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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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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 ...
Sidharth Ghoshal's user avatar
13 votes
0 answers
534 views

Lock-free, constant update-time concurrent tree data-structures?

I've been reading a bit of the literature lately, and have found some rather interesting data-structures. I have researched various different methods of getting update times down to $\mathcal{O}(1)$ ...
A T's user avatar
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13 votes
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Applications of an access lemma for dynamic forests?

Sleator and Tarjan's amortized analysis of splay trees builds on their so-called Access Lemma. For purposes of analysis, assign an arbitrary weight to each node $v$, and let $size(v)$ denote the sum ...
Jeffε's user avatar
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7 votes
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Original formulation of Spira's Lemma

I'm currently reading the book "Proof Complexity" by Jan Krajíček (2019), where Spira's Lemma is mentioned: Let $T$ be a finite $k$-ary tree and $|T| > 1$. Then there is a node $a \in T$ ...
user11718766's user avatar
7 votes
0 answers
285 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 ...
J..y B..y's user avatar
  • 2,776
7 votes
0 answers
200 views

Tree search guided by a probabilistic oracle

I'm trying to find a solution for the following problem. I have a tree $T$ of branching factor $b$ and depth $d$. For the moment, I only care about the case where I restrict $b=2$, but I would be ...
Foo Barrigno's user avatar
6 votes
0 answers
362 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 ...
AmeerJ's user avatar
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5 votes
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166 views

Complexity of a problem related to Friedman's TREE(k) function?

Background Given two rooted, vertex-colored trees $T_1, T_2$, $T_1$ is color-preserving inf-embeddle in $T_2$, which we'll denote $T_1 \leq T_2$, if there is an injective $f \colon V(T_1) \to V(T_2)$ ...
Joshua Grochow's user avatar
5 votes
0 answers
166 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 ...
Binou's user avatar
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5 votes
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163 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 $A\in N$, $t\in T(N\cup T)$, $t_x\in T(N\cup T\cup \{x_1,\dots,x_n\})$, $T(Z)$ ...
Andrey Lebedev's user avatar
5 votes
0 answers
317 views

Geometric / Visual explanation that the average height of a random binary tree of given size $n$ is asymptotically $2\sqrt{\pi n}$

I just finished reading the proof that the average height of a random binary of given size $n$ is asymptotically $2\sqrt{\pi n}$. I'm now searching for an intuitive, or geometric, or visual proof of ...
Clément's user avatar
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4 votes
0 answers
170 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 ...
domotorp's user avatar
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4 votes
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215 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 $d_i(x,y)$ ...
newbie's user avatar
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4 votes
1 answer
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Given a B-Tree, determine the order keys were inserted

Given a B-tree, determine what order the keys were inserted in. There may be multiple answers: I'd like to generate them all. Is there any known method for this? Or similar problems? Clarification:...
SRobertJames's user avatar
3 votes
0 answers
75 views

Load-balancing; Alternate methods of keeping track of nodes?

Reading various articles in the literature have given me only a few decent methods of keeping track of nodes before->after load-balancing them on a very large network. One popular method uses virtual-...
user1438003's user avatar
2 votes
0 answers
38 views

Homeomorphic subtree extraction in integer sorting time

Background Given a rooted, binary tree $T$ with leaves bijectively labeled by $\{1, \ldots, n\}$ (a "phylogenetic tree"). Let $L \subseteq \{1, \ldots, n\}$, and $|L| = k$. The homeomorphic ...
StubbornSnail's user avatar
2 votes
0 answers
54 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 $n$'...
shahin kamali's user avatar
2 votes
0 answers
74 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,...
Joey Eremondi's user avatar
2 votes
0 answers
122 views

How to find a merge tree for a set of words?

Consider a set $S \subseteq \Sigma^n$ where $\Sigma$ is a finite alphabet and $p : \Sigma \rightarrow [0,1]$ is a probability function. Let $T$ be a tree leaf-labeled by the elements of $S$. Consider ...
NisaiVloot's user avatar
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2 votes
0 answers
226 views

Quadratic Binary Optimization formulation of Steiner Tree problem

can someone point out to me a solution or give advice on how to formulate as efficiently as possible in terms of number of bits the minimum Steiner tree problem as a 0-1 quadratic optimization problem?...
Davide Venturelli's user avatar
2 votes
0 answers
431 views

Notation for drawing rooted trees with isomorphic subtrees and recursive parts

I need to specify small rooted trees with a lot of repeated parts and some recursive definitions. To illustrate the need, let's use $s(T_1, T_2, ..., T_k)$ to denote a tree that consists of a degree-$...
Jukka Suomela's user avatar
1 vote
0 answers
69 views

Directed tree decompositions on subtrees of DAGs

Given a DAG, is the arboreal decomposition of the DAG with the guarantee that given a node $x$, $v$ such that $x$ is reachable from $v$ are in the subtree of $x$? If not, is there a similar ...
shgr1092's user avatar
1 vote
0 answers
101 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 ...
Skrodde's user avatar
  • 111
1 vote
0 answers
113 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 ...
delete000's user avatar
  • 828
1 vote
0 answers
84 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 ...
Kai's user avatar
  • 11
1 vote
0 answers
41 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, ...
Vladimir Panteleev's user avatar
1 vote
0 answers
287 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 ...
Johnson's user avatar
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1 vote
0 answers
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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 ...
Super8's user avatar
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1 vote
0 answers
58 views

Repartitioning a binary tree

Suppose I have a binary tree $G = (V, E)$ (with undirected edges) that is partitioned into sets of k vertices, where each set of vertices is a connected subgraph of $G$. Additionally, if there are ...
zaloo's user avatar
  • 393
1 vote
0 answers
296 views

How to quantify the tree-like-ness of a graph?

What are good measures of tree-like-ness of a graph and algorithms for calculating them?
mwhite's user avatar
  • 119
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0 answers
36 views

Why LOUDS is considered succinct coding?

LOUDS encoding is widely cited as succinct, e.g., in the papers, in Github repos, etc. It is straightforward that LOUDS takes 2n bits for encoding an ordered tree ...
Xavier Z's user avatar
  • 113
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0 answers
69 views

What work on min max connectivity problems has there been?

For instance has min max spanning/steiner/prize-collecting tree been studied. i.e. each edge $e$ has costs $c_{v,e}$ of each resource $i$. And we wish to find a spanning tree minimizing the maximum ...
Hao S's user avatar
  • 228
0 votes
0 answers
65 views

Has multiobjective prize collecting steiner tree or TSP been studied?

Suppose we have a graph $G$ a root $r$ and each node $v$ has some amount of $c_{v,i}$ of each resource $i$. I connect a set of nodes to the root that maximizes the minimum amount of any resource using ...
Hao S's user avatar
  • 228
0 votes
0 answers
116 views

Given $a_i$ -$r$ paths $P_i$ in a planar graph construct a tree spanning $a_i$ such that each root to leaf path intersects few $P_i$

Suppose I am given distinct nodes $a_1,a_2,.., a_l, r$ and edge disjoint $a_i$-$r$ paths $P_i$ for each $i$ in a planar graph $G$. I wish to construct a tree $T$ connecting $a_1,a_2,.., a_l, r$ ...
Hao S's user avatar
  • 228
0 votes
0 answers
66 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 ...
Krzysztof Lewko's user avatar
0 votes
0 answers
242 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 ...
Paul's user avatar
  • 11
0 votes
0 answers
75 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 $...
Indranil B's user avatar
-1 votes
1 answer
169 views

Representation suitable for reconstruction of a tree with bounded degree

I am dealing with reconstruction of molecular graphs for which unlabelled rooted trees with maximum degree 4 are fair approximations. In particular, I would like to encode a small tree (assume number ...
DurgaDatta's user avatar
  • 1,291
-4 votes
1 answer
525 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: ...
stuckyp's user avatar