We suggest a new non-recursive algorithm for constructing a binary search tree given an array of numbers. The algorithm has $O(N)$ time and $O(1)$ memory complexity if the given array of $N$ numbers is sorted. The resulting tree is of minimal height and can be transformed to a complete binary search tree (retaining minimal height) with $O(\log N)$ time and $O(1)$ memory.
The algorithm allows simple and effective parallelization.
Weight-balanced trees are a popular form of self-balancing binary search trees. Their popularity is due to desirable guarantees, for example regarding the required work to balance annotated trees. While usual weight-balanced trees perform their balancing operations in a bottom-up fashion after a modification to the tree is completed, there exists a top-down variant which performs these balancing operations during descend. This variant has so far received only little attention. We provide an in-depth analysis and engineering of these top-down weight-balanced trees, demonstrating their superior …
We present an algorithm that allows for find-closest-point and kNN-style traversals of left-balanced k-d trees, without the need for either recursion or software-managed stacks; instead using only current and last previously traversed node to compute which node to traverse next.
The original description of the k-d tree recognized that rebalancing techniques, such as are used to build an AVL tree or a red-black tree, are not applicable to a k-d tree. Hence, in order to build a balanced k-d tree, it is necessary to obtain all of the data prior to building the tree then to build the tree via recursive subdivision of the data. One algorithm for building a balanced k-d tree finds the median of the data for each recursive subdivision of the data and builds the tree in O(n log n) time. A new algorithm builds a balanced k-d tree by presorting the data in each of k dimensions …
We present a method that maintains a balanced binary search tree without using any tree balance criterion at all, with the ultimate aim of maximum simplicity. In fact, our method is highly intuitive, and we only need to add minimal extra code and a simple partial-rebuilding algorithm to a naive binary search tree. Our method will be suitable as a highly simple and short solution when amortized logarithmic costs are enough.
University of British Columbia（不列颠哥伦比亚大学）；ICS-FORTH（基础与研究学院-福斯研究所）；Massachusetts Institute of Technology（麻省理工学院）；University of Crete（克里特大学）；Dartmouth College（达特茅斯学院）；York University（约克大学）
Tree alignment graphs (TAGs) provide an intuitive data structure for storing phylogenetic trees that exhibits the relationships of the individual input trees and can potentially account for nested taxonomic relationships. This paper provides a theoretical foundation for the use of TAGs in phylogenetics. We provide a formal definition of TAG that - unlike previous definition - does not depend on the order in which input trees are provided. In the consensus case, when all input trees have the same leaf labels, we describe algorithms for constructing majority-rule and strict consensus trees using…
Efficient implementations of sets and maps (dictionaries) are important in computer science, and balanced binary search trees are the basis of the best practical implementations. Pedagogically, however, they are often quite complicated, especially with respect to deletion. I present complete code (with justification and analysis not previously available in the literature) for a purely-functional implementation based on AA trees, which is the simplest treatment of the subject of which I am aware.
In this paper we generalize the definition of "Search Trees" (ST) to enable reference values other than the key of prior inserted nodes. The idea builds on the assumption an $n$-node AVL (or Red-Black) requires to assure $O(\log_2n)$ worst-case search time, namely, a single comparison between two keys takes constant time. This means the size of each key in bits is fixed to $B=c\log_2 n$ ($c\geq1$) once $n$ is determined, otherwise the $O(1)$-time comparison assumption does not hold. Based on this we calculate \emph{ideal} reference values from the mid-point of the interval $0..2^B$. This idea …
This paper deals with the facility location problems with balancing on allocation clients to servers. Two bi-objective models are considered, in which one objective is the traditional p-median or p-maxian objective and the second is to minimize the maximum demand volume allocated to any facility. An edge deletion method with time complexity O(n^2) is presented for the balanced $2$-median problem on a tree. For the balanced 2-maxian problem, it is shown the optimal solution is two end vertices of the diameter of the tree, which can be obtained in a linear time.
The article addresses balanced clustering problems with an additional requirement as a tree-like structure over the obtained balanced clusters. This kind of clustering problems can be useful in some applications (e.g., network design, management and routing). Various types of the initial elements are considered. Four basic greedy-like solving strategies (design framework) are considered: balancing-spanning strategy, spanning-balancing strategy, direct strategy, and design of layered structures with balancing. An extended description of the spanning-balancing strategy is presented including fou…
Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P, a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are polygonal chains surrounded by P. A special case of the embedding problem is that in which G is a balanced binary tree. In this paper, we present a new algorithm for embedding an n-vertex balanced binary tree BBT on a set S of n points bounded by a simple m-gon P in O(m^2 + n(log n)^2 + mn) time with at most O(m) bends per edge.
The ordered set is one of the most important data type in both theoretical algorithm design and analysis and practical programming. In this paper we study the set operations on two ordered sets, including Union, Intersect and Difference, based on four types of balanced Binary Search Trees (BST) including AVL trees, red-black trees, weight balanced trees and treaps. We introduced only one subroutine Join that needs to be implemented differently for each balanced BST, and on top of which we can implement generic, simple and efficient parallel functions for ordered sets. We first prove the work-e…
Karloff? and Shirley recently proposed summary trees as a new way to visualize large rooted trees (Eurovis 2013) and gave algorithms for generating a maximum-entropy k-node summary tree of an input n-node rooted tree. However, the algorithm generating optimal summary trees was only pseudo-polynomial (and worked only for integral weights); the authors left open existence of a olynomial-time algorithm. In addition, the authors provided an additive approximation algorithm and a greedy heuristic, both working on real weights. This paper shows how to construct maximum entropy k-node summary trees i…
We revisit multipass pairing heaps and path-balanced binary search trees (BSTs), two classical algorithms for data structure maintenance. The pairing heap is a simple and efficient "self-adjusting" heap, introduced in 1986 by Fredman, Sedgewick, Sleator, and Tarjan. In the multipass variant (one of the original pairing heap variants described by Fredman et al.) the minimum item is extracted via repeated pairing rounds in which neighboring siblings are linked. Path-balanced BSTs, proposed by Sleator (Subramanian, 1996), are a natural alternative to Splay trees (Sleator and Tarjan, 1983). In a p…
Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of genes or species, and aims at finding a single tree that minimizes the leaf-removal distance to the input trees. This problem is a specific instance of the general consensus/supertree problem, widely used to combine or summarize discordant evolutionary trees. The problem we introduce is specifically tailored to address the case of discrepancies between the inp…
In this paper we study the question of whether or not a static search tree should ever be unbalanced. We present several methods to restructure an unbalanced k-ary search tree $T$ into a new tree $R$ that preserves many of the properties of $T$ while having a height of $\log_k n +1$ which is one unit off of the optimal height. More specifically, we show that it is possible to ensure that the depth of the elements in $R$ is no more than their depth in $T$ plus at most $\log_k \log_k n +2$. At the same time it is possible to guarantee that the average access time $P(R)$ in tree $R$ is no more th…
The traditional multi-commodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multi-commodity flow network-design problem: given a set of multi-commodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the m…
We develop a time-optimal $O(mn^2)$-time algorithm to construct the subtree prune-regraft (SPR) graph on a collection of m phylogenetic trees with n leaves. This improves on the previous bound of $O(mn^3)$. Such graphs are used to better understand the behaviour of phylogenetic methods and recommend parameter choices and diagnostic criteria. The limiting factor in these analyses has been the difficulty in constructing such graphs for large numbers of trees. We also develop the first efficient algorithms for constructing the nearest-neighbor interchange (NNI) and tree bisection-and-reconnection…
Two kinds of approximation algorithms exist for the k-BALANCED PARTITIONING problem: those that are fast but compute unsatisfying approximation ratios, and those that guarantee high quality ratios but are slow. In this paper we prove that this tradeoff between runtime and solution quality is necessary. For the problem a minimum number of edges in a graph need to be found that, when cut, partition the vertices into k equal-sized sets. We develop a reduction framework which identifies some necessary conditions on the considered graph class in order to prove the hardness of the problem. We focus …
We study the load balanced capacitated vehicle routing problem (LBCVRP): the problem is to design a collection of tours for a fixed fleet of vehicles with capacity Q to distribute a supply from a single depot between a number of predefined clients, in a way that the total traveling cost is a minimum, and the vehicle loads are balanced. The unbalanced loads cause the decrease of distribution quality especially in business environments and exibility in the logistics activities. The problem being NP-hard, we propose two approximation algorithms. When the demands are equal, we present a (1-1/Q)p+3…
The suffix trees are fundamental data structures for various kinds of string processing. The suffix tree of a string $T$ of length $n$ has $O(n)$ nodes and edges, and the string label of each edge is encoded by a pair of positions in $T$. Thus, even after the tree is built, the input text $T$ needs to be kept stored and random access to $T$ is still needed. The linear-size suffix tries (LSTs), proposed by Crochemore et al. [Linear-size suffix tries, TCS 638:171-178, 2016], are a `stand-alone' alternative to the suffix trees. Namely, the LST of a string $T$ of length $n$ occupies $O(n)$ total s…
For any fixed measure $H$ that maps graphs to real numbers, the MinH problem is defined as follows: given a graph $G$, an integer $k$, and a target $τ$, is there a set $S$ of $k$ vertices that can be deleted, so that $H(G - S)$ is at most $τ$? In this paper, we consider the MinH problem on trees. We call $H$ "balanced on trees" if, whenever $G$ is a tree, there is an optimal choice of $S$ such that the components of $G-S$ have sizes bounded by a polynomial in $n/k$. We show that MinH on trees is FPT for parameter $n/k$, and furthermore, can be solved in subexponential time, and polynomial spac…
One of the key problems in tensor network based quantum circuit simulation is the construction of a contraction tree which minimizes the cost of the simulation, where the cost can be expressed in the number of operations as a proxy for the simulation running time. This same problem arises in a variety of application areas, such as combinatorial scientific computing, marginalization in probabilistic graphical models, and solving constraint satisfaction problems. In this paper, we reduce the computationally hard portion of this problem to one of graph linear ordering, and demonstrate how existin…
The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph $G$ whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of $G$ having the same number of blue vertices and red vertices. They showed that this problem is NP-hard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in $O(n^3)$ time for trees and $O(n + m)$ time for split graphs and properly colored bipartite graphs, where $n$ is the number of vertices and…
The rapid growth of big spatial data urged the research community to develop several big spatial data systems. Regardless of their architecture, one of the fundamental requirements of all these systems is to spatially partition the data efficiently across machines. The core challenges of big spatial partitioning are building high spatial quality partitions while simultaneously taking advantages of distributed processing models by providing load balanced partitions. Previous works on big spatial partitioning are to reuse existing index search trees as-is, e.g., the R-tree family, STR, Kd-tree, …
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge $e$ in $G$ we would like to decide whether $e$ belongs to a connected subgraph $G'$ consisting of $(1+ε)n$ edges (for a prespecified constant $ε>0$), where the decision for different edges should be consistent with the same subgraph $G'$. Can this task be performed by inspecting only a {\em constant} number of edges in $G$? Our main result…
We present a new algorithm, which solves the problem of distributively finding a minimum diameter spanning tree of any (non-negatively) real-weighted graph $G = (V,E,ω)$. As an intermediate step, we use a new, fast, linear-time all-pairs shortest paths distributed algorithm to find an absolute center of $G$. The resulting distributed algorithm is asynchronous, it works for named asynchronous arbitrary networks and achieves $\mathcal{O}(|V|)$ time complexity and $\mathcal{O}\left(|V|\,|E|\right)$
In this paper we consider the problem of efficiently constructing in a fully distributed manner multicast trees which are embedded into P2P overlays using virtual geometric node coordinates. We consider two objectives: to minimize the number of messages required for constructing a multicast tree by using the geometric properties of the P2P overlay, and to construct stable multicast trees when the lifetime durations of the peers are known.
Given a set S of n \geq d points in general position in R^d, a random hyperplane split is obtained by sampling d points uniformly at random without replacement from S and splitting based on their affine hull. A random hyperplane search tree is a binary space partition tree obtained by recursive application of random hyperplane splits. We investigate the structural distributions of such random trees with a particular focus on the growth with d. A blessing of dimensionality arises--as d increases, random hyperplane splits more closely resemble perfectly balanced splits; in turn, random hyperplan…
The M-tree is a paged, dynamically balanced metric access method that responds gracefully to the insertion of new objects. To date, no algorithm has been published for the corresponding Delete operation. We believe this to be non-trivial because of the design of the M-tree's Insert algorithm. We propose a modification to Insert that overcomes this problem and give the corresponding Delete algorithm. The performance of the tree is comparable to the M-tree and offers additional benefits in terms of supported operations, which we briefly discuss.
It is shown that every tree of size $n$ over a fixed set of $σ$ different ranked symbols can be decomposed (in linear time as well as in logspace) into $O\big(\frac{n}{\log_σn}\big) = O\big(\frac{n \log σ}{\log n}\big)$ many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size $O\big(\frac{n}{\log_σn}\big)$, which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case …
