线段树的论文,相当的好.不过是英文的,资源-CSDN下载

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### 线段树与动态线段树：深入理解与应用 #### 一、引言在计算机科学领域，数据结构的设计与优化始终是研究的热点。其中，**线段树**(Segment Tree)作为一种高效的数据结构，被广泛应用于解决区间查询和更新问题。本文将深入探讨线段树的概念，特别是其动态版本——**动态线段树**（Dynamic Segment Tree），以及一种创新的数据结构——**并集复制结构**(Union-Copy Structure)，该结构为动态线段树的实现提供了坚实的基础。 #### 二、线段树概述线段树是一种用于处理区间操作的数据结构，主要用于解决以下两类问题： 1. **区间查询**：如求解一个给定区间的最大值、最小值或总和。 2. **区间更新**：如对一个给定区间的所有元素进行加法或乘法等操作。线段树的基本思想是在一棵二叉树中存储区间信息，树中的每个节点对应原数组的一个区间。对于叶子节点，它们通常对应原数组中的单个元素；而对于非叶子节点，则表示由其子节点所代表的区间组成的更大区间。 #### 三、动态线段树：超越经典动态线段树与经典的线段树相比，最大的不同在于它支持区间插入、删除、合并和分割等操作，而这些操作在传统线段树中通常是不可行的。这使得动态线段树成为处理动态数据集的理想选择，尤其适用于那些数据集可能随时间变化的场景。 #### 四、并集复制结构：动态线段树的核心 **并集复制结构**(Union-Copy Structure)是动态线段树实现的关键。它不仅继承了经典的**并查集**(Union-Find Structure)的特性，还引入了复制操作，使得数据结构能够高效地创建集合副本，这对于动态线段树的构建和维护至关重要。具体来说： 1. **并集操作**(Union)：将两个不相交的集合合并为一个集合。 2. **查找操作**(Find)：确定一个元素所属的集合。 3. **复制操作**(Copy)：创建一个现有集合的完全副本。 #### 五、动态线段树的效率与应用场景动态线段树通过利用并集复制结构，实现了高效的区间操作，包括插入、删除、合并和分割，每项操作的时间复杂度仅为O(log n)，其中n为数据集大小。这种效率使动态线段树成为解决一系列复杂问题的强大工具，例如： - **几何计算**：动态线段树可用于处理涉及线段或区间的一系列几何问题，如求解多边形的重叠区域、计算区间覆盖等。 - **动态规划**：在动态规划算法中，动态线段树可以加速状态转移过程，尤其是在涉及区间操作的场景下。 - **在线算法**：面对实时数据流，动态线段树能够快速响应数据的更新，适用于网络流量监控、股市分析等实时系统。 #### 六、总结线段树及其动态版本不仅是理论上的有趣课题，更是解决实际问题的有力武器。通过深入理解并集复制结构的原理与动态线段树的实现机制，我们不仅能提升算法设计的能力，还能在面对各种复杂数据处理需求时，找到更加高效和优雅的解决方案。未来的研究方向可能集中在进一步优化动态线段树的操作效率，探索其在更广阔的应用领域中的潜力。

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Union-Copy Structures and Dynamic Segment Trees

MARC J. VAN KREVELD AND MARK H. OVERMARS

Utrecht University, Utrecht, the Netherlands

Abstract. A new data structure—the union-copy structure —is introduced, which generalizes the

well-known union-find structure. Besides the usual union and find operations, the new structure

also supports a copy operation, that generates a duplicate of a given set. The structure can

enumerate a given set, find all sets that contain a given element, insert and delete elements, etc.

All these operations can be performed very efficiently. The structure can be tuned as to obtain

different trade-offs in the efficiency of the different operations.

As an application of the union-copy structure, we give a dynamic version of the segment tree.

Contrary to the classical semi-dynamic segment trees, the dynamic segment tree is not restricted

to a fixed universe, from which the endpoints of the segments must be chosen. The tree allows for

insertions, splits, and concatenations in O(log n)-time each. Deletions can be performed in

slightly more time.

Categories and Subject Descriptors: E. 1 [Data Structures]:

trees: F.2.2 [Analysis of Algorithms and

Problem Complexity]: Nonnumerical Algorithms and Problems—geometrica lproblems

and compu-

tutlons

General Terms: Algorithms, Data Structures

Additional Key Words and Phrases: Dynamic segment tree, union-copy structure, union-find

structure

1. Introduction

A well-known data structure problem is the union-find problem (also called

equivalence problem), which dates back to the sixties [2, 7]. A data structure

for the union-find problem stores a collection of disjoint sets {Sl,..., S.}, such

that the following two operations can be performed efficiently:

UNION(Si, ~,),

which takes two (disjoint) sets as its arguments, and unites them into a single

set, and

FIND(x), which takes an element x as its argument, and returns the

name of the set that contains X. Tarjan showed that, starting with n sets that

all contain one unique element, a mixed sequence of n – 1 union operations

and nz > n find operations can be performed in O(WZ . a (nz, n)) time [16]

(where a(wz, n) is the extremely slowly growing functional inverse of

Ackermann’s function). Later, Tarjan showed that this result is optimal in the

The research for this paper was partially supported by the ESPIRIT Basic Research Action No.

3075 (Project ALCOM).

Authors’ address: Department of Computer Science, Utrecht University, P.O. Box 80.089, 3508

TB Utrecht, the Netherlands.

Permission to copy without fee all or part of this material is granted provided that the copies are

not made or distributed for direct commercial advantage, the ACM copyright notice and the title

of the publication and its date appear, and notice is given that copying is by permission of the

Association

for Computing Machinery. To copy otherwise, or to republish, requires a fee imd/or

specific permission.

01993 ACM 0004-541 1/93/0700-0635 $01.50

Journal of the Aswclatmn for Cumputmg Machmc~, Vol. 40, No. 3. July 1993, pp 635-652

636

M. J. VAN KREVELD AND M. H. OVERMARS

pointer machine model [17] (see also [12]). Since then, the union-find problem

has received considerable attention (see, e.g., [41, [61, [111, [121, [15], [181, and

[19]), and several different solutions and variations have been described.

In this paper, we study a generalization of the union-find problem, in which

an additional operation,

coPY, k supported. The operation COPY takes one set

as its argument, and returns an exact copy of it. For these operations, we

introduce the union-copy structure, which represents a collection of sets

{s,,..., S,l} and a collection {X I,..., x,,,} of elements. Although the sets are not

necessarily disjoint, we still require that two sets that are united, are disjoint.

The structure should also support

d14441 operations. Dual in the sense that the

roles of sets and elements are reversed. Such an operation is, for example,

ELEMENT-UNION( .xZ, XJ), which takes two elements that do not share a set, and

turns them into a single element that is member of all the sets xl or xl were in.

Likewise, the procedures

ELEMENT-COPY and SET-FIND(S) are supported. Note

that a structure for the normal union-find problem only supports the opera-

tions

SET-UNION and ELEMENT-FIND.

The structure can be visualized as a representation of a bipartite graph with

two node sets VI and Vz and edges running between a node of VI and a node

of Vz only. The nodes in V, form the sets, the nodes in Vz from the elements.

An edge between an element and a set indicates that the element is a member

of the set. The operations can now be described as follows:

SET-CREATE

Create a new node in V1.

SET-INSERT Add an edge between a node in VI and a node in Vz.

SET-DESTROY Remove a node from VI together with its edges.

SET-FIND Return all nodes in Vz that are connected to the node in V,.

SET-UN1ON Take two nodes in VI (not connected to the same node in Vz)

and join them into a single node.

SET-COPY Make a copy of a node in VI connected to the same nodes in

V2.

The dual operations are exactly the same for nodes in Vz. Note that it is not

possible to delete an individual edge. This would highly increase the time

bounds.

The union-copy structure is based on a mixture of two distinct union-find

structures for its implementation. The efficiency of the operations depends on

the union-find structures chosen. For example, using the structure of TarJan

[16], we obtain the following performance: SET-CREATE and ELEMENT-CREATE,

SET-INSERT, and ELEMENT-INSERT (which are in fact the same), SET-UNION, and

ELEMENT-UNION all take constant time, SET-FIND, ELEMENT-FIND, SET-DESTROY,

and ELEMENT-DESTROY take time 0(1 + k “ a (m, n)) amortized, where k is the

number of reported answers or deleted edges, and

SET-COPY and ELEMENT-COPY

take time 0( a ( nz, n)) amortized. Another version achieves constant time

SET-COPY operations at the cost of an increase in the time bound for

ELEMENT-UNION. A precise statement of the results is given in Theorem 1.

Union-copy structures are useful, for instance, for persistent set mainte-

nance [5]. If we make a copy of a set prior to each update, we can always

retrieve and use old versions of the sets. With the appropriate implementation,

such a copy takes only constant time and constant additional storage.

Union-copy structures can also be considered from the point of view of the

maintenance of bipartite graphs. It describes the relations between two types of

Union-Copy Structures and Dynamic Segment Trees

637

data, for example, suppliers and products or funds and tasks. The operations

are as indicated above. Union operations occur when funds or tasks get

merged. Copy operations are necessary when funds or tasks get split.

A third view is as a dynamic graph structure. The sets and elements of the

structure both correspond to the vertices, where an element vertex w is in a set

vertex u when (u, w) is an edge of the graph. The copy operations corresponds

to multiplication of vertices (see [8]); add a new vertex and let it have all

neighbors of some specified vertex. A union operation on two vertices results

in one new vertex, which has all the neighbors of the former two vertices. It can

be performed on two vertices of distance at least three, because then their sets

of neighbors are disjoint.

One important application, that will be described in detail in this paper, is an

almost fully dynamic version of the segment tree. Segment trees were intro-

duced by Bentley in [3], and have been used for solving a variety of problems in

computational geomet~. A segment tree stores a set of possibly overlapping

segments on a line and can answer stabbing queries efficiently (i.e., which

segments contain a given query point). In the most common description of

segment trees (see, e.g., [14]), a finite subset U = {xl, ..., XN} (universe) of R is

defined, from which all endpoints of the segments must be chosen. Such a

segment tree, storing n segments, requires O(n log IV) storage, updates take

O(log IV) time, and stabbing queries can be performed in O(k + log iV) time,

where k is the number of segments reported.

In applications where IV-the size of the universe—is O(n), these bounds

are good, but when N is considerably larger than n, or when U is not known at

all, the classical solution is not applicable. It is believed that the segment tree

can be made into a dynamic structure without most of these shortcomings, but

an actual description has not yet been given. In this paper, we develop dynamic

segment trees that can store any collection of segments, using the union–copy

structure. Our structure requires O(n log

n) space. Furthermore, insertions

take O(log n) time, deletions take O(log

n “ a(i, n)) time, where a(i, n) is

related to the functional inverse of Ackermann’s function (see Appendix A for

a precise definition), and stabbing queries take O(k + log

n) time. Our struc-

ture also allows for split and concatenate operations, which take O(log n) time

each. This dynamic segment tree, using the split and concatenate operations,

has recently been proven useful for solving a motion planning problem (see

[10]).

The paper is organized as follows: In Section 2, we give a precise description

of the operations we wish to perform, and describe the union-copy structure

and the algorithms in detail. The complexity of the operation depends on the

choice of two underlying union-find structures. Theorem 1 summarizes the

main result of the paper. In Section 3, we describe the dynamic segment tree

and the operations that can be performed on it. Finally, in Section 4, we give

some conclusions and directions for further research.

2. Union-Copy Structures

In this section, we give a precise description of the operations we wish to

perform on a union–copy structure, and describe the data structure and

algorithms that carry out the operations.

Let &’={Sl,..., SW,} be a collection of sets. Let .%’= {xl,..., x,,} be a

collection of elements. The sets in % are subsets of % Elements in X are

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zdqzeros

2014-10-14

英文的比较易懂
Cabbage123453654

2014-01-21

不错挺好的

MasterLuo

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