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When I am trying to print level Order of BST, this question prompted me.
Here is a
Pre-Order Sequence: 4, 1, 2, 3, 5, 6, 7, 8
In_order Sequence : 1, 2, 3, 4, 5, 6, 7, 8
A level order sequence for a BST with above pre_order and In_order is
[4, 2, 6, 1, 3, 5, 7, 8]
However, for the same Pre-order an In-order sequence this level order sequence seems possible. [4, 1, 5, 2, 6, 3, 7, 8]. I don't know how. I am trying to figure this out.
I am unable to construct BST in paper (drawing) that satisfies all the pre_order, In-order and level order sequences.
If you have in-order traversal together with one of pre/post-order, that is enough to reconstruct a binary tree. Moreover, in case of BST (binary search tree), post-order or pre-order alone is sufficient.
In your case, reconstructing a BST from pre-order 4, 1, 2, 3, 5, 6, 7, 8 gives the following BST:
4
/ \
1 5
\ \
2 6
\ \
3 7
\
8
which gives, again unique, level-order traversal [4,1,5,2,6,3,7,8].
See also:
Reconstructing binary trees from tree traversals
Following combination will generate unique binary tree(which can be BST).
Inorder and Preorder.
Inorder and Postorder.
Inorder and Level-order.
So in your case inorder & pre order are given which will generate unique binary tree which is BST in your case so level order will be unique for that tree.
Pre-Order Sequence: 4, 1, 2, 3, 5, 6, 7, 8
In_order Sequence : 1, 2, 3, 4, 5, 6, 7, 8
SO tree will be
level 0- 4
level 1- 1,5
level 2- 2,6
level 3- 3,7
level 4- 8
Level order is
4,1,5,2,6,3,7,8
in sort there will always unique level order traversal
According to this post on Wikipedia, given a tree with distinct elements, either pre-order or post-order paired with in-order is sufficient to describe the tree uniquely. However, pre-order with post-order leaves some ambiguity in the tree structure.
I am looking for a quick example that would demonstrate this claim.
So given the following tree:
The orderings are:
Pre Order: 1, 2, 4, 3, 5, 7, 8, 6
In Order: 4, 2, 1, 7, 5, 8, 3, 6
Post Order: 4, 2, 7, 8, 5, 6, 3, 1
How can I deserialize this tree using its Pre Order and In Order or Post Order and In Order?
Thanks
So the claim is that given a pre-order and in-order traversal of the tree we can uniquely construct the tree.
In your example
Preorder : 1, 2, 4, 3, 5, 7, 8, 6
In Order : 4, 2, 1, 7, 5, 8, 3, 6
Now the root node in the tree will be the first node in the pre-order traversal. Moreover, all the numbers appearing before a value in the inorder traversal will lie in the left subtree rooted at that node. And similarly all values appearing after a value will lie in the right sub-tree. So continuing in this recursive fashion will deserialize the tree. You can read more about it here http://www.geeksforgeeks.org/construct-tree-from-given-inorder-and-preorder-traversal/
Similarly, you can deserialize the tree given its postorder and inorder traversal. Here the last value appearing in the post order traversal will be the root of the tree. You can read about the deserialization from post-order and in-order from here http://www.programcreek.com/2013/01/construct-binary-tree-from-inorder-and-postorder-traversal/
I am trying to solve above Longest Monotonically Increasing Subsequence problem using javascript. In order to do that, I need to know about Longest Monotonically Subsequence. Current I am following wikipedia article. The thing I am not understanding this example is that the longest increasing subsequence is given as 0, 2, 6, 9, 13, 15 from 0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15, … list. The question is Why the answer does not have 3 in between 2 and 6, and 8 between 6 and 9 etc? How does that answer come from that list?
Ist of all , consider the name "Longest Monotonically Increasing Subsequence" . So , from the given array you need to figure out the largest sequence where the numbers should be appeared in a strictly increasing fashion. There can be many sequence, where the sub array can be strictly increasing but you need to find the largest sub-Array.
So. lets debug this array. a[] = {0, 8, 4, 12, 2, 10, 6, 14, 1, 9, 5, 13, 3, 11, 7, 15}
In here the some monotonously increasing sub-arrays are :
{0,8,12,14,15} Length = 5
{0,4,12,14,15} Length = 5
{0,1,9,13,15} Length = 5 and so on.
But if you calculate like this , you can find the largest sub-array will be :
{0, 2, 6, 9, 13, 15} , Length = 6, so this is the answer.
Every single little time you pick any number , the next number should be large than the previous one and must be present in the array. say {0, 2, 6, 9, 13, 15} this list, when you pick 9 , then the next number should be larger than 9. the immediate sequence shows 13>9, so you can pick 13. You can also pick 11. But that will create another branch of sub-array. Like :
{0, 2, 6, 9, 11, 15} which is another solution.
Hope this explanation will help you to understand the LIS (Longest Increasing Subsequence).Thanks.
First of all, the title of your question says: Longest increasing CONTIGUOUS subsequence which is a slight variation of the original problem of LIS in which the result need not have contiguous values from original array as pointed out in above examples. Follow this link for a decent explanation on LIS algorithm which has O(n^2) solution and it can be optimized to have a O(nlogn) solution:
http://www.algorithmist.com/index.php/Longest_Increasing_Subsequence
for the contiguous variant of LIS, here is a decent solution:
http://worldofbrock.blogspot.com/2009/10/how-to-find-longest-continuous.html
I have been trying to solve Maximum clique problem with the algorithm mentioned below and so far not been able to find a case in which it fails.
Algorithm:
For a given graph, each node numbered from 1 to N.
1. Consider a node as permanent node and form a set of nodes such that each node is connected to this permanent node.(the set includes permanent node as well)
2. Now form a subgraph of the original graph such that it contains all the nodes in the set formed and only those edges which are between the nodes present in the set.
3. Find degree of each node.
4. If all the nodes have same degree then we have a clique.
5. Else delete the least degree node from this subgraph and repeat from step 3.
6. Repeat step 1-5 for all the nodes in the graph.
Can anyone point out flaw in this algorithm?
Here is my code http://pastebin.com/tN149P9m.
Here's a family of counterexamples. Start with a k-clique. For each node in this clique, connect it to each node of a fresh copy of K_{k-1,k-1}, i.e., the complete bipartite graph on k-1 plus k-1 nodes. For every permanent node in the clique, the residual graph is its copy of K_{k-1,k-1} and the clique. The nodes in K_{k-1,k-1} have degree k and the other clique nodes have degree k - 1, so the latter get deleted.
Here's a 16-node counterexample, obtained by setting k = 4 and identifying parts of the K_{3,3}s in a ring:
{0: {1, 2, 3, 4, 5, 6, 7, 8, 9},
1: {0, 2, 3, 7, 8, 9, 10, 11, 12},
2: {0, 1, 3, 10, 11, 12, 13, 14, 15},
3: {0, 1, 2, 4, 5, 6, 13, 14, 15},
4: {0, 3, 7, 8, 9, 13, 14, 15},
5: {0, 3, 7, 8, 9, 13, 14, 15},
6: {0, 3, 7, 8, 9, 13, 14, 15},
7: {0, 1, 4, 5, 6, 10, 11, 12},
8: {0, 1, 4, 5, 6, 10, 11, 12},
9: {0, 1, 4, 5, 6, 10, 11, 12},
10: {1, 2, 7, 8, 9, 13, 14, 15},
11: {1, 2, 7, 8, 9, 13, 14, 15},
12: {1, 2, 7, 8, 9, 13, 14, 15},
13: {2, 3, 4, 5, 6, 10, 11, 12},
14: {2, 3, 4, 5, 6, 10, 11, 12},
15: {2, 3, 4, 5, 6, 10, 11, 12}}
What you propose looks very much like the following sorting algorithm combined with a greedy clique search:
Consider a simple undirected graph G=(V,E)
Initial sorting
Pick the vertex with minimum degree and place it first in the new list L. From the remaining vertices pick the vertex with minimum degree and place it in the second position in L. Repeat the operations until all vertices in V are in L.
Find cliques greedily
Start from the last vertex in L and move in reverse order. For each vertex v in L compute cliques like this:
Add v to the new clique C
Compute the neighbor set of v in L: N(v)
Pick the last vertex in N(v)
v=w; L=L intersection with N(v);
Repeat steps 1 to 4
Actually the proposed initial sorting is called a degeneracy ordering and decomposes G in k-cores (see Batagelj et al. 2002 ) A k-core is a maximal subgraph such that all its vertices have at least degree k. The initial sorting leaves the highest cores (with largest k) at the end. When vertices are picked in reverse order you are picking vertices in the highest cores first(similar to your step 4) and trying to find cliques there. There are a number of other possibilities to find cliques greedily based on k-cores but you can never guarantee an optimum unless you do full enumeration.
The proposed initial sorting is used, for example, when searching for exact maximum clique and has been described in many research papers, such as [Carraghan and Pardalos 90]
I am looking at implementing code to work out binary search tree.
Before I do this I was wanting to verify my input data in postorder and preorder.
I am having trouble working out what the following numbers would be in postorder and preorder
I have the following numbers 4, 3, 14 ,8 ,1, 15, 9, 5, 13, 10, 2, 7, 6, 12, 11, that I
am intending to put into an empty binary tree in that order.
The order I arrived at for the numbers in POSTORDER is
2, 1, 6, 3, 7, 11, 12, 10, 9, 8, 13, 15, 14, 4. Have I got this right?
I was wondering if anyone here would be able to kindly verify if the postorder sequence I came up with is indeed the correct sequence for my input i.e doing left subtree, right subtree and
then root.
The order I got for pre order (Visit root, do left subtree, do right subtree) is
4, 3, 1, 2, 5, 6, 14 , 8, 7, 9, 10, 12, 11, 15, 13. I can't be certain I got this right.
Very grateful for any verification.
Many Thanks
You haven't specified the structure of your tree (is it balanced?) nor the order in which you are inputting the data originally. If you do that we can verify your results