Questions tagged [trees]
Questions about a special kind of graphs, namely connected and cycle-free ones.
48
questions
45
votes
4answers
55k views
Longest path in an undirected tree with only one traversal
There is this standard algorithm for finding longest path in undirected trees using two depth-first searches:
Start DFS from a random vertex $v$ and find the farthest vertex from it; say it is $v'$.
...
120
votes
2answers
38k views
BIT: What is the intuition behind a binary indexed tree and how was it thought about?
A binary indexed tree has very less or relatively no literature as compared to other data structures. The only place where it is taught is the topcoder tutorial. Although the tutorial is complete in ...
3
votes
1answer
478 views
Expected distance between tree nodes
I have been given a tree with n nodes and n-1 edges with it's weight.
There are two people A and B.
I have been given a list of nodes of size k.
A will pick a random node x from this list and B will ...
2
votes
1answer
4k views
Number of Different AVL Tree
I studying the related question.
https://stackoverflow.com/questions/13500560/number-of-ways-to-create-an-avl-tree-with-n-nodes-and-l-leaf-node
but it's not so general.
In-fact, We want to know ...
34
votes
9answers
49k views
Algorithm to find diameter of a tree using BFS/DFS. Why does it work?
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 ...
4
votes
2answers
3k views
How to query and update ranges of arrays?
I have an array of size $N$ $(N \leq 10^5)$. I need to perform two types of operations on the array.
Decrease elements in range $[L,R]$ by $X$.
Count the number of negative elements in range $[L,R]$.
...
39
votes
2answers
24k views
What is the difference between radix trees and Patricia tries?
I am learning about radix trees (aka compressed tries) and Patricia tries, but I am finding conflicting information on whether or not they are actually the same. A radix tree can be obtained from a ...
10
votes
3answers
48k views
Why is the minimum height of a binary tree $\log_2(n+1) - 1$?
In my Java class, we are learning about complexity of different types of collections.
Soon we will be discussing binary trees, which I have been reading up on. The book states that the minimum height ...
1
vote
1answer
190 views
Minimum cost of “signal” cover in a tree with DP
I'm given a (not necessarily binary) tree. Now every node can have a signal with range $i$, reaching all nodes being at most $i$ edges away. The cost of a signal is determined by a function $f(n, i)$ ...
17
votes
3answers
15k views
Can a tree be traversed without recursion, stack, or queue, and just a handful of pointers?
Half a decade ago I was sitting in a data structures class where the professor offered extra credit if anyone could traverse a tree without using recursion, a stack, queue, etc. (or any other similar ...
14
votes
2answers
13k views
Correctness-Proof of a greedy-algorithm for minimum vertex cover of a tree
There is a greedy algorithm for finding minimum vertex cover of a tree which uses DFS traversal.
For each leaf of the tree, select its parent (i.e. its parent is in minimum vertex cover).
For each ...
11
votes
2answers
3k views
What are the applications of Rose trees?
I recently found out about the Rose tree data structure, but just going off of a Haskell data definition and the tiny Wikipedia description of it, I've got some ...
9
votes
1answer
130 views
Generate random labeled tree with constrained edge lengths
Let $T$ be a labeled tree with vertices $V = \{1, \dots, n\}$ and edges $E$. Define the length of an edge $e = \{ u, v \}, u \in V, v \in V$ to be $l(e) = |u - v|$, i.e. the distance between the nodes ...
8
votes
5answers
13k views
Why does the formula 2n + 1 find the child node in a binary heap?
I learned that when you have a binary heap represented as a vector / list / array with indicies [0, 1, 2, 3, 4, 5, 6, 7, 8, ...] the indicies of the children of element at index n can be found with
$...
7
votes
3answers
4k views
Is the height of the tree the number of edges or number of nodes?
I'm so confused by some of the theorems online about tree heights. Does tree height mean the number of edges or nodes? if nodes, does it include the node it is counting from? Can the height of a tree ...
5
votes
3answers
7k views
how is data in a tree stored in memory?
I am not so familiar with trees.
How is a tree stored in memory? what is the data structure type used for storing it?
As a string?linked list? stack (!)?
Or is there some kind of data storage model ...
20
votes
5answers
545 views
Efficient compression of unlabeled trees
Consider unlabeled, rooted binary trees. We can compress such trees: whenever there are pointers to subtrees $T$ and $T'$ with $T = T'$ (interpreting $=$ as structural equality), we store (w.l.o.g.) $...
3
votes
1answer
534 views
Queries on Tree
We have a tree with $N$ nodes. $N \le 10^5.$ Each node has a value $V$ associated with it. Now we have $Q$ $(\le 10^5)$ queries. There are two types of queries:
Q X Y: in this type of query we have ...
2
votes
0answers
74 views
Using the random forest algorithm to predict vectors [duplicate]
I know this might sound like a newbie question, but bear with me.
I have read a paper where researchers use a random forest to predict species distribution, but in their study, they only predict a ...
2
votes
1answer
225 views
Connecting an unconnected forest of subtrees in a graph?
If I have a weighted graph $G=(V,E)$ and three subgraphs $T_1$, $T_2$ and $T_3$ in $G$ which are trees and all unconnected from each other.
What is the best way to connect these three trees such that ...
5
votes
2answers
747 views
MST with half the edges the maximum weight
I have been cracking my head over the following question -
You are given an undirected connected graph with an even number of
edges. Half of the edges have weight less than C (possibly with
...
4
votes
0answers
278 views
Minimal Steiner Tree in unweighted directed graph
I have an unweighted directed graph $(V, E)$ and a subset $T \subseteq V$ of these vertices. I want to find the minimum tree $(V',E')$ that contains all these $T$ vertices (minimize in number of nodes ...
4
votes
4answers
1k views
Find node that splits tree in half
Given a tree $T = (V , F)$, find an algorithm which finds $u \in V$, so in the graph $T = (V \setminus \{u\} , F)$ the size of each connected component is $\lceil |V| / 2 \rceil$ at most. What is the ...
3
votes
1answer
142 views
Finding the shortest path for synchronized pawns in a maze
I have been trying to wrap my head around this problem, and I just can't get it.
We have an $a \times b$ matrix where every cell corresponds to either an empty space, denoted with a dot, or a wall, ...
3
votes
1answer
1k views
Finding the minimum number of calls in a tree
I was asked this question in an interview and struggled to answer it correctly in the time allotted. Nonetheless, I thought it was an interesting problem, and I hadn't seen it before.
Suppose you ...
3
votes
0answers
407 views
Facility location on a tree
Question:
Given a tree representing a neighbourhood where each node is a house.
Assign an antenna to each node such that the whole tree is covered.
An antenna of strength 0 can only ...
2
votes
2answers
123 views
Prove that given a number we can find whether there're 2 elements in a red/black tree that their sum equals that number in $\Theta(n)$ time
Prove that given a number we can find whether there're 2 elements in a red/black tree that their sum equals that number in $\Theta(n)$ time and constant space.
The original problem appears here,
...
0
votes
1answer
327 views
How to answer multiple queries for a tree?
I encountered an interesting problem based on tree-data-structure.
We are given a tree which has N nodes, with 1ā¤Nā¤105.
Time starts from second 1 and it continues for q seconds.
At each second, the ...
0
votes
3answers
231 views
Intuitive proof for a tree with n nodes, has n-1 edges
I am interested in an intuitive proof for "any binary tree with $n$ nodes has $n-1$ edges", that goes beyond proof by strong induction.
4
votes
2answers
251 views
Minimally extend a tree such that there are no bridges in the new graph
We are given an undirected tree on which we should add the minimum number of edges such that there are no bridges in the new graph.
An edge $e$ is a bridge if the graph with that edge removed is no ...
4
votes
2answers
514 views
Counting trees (order matters)
As a follow up to this question (the number of rooted binary trees of size n), how many possible binary trees can you have if the nodes are now labeled, so that abc is different than bac cab etc ? In ...
4
votes
2answers
64 views
How to make efficient path minimum queries in a tree?
Given a tree in which each node has a given value, I want to process "Path Minimum Queries": given two nodes, what is the minimal value of any node on the shortest path between them?
My ...
2
votes
1answer
533 views
How to read off the set represented by a van-Emde-Boas tree?
I'm reviewing my background in Algorithms and DS design. Specifically I never went through the van Emde Boas Tree. Though I can undestand the proto-vEB with related picture. I'm struggling to ...
2
votes
1answer
538 views
How does insertion work in an AVL tree?
From the above image, while trying to maintain an AVL tree data structure, how would the tree look after inserting the value 10? Also, if anyone has any suggestions or simple method of rotating, feel ...
2
votes
1answer
38 views
Prove that there is a sequence of k minimum spaning trees between two distinct minimum spanning trees that each one is different in only 1 edge [duplicate]
I'm pracitcing exams towards finals,
Given an undirected graph $G(V,E)$ , we denote 2 MST $T,T'$ neighbours if by deleting one edge from $T$ and add another one we get $T'$.
Prove :
for every 2 ...
2
votes
2answers
74 views
Distance queries on Tree with hotspots
We are given a tree with $n$ vertices and some of the vertices act as a "hotspot".
We have to answer multiple queries of type $(a,b,c)$, which means we have to find the distance to the nearest ...
2
votes
1answer
430 views
Correctness proof for finding weighted maximum independent set for a tree
The maximum weighted independent set for a tree can found out using the following dynamic programming approach.
Min[u] = wt(u) + Ī£ Mout[v] where v ā children(u) Mout[u] = Ī£ max { Min[v], Mout[v] } ...
2
votes
1answer
141 views
What is the name of this function of a tree?
I've written a recursive function of a tree, and I would like to know what it's called! It's not quite the same as the height or the width of a tree, but it seems kind of like a width.
Assuming the ...
1
vote
0answers
61 views
Given a tree find path that maximizes the median of the costs of edges
We have given tree with $N$ nodes and $N-1$ edges, such that each edges is assigned positive weight. We need to find path of length between $L$ and $R$ inclusively, with maximum cost. Cost of a path ...
1
vote
0answers
290 views
Calculating maximum number of splits that can occur during insertion of $n$ keys in B Tree of order $m$
I can calculate this by trying out manually inserting $n$ keys in $m$ order B Tree as follows:
Assume median to be selected for split be left biased. That is $m/2$. For example, if $m=4$, then a ...
1
vote
1answer
130 views
Why not use large $k$ in a $k$-ary tree?
Obviously binary trees are great because of $O(\log_2 n)$ search, inserts, and deletes in best case.
To "maximize" occurrence of best case, we can use self-balancing trees like red-black trees, AVLs, ...
1
vote
1answer
1k views
Best way to merge 2 max heaps into a min heap
Assume we have 2 max heaps, each with n nodes. We want to merge these 2 heaps and build a min heap. What is the best way to do this?
The easiest way is to consider 2 max heaps an array with $2n$ ...
1
vote
0answers
81 views
Conceptualizing a balance in a DFS traversal [closed]
I'm trying to use the concept of DFS traversal to go through a cycle, and attempt to get a balance of 0 in the end. Each student either owes or is owed some money, so I'm trying to go through all of ...
1
vote
2answers
679 views
Why can't we just use preorder traversal to check if a tree is subtree of binary tree?
Is preorder traversal enough to check if a tree is subtree of a binary tree?
Are there any scenarios which I can miss if I use just the preorder traversal?
What other methods can be used to check if ...
1
vote
0answers
127 views
Why the update low value in Tarjan's for the ancestors is not it's low value instead of it's discovery value? [closed]
In Tarjan's algorithm for finding SCC/AP/Bridges, we update the value of the low[u] to be the min ( low[u], desc[v] ) given that v is a neighbor and has been discovered before, why it's not like this ...
0
votes
2answers
168 views
How to convert a Complete Binary Tree to a Priority Search Tree in O(n)
There doesn't seem to be any resources on this.
I would like to know if there is a linear-time algorithm to convert a Complete Binary Tree with data left-to-right increasing stored in external nodes, ...
0
votes
0answers
56 views
Can't reach a balance in a DFS search?
Below is a question based on CLRS, about using an algorithm to reach a balance between a group of friends. I figured the best way to do this, is through the use of a DFS algorithm. Below the question ...
-2
votes
1answer
467 views
Obtain data structure able to do reverse range updates
For given array $A$ of size $N$, note that the array is going to be permutation of the numbers from 1 to N, each number will be there exactly once, we want to obtain data structure being able to ...
