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I couldn't figure this one out. Given an unsorted list, we want to build a balanced binary tree (not a search tree, namely - left child is of lower key, and right child of higher key). Can we do it in linear time? Thanks!

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    $\begingroup$ What do you think? What have you tried? Please edit the question to show your thoughts, partial progress or where you got stuck. $\endgroup$ – Apass.Jack Mar 5 at 13:05
  • $\begingroup$ Are you over there? $\endgroup$ – Apass.Jack Mar 6 at 14:27
  • $\begingroup$ I thought maybe building a heap out of the list in O(n), and then start to switch nodes between parent nodes and it's children accordingly until we get a balanced binary tree, but i think this "solution" misses a lot of small points. $\endgroup$ – Yariv Levy Mar 7 at 9:50
  • $\begingroup$ If this is not a search tree, and you just need a balanced binary tree then this is rather simple. Assume $n = 2^k - 1$ w.l.o.g., then have element $A[i]$ be the left child of $A[\lfloor i / 2 \rfloor ]$ if $i$ is even, otherwise it is the right child if $i$ is odd. If you want an ordered tree, things become more difficult. This assumes $A$ is 1-indexed. $\endgroup$ – ryan Mar 19 at 21:45
  • $\begingroup$ @ryan Thanks, but it's not what I meant. I wanted a binary tree (namely - left child smaller and right child bigger), which is not necessarily a BST. $\endgroup$ – Yariv Levy Mar 19 at 21:47
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Answer #2.

An alternative approach (arguably more elegant). This basically does the same operations as my other answer, just in a different order. It is as follows:

build-tree(list A of length n):
  if n == 1:
    return singleton tree of the 1 element in A.

  let m be the median element of A  # linear time select

  let L be all elements in A less than m
  let R be all elements in A greater than m

  construct a max-heap HL from L in linear time
  construct a min-heap HR from R in linear time

  let m be the root node of tree T
  let the structure of HL be the structure of all  left-children in T
  let the structure of HR be the structure of all right-children in T

  return T

Let's try an example on $[8,12,9,15,5,1,10,3,11,14,2,4,13,7,6]$.

  1. First we find the median $m = 8$.
  2. We find $L = [5,1,3,2,4,7,6]$ and $R = [12,9,15,10,11,14,13]$.
  3. Then we have max heap HL as shown below:

maxheap

  1. Then we have min heap HR as shown below:

minheap

  1. Then we fill in the structure of T as follows:

mheap

The dotted lines represent the underlying structure of each original heap.


Analysis

The running time is as follows:

  • $O(n)$ for finding the median.
  • $O(n)$ for constructing the two heaps.
  • $O(n)$ for merging the two heaps.

Thus, the total time is still $O(n)$.

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  • $\begingroup$ That's very elegant, thanks! $\endgroup$ – Yariv Levy Mar 20 at 6:39
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We can do this similar to the linear construction of a heap.

We will start with the leaves. We need to determine what elements will be left child leaves and which will be right child leaves.

Assume w.l.o.g. that $n = 2^k - 1$. There will be $2^{k-1}$ leaves where $2^{k-2}$ will be left children, and the other $2^{k-2}$ will be right children.

Since we know every left child must be less than it's parent, we can guarantee this by using the smallest $2^{k-2}$ elements for left child leaves. Similarly we use the largest $2^{k-2}$ elements for right child leaves. The recursive procedure is as follows (building from the leaves to the root).

build-tree(list A of length n):
  if n == 1:
    return singleton tree of the 1 element in A.

  let n = 2^k - 1
  let l = 2^(k-2)         # rank of  left partition element
  let r = n - 2^(k-2) + 1 # rank of right partition element

  l_elem = SELECT(l, A)   # linear time select
  r_elem = SELECT(r, A)   # linear time select

  let L be all elements in A <= l_elem
  let R be all elements in A >= r_elem
  let C be all elements in A that are not in L or R

  let tree T = build-tree(C)

  for i in {0 ... l}:
    let L[i] be the  left child of leaf i in T
    let R[i] be the right child of leaf i in T

  return T

Let's try an example on $[8,12,9,15,5,1,10,3,11,14,2,4,13,7,6]$. The L, R, and Cs would be as follows:

  1. $L = [1, 3, 2, 4]$, $C = [8, 9, 5, 10, 11, 7, 6]$, $R = [12, 15, 14, 13]$
  2. $L = [5, 6]$, $C = [8, 9, 7]$, $R = [10, 11]$
  3. $L = [7]$, $C = [8]$, $R = [9]$
  4. return TreeNode(8).

Then it would recursively build the following tree:

tree


Analysis

We get the following recurrence:

$$T(2^k - 1) = T(2^{k-1} - 1) + O(n)$$

This comes out to be $O(n)$.

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