Queries to find distance between two nodes of a Binary tree – O(logn) method
Given a binary tree, the task is to find the distance between two keys in a binary tree, no parent pointers are given. Distance between two nodes is the minimum number of edges to be traversed to reach one node from other.
This problem has been already discussed in previous post but it uses three traversals of the Binary tree, one for finding Lowest Common Ancestor(LCA) of two nodes(let A and B) and then two traversals for finding distance between LCA and A and LCA and B which has O(n) time complexity. In this post, a method will be discussed that requires the O(log(n)) time to find LCA of two nodes.
The distance between two nodes can be obtained in terms of lowest common ancestor. Following is the formula.
Dist(n1, n2) = Dist(root, n1) + Dist(root, n2) - 2*Dist(root, lca) 'n1' and 'n2' are the two given keys 'root' is root of given Binary Tree. 'lca' is lowest common ancestor of n1 and n2 Dist(n1, n2) is the distance between n1 and n2.
The above formula can also be written as:
Dist(n1, n2) = Level[n1] + Level[n2] - 2*Level[lca]
This problem can be breakdown into:
- Finding levels of each node
- Finding the Euler tour of binary tree
- Building segment tree for LCA,
These steps are explained below :
- Find the levels of each node by applying level order traversal.
- Find the LCA of two nodes in binary tree in O(logn) by Storing Euler tour of Binary tree in array and computing two other arrays with the help of levels of each node and Euler tour.
These steps are shown below:
- First, find Euler Tour of binary tree.
- Then, store levels of each node in Euler array in a different array.
- Then, store First occurrences of all nodes of binary tree in Euler array. H stores the indices of nodes from Euler array, so that range of query for finding minimum can be minimized and their by further optimizing the query time.
- Then build segment tree on L array and take the low and high values from H array that will give us the first occurrences of say Two nodes(A and B) . Then, we query segment tree to find the minimum value say X in range (H[A] to H[B]). Then we use the index of value X as index to Euler array to get LCA, i.e. Euler[index(X)].
Let, A = 8 and B = 5.
- H = 1 and H =2
- Querying on Segment tree, we get min value in L array between 1 and 2 as X=0, index=7
- Then, LCA= Euler, i.e LCA = 1.
- Finally, we apply distance formula discussed above to get distance between two nodes.
Dist(4, 5) = 2 Dist(4, 6) = 4 Dist(3, 4) = 3 Dist(2, 4) = 1 Dist(8, 5) = 5
- Time Complexity: O(Log N)
- Space Complexity: O(N)