Questions tagged [tree]

A tree is a special type of graph which only allows for a hierarchical set of edges similar to a tree . Mathematically it is actually an arborescence. Trees have a root node and children nodes. In formal terms it is described as an acyclic connected graph.

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3
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2answers
365 views

Algorithm for computing unordered tree edit distance

I am trying to compute the edit distance between two dendrograms, one produced from hierarchical clustering, and the other manually constructed from some tree structure. In this setting, the rename ...
5
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0answers
81 views

Height of AVL tree with random elements

I know that for an AVL tree of N nodes, the depth of the tree is bounded by $$ \log_2(N + 1) -1 \leq height \leq c \log_2(N + 2) + b$$ where $c,b$ are taken from the golden ratio linked to the worst ...
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0answers
42 views

The number of rooted ordered trees of max-out degree $k$

An ordered tree (also known as ordinal tree and plane tree) is a rooted tree in which the children of each node are ordered. It is known that the number of the ordered tree with $n$ edges is the $n$'...
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1answer
73 views

Distinguish Graph from Tree using Adjacency Matrix

Given an adjacency matrix, is there a way to determine if the graph will be a tree or a graph (whether or not there is a cycle). For example, given the adjacency matrix: ...
1
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1answer
57 views

Reachability Query for Tree

What is the best complexity for reachability queries on trees so far please? There is no constraint on the directions of the edges in the tree. According to Mikkel Thorup, there is an oracle of size $...
6
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2answers
2k views

Improving Bloom filter - can we distinguish elements of a database using less than 2.33275 bits/element?

While we usually use large e.g. 64 bit hashes, there are many techniques to reduce this size, e.g. for savings in storage and transmission. Popular Bloom filter instead of marking just 1 hash ...
48
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20answers
7k views

NP-hard problems on trees

Several optimization problems that are known to be NP-hard on general graphs are trivially solvable in polynomial time (some even in linear time) when the input graph is a tree. Examples include ...
2
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0answers
103 views

Notion of “quotient” or “inverse” for recognizable tree languages?

Related to my previous question but this time I have a better idea of what I'm actually asking. I'm looking at the following operation on recognizable tree languages (i.e. regular tree grammars, ...
2
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0answers
56 views

Regular Tree Languages are closed under quotient?

The Wikipedia page for Regular Tree Grammars notes that if $L_1$ and $L_2$ are regular tree languages, than $L_1 \setminus L_2$ is as well. However, it doesn't define this quotient operation for trees,...
7
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2answers
5k views

How can I find the second cheapest spanning tree?

The classic Mininum Spanning Tree (MST) algorithms can be modified to find the Maximum Spanning Tree instead. Can an algorithm such as Kruskal's be modified to return a spanning tree that is strictly ...
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0answers
96 views

Parallel building time of a k-d tree on n points with n processors

Given a point set with $n$ points to build a k-d tree on. We have $n$ processors available. What is the time-optimal building time for the k-d tree? A straight forward parallelization would be as ...
10
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2answers
367 views

Exact formula for the number of spanning trees of a rectangle

This blog talks about generating "twisty little mazes" using a computer an enumerating them. The enumeration can be done using Wilson's algorithm to get the UST, but I don't remember the formula for ...
20
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1answer
3k views

Finding the distance between two polynomials (represented as trees)

A colleague who works on genetic programming asked me the following question. I first tried to solve it based on a greedy approach, but on a second thought, I found a counterexample to the greedy ...
6
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0answers
159 views

Optimal set union tree

Suppose we have a ground set of $n$ elements and $m$ sets are defined over them $S_i \subseteq [n]$. Think of the following procedure: At each step take two of the sets, take the union, and add the ...
4
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0answers
155 views

What is the exact communication complexity of subtree disjointness?

A classic textbook example for communication complexity is when A and B both receive a subtree of a an $n$-node tree (that they both know), and they need to output whether their subtrees are disjoint ...
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0answers
104 views

Growth of random square lattice trees

Consider the problem of growing a random tree on a $L\times L$ square lattice of initially disconnected vertices, starting from an isolated vertex on one of the corners of the lattice and proceeding ...
5
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1answer
82 views

If I naively generalize the homeomorphic embedding relation for labeled finite trees in this way, do I still have a wqo?

The homeomorphic embedding relation for trees as I understand it is a well-quasi-order (wqo) on trees when the label of a node determines the number of children of that node, and there are a finite ...
6
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1answer
145 views

Example of context-free tree language which can not be generated by monadic CFTG

Assuming that a context-free tree language (CFTL) is that which is generated by a context-free tree grammar (CFTG), I am looking for an example of CFTL which can not be generated by a monadic CFTG (...
26
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1answer
667 views

Is there a regular tree language in which the average height of a tree of size $n$ is neither $\Theta(n)$ nor $\Theta(\sqrt{n})$?

We define a regular tree language as in the book TATA: It is the set of trees accepted by a non-deterministic finite tree automaton (Chapter 1) or, equivalently, the set of trees generated by a ...
7
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1answer
227 views

How fast can we find and disconnect roots in a forest?

Consider a forest of rooted trees. The problem is to support two operations: disconnect(v): if v is the root of some tree in the forest, remove all edges of v; findroot(v): find root of the tree ...
4
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1answer
118 views

What are “unranked trees”?

Recently I have some dispute with my colleagues and would like to clarify the following question. It is clear what are "ranked" trees. They are those, which produced by tree grammars, where each ...
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1answer
78 views

Proving liveness-like property on an infinite tree

(Context: this is a problem from the middle of my dissertation work, abstracted into a more general problem. It's related to showing that some liveness properties hold, so perhaps work from something ...
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0answers
63 views

Generating random labelled trees

I am looking for a simple rejection-free algorithm to uniformly sample random labelled trees (i.e. to generate each of them with the same probability). One possibility is to generate Prüfer sequences ...
4
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0answers
199 views

Find a pair of nodes with maximum sum of distances in k given trees

For k edge-weighted trees $T_1,T_2...T_k$ which contain the same set of nodes $\{1,2,... n \}$, I want to find a pair of nodes $(x,y)$ which maxifies $$\sum_{i=1}^k d_i(x,y)$$ where $d_i(x,y)$ ...
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0answers
160 views

“Context” understanding in tree grammars

The Context-Free tree grammar has rules of the form: $A\rightarrow t$ or $A(x_1,\dots,x_n)\rightarrow t_x$, where $A\in N$, $t\in T(N\cup T)$, $t_x\in T(N\cup T\cup \{x_1,\dots,x_n\})$, $T(Z)$ ...
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1answer
146 views
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2answers
3k views

What is a zipper, and how does it relate to a tree-like structure?

I was reading a chapter in LYAH which didn't really make sense to me. I understand that zippers can arbitrarily traverse a tree-like structure, but I need some clarification on it. Also, can zippers ...
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0answers
38 views

BK-Tree intersection

I'm looking to calculate the approximate intersection (proximity under a certain distance) of two sets of points in a discrete metric space. In other words, given a metric space $(M, d)$, subsets $A, ...
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2answers
136 views

How to continue this algorithm? [closed]

I want to create an algorithm to fill a fixed-size big rectangle (W,H) with the maximum number of fixed-size smaller rectangles (w,h) (I can rotate the small rectangles 90º). I have thought about ...
3
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1answer
103 views

Place of tree-adjoining grammars in the hierarchy of tree grammars

As tree-adjoining grammars operate with trees, I suppose they can be considered as a kind of tree grammars. If this assumption is correct, I'm wondering: where should we place them in the tree grammar ...
5
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1answer
137 views

Regarding proper form of production rules of Context-free tree grammars

Is it possible to describe Context-free Tree Grammar $G_t$ such that set of yields of its trees will coincide with Context-sensitive word language $a^nb^nc^n$? $\{a^nb^nc^n | n>0\}=\{Yield(t)|t\...
2
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1answer
194 views

Efficient enumeration of the reachable leaves of nodes in a polytree

A polytree is a directed acyclic graph which does not have any undirected cycles, i.e., it is a tree when we replace each directed edge by its undirected counterpart. Given a polytree $T$ and a node $...
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0answers
186 views

Maximize the weight of MST + sum of vertex weights

I am considering a problem where the goal is to choose a subset of size $k$ of the vertices in a graph, such that the weight of their minimum spanning tree + the sum of their vertex weights is ...
7
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1answer
320 views

Minimum Spanning tree on a complete “random” graph

Consider a complete undirected graph with $n$ vertices, $K_n$. Let weight of an edge between vertices $i\; \& \;j$ be a random variable $E_{ij}$. Let $E_{ij} \sim exp(\lambda)$, where $exp(\lambda)...
7
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1answer
310 views

For a given binary-search tree obtain an isomorphic splay tree

I will assume that the reader is familiar with some undergraduate algorithms and data structures. To people who are not familiar with splay trees I recommend to read through this link : https://en....
2
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2answers
293 views

Finding a minimum tree which is isomorphic to a subtree of $T_1$ but not to a subtree of $T_2$

Consider the problem that receives two trees $T_1$, $T_2$, and asks to find a minimum size tree $T$ such that there exists a subtree of $T_1$ which is isomorphic to $T$, but there is no such ...
12
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0answers
166 views

What is the curve of “search vs. insert”

Consider a collection of numbers (of arbitrary size), and an oracle that is able to accept two such numbers $a,b$ and answer queries of the form $a<b, a>b, a=b$ in constant time. With this ...
7
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1answer
374 views

Complexity of “destroying” the graph's minimum spanning tree weight

Assume we have a connected input graph $G=(V,E)$ and a weight function $w:E\to\mathbb N$. Denote by $w(G)$ the weight of a minimum spanning gree for a graph $G$. For this purpose, define $w(G')$ as $\...
7
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0answers
260 views

Number of ordinal trees with n nodes, of depth d, with l leaves

What is the number of ordinal trees (aka rose trees) with $n$ nodes, of depth $d$, with $l$ leaves? I thought that it was a known results but I could not find it, and neither did the various ...
7
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1answer
267 views

Inexact labelled binary tree matching

Does anyone recognise the following problems? Do they have names? Are they hard? If we were looking for an exact match (0 mismatches), these would be solvable in polynomial time (using e.g. standard ...
3
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5answers
914 views

How to constrain a finite automaton (NFA and DFA) to a tree?

I have a finite automaton by the standard model Hopcroft & Ullman define: $$ M = (Q, \Sigma, \delta, q_0, F) $$ Where $\delta$ is the transition function mapping $Q \times \Sigma \mapsto Q$, such ...
6
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1answer
3k views

How does Camerini's algorithm for minimum-bottleneck-spanning-tree run in linear time?

I'm having a difficult time understanding Camerini's algorithm because there are very few clear explanations online. The goal is to find a minimum-bottleneck spanning tree in linear time. Camerini's ...
3
votes
1answer
273 views

Graph (Forest) representation that supports edge deletion and efficient traversal

I am trying to write a data structure that given a general tree (or forest) will support the following operations: Edge deletion Connected(u,v) queries This problem is addressed in section two of ...
0
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0answers
46 views

Suffix tree and searching for longest subword that appears two times and two occurencies are not overlapping

I'm learning basics of text algorithms, so my question might seem simple. Let's have word $S$, i want to find longest subword $x$ such that it appears in $S$ two times and those two occurencies are ...
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0answers
225 views

Steiner Tree and minimum spanning tree

If I must connect: $$2^k$$ terminals in a Steiner Tree choosen randomly and connect them with the cheapest component; "loss - contracting algorithm" is a good way? Or is an "Iterative Randomized ...
1
vote
1answer
257 views

Alternating tree automata for arbitrary arity tree

Could alternating tree automata be used for recognizing set (language) of arbitrary-arity trees? More specifically, as an example: let $\Sigma = \{a,b,c\}$ - labels for tree nodes. Trees from $T$ ...
9
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2answers
421 views

Efficient algorithms for searching a collection of trees

I have a large dataset of trees and I would like to search it by specifying a treelet (connected subgraph). The query should return all the occourrences of the treelet in the dataset. Are there ...
0
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0answers
68 views

On a property of random rooted trees with $n$ nodes and of height $h$

I am working on a proof that require the result of the following problem: Let, $T$ be a rooted directed tree with height $h (\ge \lceil{log_d{n}}\rceil )$ and having $n$ nodes. Each internal node of $...
0
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2answers
829 views

Pre order traversal of an array [closed]

I am wondering if there is an algorithm that, given a sorted array, allows you to build a binary search tree in linear time? I am facing a problem where I have about 8 million elements in a file that ...
1
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0answers
88 views

Are there any polynomial cases of Balanced Minimum Evolution?

The BME problem has an interest in computational biology, for the reconstruction of phylogenetic trees from a distance matrix. Let me provide some context before defining the problem. Suppose that we ...