                     /                      \
                    /                        \
                   /                          \
                 (v)                          (y)

and

                          (x)        (u)  
                          /            \  
                         /              \ 
                        /                \
     ( s )---------( z_xy )----------( z_uv )
                        \                /          
                         \              /           
                          \            /            
                          (y)        (v)            

these are all possible configurations. in particular, $x \not\in path(s,u), x \not\in path(s,v)$ due to the result of stage 1 of the algorithm and $y \not\in path(x,u), y \not\in path(x,v)$ due to stage 2.

Answer 2

The intuition behind is very easy to understand. Suppose I have to find longest path that exists between any two nodes in the given tree.

After drawing some diagrams we can observe that the longest path will always occur between two leaf nodes( nodes having only one edge linked). This can also be proved by contradiction that if longest path is between two nodes and either or both of two nodes is not a leaf node then we can extend the path to get a longer path.

So one way is to first check what nodes are leaf nodes, then start BFS from one of the leaf node to get the node farthest from it.

Instead of first finding which nodes are leaf nodes , we start BFS from a random node and then see which node is farthest from it. Let the node farthest be x. It is clear that x is a leaf node. Now if we start BFS from x and check farthest node from it, we will get our answer.

But what is the guarantee that x will be a end point of a maximum path?

Let's see by an example :-

       1   
    / /\ \
   6 2  4 8
         \ \
          5 9
           \
            7

Suppose I started my BFS from 6. The node at maximum distance from 6 is node 7. Using BFS we can get this node. Now we star BFS from node 7 to get node 9 at maximum distance. Path from node 7 to node 9 is clearly the longest path.

What if BFS that started from node 6 gave 2 as the node at maximum distance. Then when we will BFS from 2 we will get 7 as node at maximum distance and longest path will be then 2->1->4->5->7 with length 4. But the actual longest path length is 5. This cannot happen because BFS from node 6 will never give node 2 as node at maximum distance.

Hope that helps.

Answer 3

Here's a proof that follows the MIT solution set linked in the original question more closely. For clarity, I will use the same notation they use so the comparison can be more easily made.

Suppose we have two vertices $a$ and $b$ such that the distance between $a$ and $b$ on the path $p(a, b)$ is a diameter, e.g. the distance $d(a, b)$ is maximum possible distance between any two points in the tree. Suppose we also have a node $s \neq a, b$ (if $s = a$, then it would be obvious that the scheme works, since the first BFS would get $b$, and the second would return to a). Suppose also that we have a node $u$ such that $d(s, u) = \max_x d(s, x)$, e.g. $u = \mathrm{argmax}_x d(s, x)$.

Lemma 0: Both $a$ and $b$ are leaf nodes.

Proof: If they weren't leaf nodes, we could increase $d(a,b)$ by extending the endpoints to leaf nodes, contradicting $d(a, b)$ being a diameter.

Lemma 1: $\max [d(s,a), d(s, b)] = d(s, u)$.

Proof: Suppose for the sake of contradiction that both $d(s, a)$ and $d(s,b)$ were strictly less than $d(s, u)$. We look at two cases:

Case 1: path $p(a, b)$ does not contain vertex $s$. In this case, $d(a, b)$ cannot be the diameter. To see why, let $t$ be the unique vertex on $p(a, b)$ with the smallest distance to $s$. Then, we see that $d(a, u) = d(a, t) + d(t, s) + d(s, u) > d(a, b) = d(a, t) + d(t, b)$, since $d(s, u) > d(s, b) = d(s, t) + d(t, b) > d(t, b)$. Similarly, we would also have $d(b, u) > d(a, b)$. This contradicts $d(a, b)$ being a diameter.

Case 2: path $p(a, b)$ contains vertex $s$. In this case, $d(a, b)$ again cannot be the diameter, since for some vertex $u$ such that $d(s, u) = \max_x d(s, x)$, both $d(a, u)$ and $d(b, u)$ would be greater than $d(a, b)$.

Lemma 1 gives the reason why we start the second Breadth-First search at the last-discovered vertex $u$ of the first BFS. If $u$ is the unique vertex with the greatest possible distance from $s$, then by Lemma 1, it must be one of the endpoints of some path with a distance equal to the diameter, and hence a second BFS with $u$ as the root unambiguously finds the diameter. On the other hand, if there is at least one other vertex $v$ such that $d(s, v) = d(s, u)$, then we know that the diameter is $d(a, b) = 2d(s, u)$, and it doesn't matter whether we start the second BFS at $u$ or $v$.

Answer 4

It works because-

1. For any randomly chosen vertex $u$, the farthest vertex from it lies on one end of the diameter.

2. If vertex $a$ is one end of the diameter, the farthest vertex from it is the other end.

The first BFS takes you to one end of the diameter and the second BFS from that end takes you to the other end.

Proof

#2 is easy to see why it's true. We will see why #1 is true by contradiction. Let's suppose, the path from vertex $a$ to $b$ is the diameter of the tree and $v$, which is not on the diameter, is the farthest vertex from $u$. Also, let's assume paths $(a,b)$ and $(u,v)$ cross over at $t$ ($t$ can more generally be a path with multiple vertices). Because we don't have any cycles, all paths between vertices $a$, $b$, $u$ and $v$ pass through $t$.

1. By our assumptions $d(a,b)$ is the diameter and $d(u,v)$ is larger than both $d(u,a)$ and $d(u,b)$.

2. Path $(u,t)$ is the common piece in all paths originating from $u$ and hence we can deduce that $d(v, t) > d(a, t)$ ($v$ is farther away from $t$ as compared to $a$).

3. This implies that $v$ is also farther away from $b$ as compared to $a$ and $d(v,b) > d(a,b)$. (add $d(t,b)$ to both sides in previous step)

4. This contradicts our initial assumption that $(a,b)$ is a diameter because we have found a path $(v,b)$ which is longer. Symmetrically, we can also see why $d(v,a) > d(a,b)$.

5. Hence, the farthest vertex from $u$ must either be $a$ or $b$ if $(a,b)$ is indeed the diameter.

Answer 5

By the definition of BFS, the distance (from the starting node) of each node explored is either equal to the distance of the previous node explored or greater by 1. Thus, the last node explored by BFS will be among those farthest from the starting node.

Thus, the algorithm of using BFS twice amounts to "Pick an arbitrary node $x$. Find the node $a$ farthest from $x$ (last node found by BFS starting from $x$). Find the node $b$ farthest from $a$ (last node found by BFS starting from $a$).", which thus finds two nodes of maximum distance from eachother.

Answer 6

One key thing to know is that a tree is always planar, which means it can be laid out on a plane, so often ordinary 2-dimensional thinking works. In this case, the algorithm says starting anywhere, go as far away as you can. The distance from that point to as far as you can get away from that point is the longest distance in the tree, and therefore the diameter.

This method would also work to find the diameter of a real, physical island if we defined that as the diameter of the smallest circle that would fully enclose the island.

Answer 7

@op, the way the cases are defined in the PDF may be a bit off.

I think that the two cases should be:

1. $p_1$ does not intersect with $p_2$, i.e. there are no common vertices between paths $p_1$ and $p_2$. In this case, define $t$ as the first node on $p_2$ discovered by the first BFS starting at $s$.

2. $p_1$ and $p_2$ have at least one common vertex. In this case, define $t$ to be the first node on $p_2$ discovered by the first BFS that is also on $p_1$.

The rest of the proof in the PDF should follow.

With this definition, the figure shown by OP falls into Case 2.

Answer 8

First run a DFS from a random node then the diameter of a tree is the path between the deepest leaves of a node in its DFS subtree: