Skip to main content
Share Your Experience: Take the 2024 Developer Survey

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.

Filter by
Sorted by
Tagged with
52 votes
20 answers
9k 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 ...
Shiva Kintali's user avatar
45 votes
4 answers
44k views

Why would one ever use an Octree over a KD-tree?

I have some experience in scientific computing, and have extensively used kd-trees for BSP (binary space partitioning) applications. I have recently become rather more familiar with octrees, a similar ...
Noldorin's user avatar
  • 867
27 votes
1 answer
727 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 ...
john_leo's user avatar
  • 431
21 votes
2 answers
885 views

maintaining a balanced spanning tree of a growing undirected graph

I am looking for ways to maintain a relatively balanced spanning tree of a graph, as I add new nodes/edges to the graph. I have an undirected graph that starts as a single node, the "root". At each ...
SuperElectric's user avatar
20 votes
2 answers
10k views

efficient diff algorithm for trees and Levenshtein distance

I've recently read this summary of the issues involved with doing diff between trees and it got me interested in learning what is the state of the art for this problem. Also, suppose that between ...
lurscher's user avatar
  • 303
20 votes
1 answer
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 ...
Sadeq Dousti's user avatar
  • 16.5k
17 votes
3 answers
5k views

Merging Two Binary Search Trees

I'm looking for an algorithm to merge two binary search trees of arbitrary size and range. The obvious way I would go about implementing this would be to find entire subtrees whose range can fit into ...
efritz's user avatar
  • 273
15 votes
2 answers
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 ...
n_ary_crazy's user avatar
15 votes
3 answers
557 views

Bob's Sale (reordering of pairs with constraints to minimize sum of products)

I've asked this question on Stack Overflow a while ago: Problem: Bob's sale. Someone suggested posting the question here as well. Someone has already asked a question related to this problem here - ...
Vladimir Panteleev's user avatar
14 votes
4 answers
2k views

Subrange of a Red and Black Tree

While trying to fix a bug in a library, I searched for papers on finding subranges on red and black trees without success. I'm considering a solution using zippers and something similar to the usual ...
Daniel C. Sobral's user avatar
14 votes
4 answers
740 views

P-complete problems on trees

This question is related to one of my previous questions, NP-hard problems on trees. I am looking for problems that are P-complete on trees.
Shiva Kintali's user avatar
13 votes
0 answers
178 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 ...
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
  • 566
13 votes
0 answers
375 views

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
  • 23.2k
11 votes
1 answer
330 views

Minimum weight subforest of given cardinality

This question was motivated by a question asked on stackoverflow. Suppose you are given a rooted tree $T$ (i.e. there is a root and nodes have children etc) on $n$ nodes (labelled $1, 2, \dots, n$). ...
Aryabhata's user avatar
  • 1,855
10 votes
2 answers
1k views

How do I choose a functional dictionary data structure?

I've read a bit about the following data structures: Bagwell's Ideal Hash Tries Larson's Dynamic hash tables Red-Black trees Patricia trees ...and I'm sure there are a lot of others out there. I've ...
Jason's user avatar
  • 397
10 votes
6 answers
843 views

A data structure for sets of trees.

Tries allow for efficient storage of lists of elements. The prefixes are shared so it is space efficient. I am looking for a similar way to efficiently store trees. I would like to be able to check ...
Abdallah's user avatar
  • 803
10 votes
2 answers
546 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 ...
john mangual's user avatar
10 votes
1 answer
344 views

Lower bound on the number of "short" paths in a rooted tree with polynomial size

Let $T$ be a rooted binary tree. Every path from the root of $T$ to a leaf has length $n$. Every node of $T$ has always a left and a right child node but it is possible that they are the same (So ...
Marc Bury's user avatar
  • 1,338
9 votes
2 answers
8k 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 ...
user3783608's user avatar
9 votes
2 answers
1k views

What is the optimal data structure for a tree of maps.

I'm looking for a data structure, that is basically a tree of maps, where the map at each node contains some new elements, as well as the elements in its parent node's map. By map here I mean a ...
phreeza's user avatar
  • 93
9 votes
2 answers
382 views

O(n)-space, polylog-time subtree sums in incremental forests?

Consider a forest $G$ of $n$ vertices $v_1, \dots, v_n$ arranged left to right with edges from child to parent always going to the left, i.e. if the parent of vertex $v_i$ is $v_j$, then $j < i$. ...
Mathias Rav's user avatar
9 votes
2 answers
432 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 ...
Antonio Valerio Miceli-Barone's user avatar
8 votes
1 answer
712 views

Minimum degree of the "tree graph"

Given a graph $G$, define the tree graph $T(G)$ as a graph whose vertices are the spanning trees of $G$, and there is an edge between two trees if one can be obtained from the other by replacing a ...
corwin's user avatar
  • 91
8 votes
2 answers
840 views

What is the initialization time of a link-cut tree?

Link-cut tree is a data structure invented by Sleator and Tarjan, which supports various operations and queries on a $n$-node forest in time $O(\log n)$. (For example, operation link combines two ...
Hsien-Chih Chang 張顯之's user avatar
7 votes
3 answers
8k views

Split or merge Binary Search Trees in O(log n)

We need to have an efficient operation of merging or splitting two binary search trees $S_1$ and $S_2$. There are given the following. The element with the largest value in $S_1$ is smaller than the ...
Dimitris Leventeas's user avatar
7 votes
1 answer
282 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 ...
Dmitri Urbanowicz's user avatar
7 votes
1 answer
394 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 $\...
R B's user avatar
  • 9,458
7 votes
1 answer
386 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....
Bartosz Bednarczyk's user avatar
7 votes
1 answer
462 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)...
Vivek Bagaria's user avatar
7 votes
1 answer
282 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 ...
Jukka Suomela's user avatar
7 votes
0 answers
97 views

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
284 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
2 answers
560 views

Caterpillar decomposition of trees

Can any tree on $n$ nodes be decomposed into a set of $O(\log n)$ caterpillars? If not, what is the maximum number of caterpillars required? Are there efficient algorithms for finding the ...
Arindam Pal's user avatar
  • 1,591
6 votes
2 answers
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 ...
Jarek Duda's user avatar
6 votes
1 answer
530 views

Searching nodes in semi-splay tree

If you search for a node in a semi-splay tree, it's basically to push certain nodes closer to the root, to reduce future search operations. My course also says that if you search for a node and the ...
Aerus's user avatar
  • 227
6 votes
1 answer
261 views

Complexity of reachability in directed rooted forests

I'm trying to figure out the complexity of the reachability problem having as input a directed rooted forest, i.e., given a set of directed rooted trees and two vertices $s$ and $t$, tell if $s$ and $...
Abel Freid's user avatar
6 votes
1 answer
348 views

Optimal Self Balancing Trees with Canonical Form?

Are any efficient [O(log n)] self balancing trees that are canonical? By canonical I mean that for any set of data inserted into the tree, inserting it after any permutation results in the same tree. ...
Jonathan's user avatar
  • 250
6 votes
1 answer
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 ...
Mark Miller's user avatar
6 votes
3 answers
485 views

Chomsky hierarchy for tree structures

I know of the Chomsky hierarchy, which concerns the expressive power of grammars to recognize languages $L \subseteq \Sigma^*$ made of words on an alphabet $\Sigma$. Is there a similar hierarchy for ...
Bruno Le Floch's user avatar
6 votes
1 answer
167 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 (...
Andrey Lebedev's user avatar
6 votes
0 answers
359 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
  • 679
5 votes
1 answer
101 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 ...
user's user avatar
  • 615
5 votes
1 answer
596 views

Dynamic Tree Marked Ancestor Queries

Assuming a rooted tree $T$ with vertices $V$, I am maintaining subsets of $V$, for example $M \subseteq V$ whose vertices are associated with particular labels or values. $V$ is dynamic in that it ...
0__'s user avatar
  • 173
5 votes
3 answers
411 views

Storage system for large quantities of unique key value pairs optimized for insert

Background I'm in the process of attempting to improve part of our data storage and analysis architecture. Without getting into a lot of details, at a certain part of our data analysis process we ...
user avatar
5 votes
1 answer
149 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\...
Andrey Lebedev's user avatar
5 votes
1 answer
1k views

Trees that structure partially ordered data

Suppose we have a binary search tree $T$ built over keys from a totally ordered set, and we want to support the standard dictionary lookup $\mbox{Find}(x)$ which returns a pointer to the node ...
Joe's user avatar
  • 1,327
5 votes
0 answers
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
164 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
  • 171