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Intuitively, you can think of a binary indexed tree as a compressed representation of a binary tree that is itself an optimization of a standard array representation. This answer goes into one possible derivation.
Let's suppose, for example, that you want to store cumulative frequencies for a total of 7 different elements. You could start off by writing ...
82
Heap just guarantees that elements on higher levels are greater (for max-heap) or smaller (for min-heap) than elements on lower levels, whereas BST guarantees order (from "left" to "right"). If you want sorted elements, go with BST.by Dante is not a geek
Heap is better at findMin/findMax (O(1)), while BST is good at all finds (O(logN)). Insert is O(logN)...
40
Summary
Type BST (*) Heap
Insert average log(n) 1
Insert worst log(n) log(n) or n (***)
Find any worst log(n) n
Find max worst 1 (**) 1
Create worst n log(n) n
Delete worst log(n) log(n)
All average times on this table are the same as their worst times except for Insert.
*: everywhere in ...
36
Both binary search trees and binary heaps are tree-based data structures.
Heaps require the nodes to have a priority over their children. In a max heap, each node's children must be less than itself. This is the opposite for a min heap:
Binary search trees (BST) follow a specific ordering (pre-order, in-order, post-order) among sibling nodes. The tree must ...
28
Tree Examples (image):
A: B:
‾‾ ‾‾
1 1
/ / \
2 2 3
/
3
This is an example that fits your scenario, Tree A root׳s value is 1, having a left child with value 2, and his left child has also a left child with value 3.
Tree B root׳s value is 1, ...
19
A Fenwick tree is a binary tree used to efficiently handle cumulative frequencies or sums in an array.
Without loss of generality we shall examine a 16-element array. Imagine a binary tree imposed on top of the array. Furthermore, label all the left edges in this tree with a "0" and all the right edges with a "1". We get something like this:
This is the so ...
18
There is nothing to be done: for instance, let $S_k$ denote the star graph with $k$ leaves. The graph $S_k$ has a unique spanning tree (which is $S_k$ itself), and it has a vertex with degree exactly $k$.
In fact, the general problem of finding a degree-constrained minimum spanning tree is NP-complete.
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If looking for the key 60 we reach a number $K$ less than 60, we go right (where the larger numbers are) and we never meet numbers less than $K$. That argument can be repeated, so the numbers 10, 20, 40, 50 must occur along the search in that order.
Similarly, if looking for the key 60 we reach a number $K$ larger than 60, we go leftt (where the smaller ...
14
Here's a simpler logical proof.
Last leaf is $n^{th}$ index. Its parent is at index $\lfloor{n/2}\rfloor$ and similarly, there is no element such that its parent is ($\lfloor n/2 +1\rfloor)^{th}$ element. Thus leaves are indexed from $\lfloor{n/2}\rfloor$ +1 to n.
Hence, total number of leaves = n- $\lfloor(n/2)\rfloor$ = $\lceil(n/2)\rceil$.
14
Hash tables can only tell you if an element is present or not.
Here are somethings you can do with a binary tree that you can't do wiht a hash table.
sorted traversal of the tree
find the next closest element
find all elements less than or greater than a certain value
See this wikipedia article on K-d trees for an example of a real world data structure ...
13
With data structure one has to distinguish levels of concern.
The abstract data structures (objects stored, their operations) in this question are different. One implements a priority queue, the other a set. A priority queue is not interested in finding an arbitrary element, only the one with largest priority.
The concrete implementation of the structures. ...
12
If for each node of a tree, the longest path from it to a leaf node is no more than twice longer than the shortest one, the tree has a red-black coloring.
Here's an algorithm to figure out the color of any node n
if n is root,
n.color = black
n.black-quota = height n / 2, rounded up.
else if n.parent is red,
n.color = black
n.black-quota = ...
12
I think that the original paper by Fenwick is much clearer.
The answer above by @templatetypedef requires some "very cool observations" about the indexing of a perfect binary tree, which are confusing and magical to me.
Fenwick simply said that the responsibility range of every node in the interrogation tree would be according to its last set bit:
E.g. as ...
12
The depth of a decision tree is the length of the longest path from a root to a leaf.
The size of a decision tree is the number of nodes in the tree.
Note that if each node of the decision tree makes a binary decision, the size can be as large as $2^{d+1}-1$, where $d$ is the depth. If some nodes have more than 2 children (e.g., they make a ternary ...
12
According to this book (Chapter 3.2), a node in a BST has rank $k$ if precisely $k$ other keys in the BST are smaller. So, if you order all the BST nodes according to their keys, then each node with rank $k$ will take $k$-th place.
11
One application domain where binary trees are better, or more easily adjustable than certain alternatives, are persistent data structures (which are often used in (purely) functional programming).
A persistent data structure is a data structure that preserves the previous version of itself when it is modified. (Data structures that do not have this property ...
9
Suppose you had an empty array:
0 0 0 0 0 0 0 0 0 0 (array)
0 0 0 0 0 0 0 0 0 0 (cumulative sums)
And you wanted to make a range update of +5 to [3..7]:
0 0 0 5 5 5 5 5 0 0 (array)
0 0 0 5 10 15 20 25 25 25 (desired cumulative sums)
How could you store the desired cumulative sums using 2 binary indexed trees?
The trick ...
9
Your algorithm runs in linear time on all inputs. The algorithm visits each node of the tree exactly once, and does $O(1)$ work per node. Therefore it runs in time $\Theta(n)$, where $n$ is the number of nodes.
The argument above is better than using recurrences, since it is more immediate. It also shows that if you had an arbitrary tree (not necessarily ...
9
A generalization of this class of problems is widely studied. See, e.g., this paper for a survey.
In your particular case, the problem can be easily solved without any asymptotic change in the computational complexity. Run the binary search three times. At least two of the three results must be equal to the hidden number. Return the majority result.
There ...
8
There is a general formula for this sum:
$$
\sum_{h=0}^m h2^h = \sum_{h=1}^m \sum_{k=1}^h 2^h = \sum_{k=1}^m \sum_{h=k}^m 2^h = \sum_{k=1}^m (2^{m+1}-2^k) = m2^{m+1} - (2^{m+1}-2).
$$
Overall, we get
$$
\sum_{h=0}^m h2^h = (m-1)2^{m+1} + 2.
$$
When $m = \lg n$, this works out to be $2n\lg n - 2n + 2$.
8
This may not be an exact answer but some information of interest related to your question
Other answers have mentioned various ways in which the binary data structure can be represented and you might want to use one of them but mostly when using databases the Adjacency List Model and Nested Set Model are used for representing Binary Trees.
Binary Trees are ...
8
Lets assume you consider trees of $n$ nodes. Now take any binary tree with $n$ nodes and name the nodes according to their pre-order numbering. Then clearly the pre-order sequence of the tree will be $1,2,\dots,n$.
This means that we can name the nodes of any binary tree structure so that it will generate the same pre-order sequence as that of another ...
8
Counting argument
The number of unlabeled binary trees of $n$ nodes is the $n^\text{th}$ Catalan number $C_n=(2n)!/(n!(n+1)!).$ For example there are 5 binary trees of 3 nodes,
o o o o o
/ / / \ \ \
o o o o o o .
/ \ ...
7
Let $a_{n,h}$ denote the number of AVL trees with $n$ nodes and height $h$. It is straightforward to get a recurrence for $a_{n,h}$:
$$a_{n,h} = \sum_{k=1}^n \bigl(a_{k-1,h-1}a_{n-k,h-1} + a_{k-1,h-1}a_{n-k,h-2} + a_{k-1,h-2}a_{n-k,h-1}\bigr),~ n\geq h > 1,$$
with the initial conditions $a_{n,h} = 0$, if $h>n$ or $h\in\{0,1\}, n\neq h$, and $a_{0,0} =...
7
There are several different concepts here. Let's start from the most general. A tree is a data structure that consists of a root and a collection of children, each of which is a tree. A node of the tree is either the root of the tree itself of a node of the children, i.e. it's the root, or the root of a child, or the root of a child's child, etc. The nodes ...
answered Dec 16 '14 at 12:39
Gilles 'SO- stop being evil'
40.4k7 gold badges108 silver badges171 bronze badges
7
I've never seen this data structure before. However, it doesn't seem like a good choice for storing a set of words, for most purposes. I see three significant disadvantages:
Performance. Looking up a word in this data structure can potentially be quite slow. In particular, when looking up a word of length $n$, checking whether it contains a particular ...
7
There are many ways to represent trees, each with their own set of advantages and disadvantages.
Here's an incomplete list:
Use any graph representation, e.g.
adjacency matrix,
incidence matrix,
adjacency list,
linked nodes,
potentially adding a label for the root (if you have one).
An array, listing nodes level by level (see e.g. heaps).
Post- and pre-...
7
Different traversals of a binary tree exist to suffice different data dependencies between the nodes.
Let's have a comparison between different traversals of a tree. Note that aside from in-fix traversal, the tree doesn't have to be binary.
In case of Postorder, we process the descendants of a node before the node itself, meaning that we want to send some ...
7
A tree is defined to be a set of nodes, with a parent-child relationship that satisfies certain properties. Thus, it doesn't make sense to ask whether a node can "appear" twice.
In your code snippet, you have constructed a DAG, not a tree.
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If normal binary search would take k questions, then you can solve this with 2k+1 questions: Ask each question twice. If you get the same answer, it was the truth. If not, a third question reveals the truth, this happens only once.
I suspect you can do better. If the number is from 1 to 100 and I check the numbers 40 and 60 then knowing that one answer is ...
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