The P-range tree A new data structure for range searching in secondary memory
小学下册U卷英语第2单元综合卷
小学下册英语第2单元综合卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1. A _____ (植物适应性) allows survival in harsh conditions.2.The process of filtration is used to separate ______ mixtures.3.I want to ________ (lead) community projects.4.The ______ teaches us about environmental science.5.What is the capital of the USA?A. New YorkB. Washington D.C. C. Los AngelesD. Chicago6.The elephant sprays water with its ______ (鼻子). It is very ______ (有趣).7. A ____(landslide) occurs when soil and rock move downhill.8. A chemical that causes a reaction to occur is called a ______.9.The Earth's surface is influenced by both human and ______ factors.10. A ____ is a tiny creature that hops in gardens.11.The tree is _______ (covered) in blossoms.12. A __________ is formed from the deposition of sediments in a lake.13.The ________ is a famous landmark in Egypt.14.My grandfather, ______ (我的祖父), tells great stories from the past.15.Chemicals can change color when they undergo a ______.16.The process of turning a gas into a liquid is called ______.17.She can _______ (speak) three languages.18.What do you use to write on paper?A. PaintB. PencilC. ScissorsD. GlueB19.How many senses do humans have?A. FourB. FiveC. SixD. SevenB20.The chemical formula for acetic acid is ______.21.The _____ (商店) opens at nine.22.What do you call the time before noon?A. AfternoonB. MorningC. EveningD. NightB23.The rock cycle illustrates how rocks can change from one form to another over ______.24.The sound of rain is very ______ (放松).25.n rainforest is home to many ________ (动物). The Amaz26.What do you call a house made of ice?A. IglooB. HutC. TentD. Cabin27.What do we call the part of the plant that absorbs water and nutrients?A. StemB. LeafC. RootD. FlowerC28.The chemical formula for sodium sulfate is ______.29.I have a ___ (sister/brother) and a dog.30.What do you call the practice of growing crops?A. AgricultureB. HorticultureC. FarmingD. Gardening31.What is the capital city of Belgium?A. BrusselsB. AntwerpC. GhentD. BrugesA32. A __________ is a chemical reaction that produces sound and light.33.I enjoy _____ (painting/drawing).34.What do we call the primary source of energy for the Earth?A. WindB. WaterC. SunD. VolcanoesC35.I can _____ my shoes by myself. (put on)36.What is the capital city of Mexico?A. CancúnB. GuadalajaraC. Mexico CityD. Monterrey37.The country known for its festivals is ________ (印度).38.The _____ (桌子) is made of wood.39.The __________ is a famous mountain in Japan.40.I enjoy _______ (散步) in the evenings.41.What do you call a person who plays an instrument?A. MusicianB. SingerC. DancerD. Actor42. (Civil Rights) Movement fought for equality in the USA. The ____43.I like to _____ (画) flowers.44.The ________ was a famous route for traders in the Middle Ages.45.What is the main source of energy for plants?A. WaterB. SoilC. SunlightD. Air46.He is flying a ___. (kite)47.I believe that everyone should have the opportunity to express their __________.48.I enjoy creating with my ________ (塑料积木).49.The _______ (The Spanish Civil War) was fought between Republicans and Nationalists.50.The __________ (元素) table organizes all known elements.51.What is the fastest land animal?A. CheetahB. HorseC. LionD. Gazelle52.The __________ was a time of significant social change in the 1960s. (社会运动)53.What is the name of the famous bear that loves honey?A. PaddingtonB. Winnie the PoohC. YogiD. Baloo54.The antelope's speed is unmatched, making it one of the swiftest creatures on ____.55.What is the name of the famous bear who loves honey?A. PaddingtonB. Winnie the PoohC. BalooD. Yogi BearB56.The process of using enzymes to break down substances is called ______.57.The ______ helps us learn about ethical practices.58.Birds make _______ (巢) in the trees.59. (Renaissance) was a period of great cultural revival in Europe. The ____60.What is the main ingredient in potato chips?A. RiceB. PotatoC. CornD. WheatB61.What is the name of the person who performs in front of an audience?A. PerformerB. AudienceC. DirectorD. Producer62.We analyze ________ (data) for insights.63.What is the capital city of Nigeria?A. LagosB. AbujaC. Port HarcourtD. Kano64. A polymer is a large molecule made of repeating ______.65.Some animals, like bears, prepare for winter by __________.66.What is the capital of India?A. MumbaiB. New DelhiC. KolkataD. Bangalore67.The __________ is a famous historical site in Cambodia. (吴哥窟)68.Moss often grows in ______ (阴湿) areas.69.The first written language was _____ (cuneiform) developed by the Sumerians.70.My teacher is __________, and she/ he helps us __________.71.The main gas that causes global warming is ______.72.The __________ is a mountain range that runs along the west coast of North America.73. A group of lions is called a ______.74.She is ___ (studying/relaxing) at home.75.What is the fastest land animal?A. CheetahB. LionC. HorseD. GazelleA76.What is the primary color of a honeydew melon?A. GreenB. YellowC. OrangeD. Red77. A ________ (航空港) is where planes take off and land.78.What is the name of the sweet dessert made from sugar and cream?A. MeringueB. PavlovaC. CheesecakeD. TiramisuC79.Herbs are often grown in ______ (窗台) pots.80.Which day comes after Monday?A. SundayB. TuesdayC. WednesdayD. ThursdayB81.Liquid water freezes at _____ ( degrees Celsius).82.What do you call the plant that grows in water?A. TreeB. FlowerC. Aquatic plantD. BushC83.What is the name of the process by which stars generate energy?A. Nuclear FusionB. FissionC. Chemical ReactionD. Thermal Emission84.I drink __________ in the morning.85. A chemical that can accept electrons is called an ______ agent.86.What is the name of the fairy tale character who lost her glass slipper?A. Snow WhiteB. CinderellaC. RapunzelD. Sleeping Beauty87.The __________ (历史的共鸣) resonates across cultures.88. A ____(sustainable product) is made with minimal environmental impact.89.Oxygen is essential for ______.90.The fish tank is ___ (full).91.What is the capital of Malaysia?A. Kuala LumpurB. GeorgetownC. MalaccaD. Johor BahruA92.What do you call a young mongoose?A. KitB. PupC. CalfD. Cub93. A __________ is a geological feature that can impact agricultural practices.94.An insulator does not allow heat to ______.95.I want to eat _______ (比萨饼) for dinner.96.How many colors are in a traffic light?A. 3B. 4C. 5D. 297.I want to _______ (去参加) a workshop.98. A parrot can ______ (说话) and mimic sounds.99.What is the name of the famous American author known for "The Great Gatsby"?A. F. Scott FitzgeraldB. Ernest HemingwayC. Mark TwainD. John SteinbeckA100.What is the name of the famous wizard in J.K. Rowling's series?A. GandalfB. DumbledoreC. Harry PotterD. FrodoC。
树上启发式合并详解
树上启发式合并详解In graph theory, a tree is a widely used data structure that consists of nodes or vertices connected by edges. One common problem in tree algorithms is the need to merge or combine subtrees efficiently. This is where the concept of heuristic merging, also known as tree merging or tree union, comes into play.Heuristic merging is a technique that allows us to efficiently merge two subtrees in a tree structure. It is particularly useful when dealing with large trees where the traditional methods of merging, such as traversing theentire tree, become inefficient. The goal of heuristic merging is to find an optimal way to merge the subtrees based on certain criteria or heuristics.One common heuristic used in tree merging is the sizeof the subtrees. The idea is to merge the smaller subtree into the larger subtree, as merging smaller subtrees is generally faster and requires fewer operations. Thisheuristic is based on the assumption that larger subtrees are more likely to have more nodes and edges, and therefore merging them would result in a more significant reductionin the overall size of the tree.Another heuristic used in tree merging is the depth or height of the subtrees. The depth heuristic suggests merging the subtree with a smaller depth into the subtree with a larger depth. This heuristic is based on the observation that merging a shallow subtree into a deeper subtree can potentially reduce the overall depth of the tree, leading to a more balanced and efficient structure.In addition to size and depth, other heuristics can also be employed in tree merging algorithms. These heuristics can be based on various properties of the subtrees, such as the number of leaves, the average distance between nodes, or even domain-specific characteristics. The choice of heuristics depends on the specific requirements and constraints of the problem at hand.To implement heuristic merging, various data structures and algorithms can be used. One common approach is to use a disjoint-set data structure, also known as a union-find data structure. This data structure allows efficient merging of subtrees by maintaining a set of disjoint sets and providing operations to merge and find the representative of each set. By utilizing heuristics, the merging operation in the disjoint-set data structure can be optimized to achieve better performance.Overall, heuristic merging is a powerful technique in tree algorithms that allows efficient merging of subtrees based on certain criteria or heuristics. By carefully choosing and implementing these heuristics, we can significantly improve the performance and efficiency of tree merging operations.。
数据结构专业英语词汇汇总
数据结构专业英语词汇汇总
- Data structure: 数据结构
- Array: 数组
- Linked list: 链表
- Stack: 栈
- Queue: 队列
- Binary tree: 二叉树
- AVL tree: AVL树 (一种自平衡二叉查找树)
- Red-black tree: 红黑树 (一种自平衡二叉查找树)
- Hash table: 哈希表
- Graph: 图
- Vertex: 顶点
- Edge: 边
- Adjacency list: 邻接表 (一种表示图的数据结构)
- Adjacency matrix: 邻接矩阵 (一种表示图的数据结构) - Heap: 堆
- Binary heap: 二叉堆 (一种特殊的堆数据结构)
- Priority queue: 优先队列 (用堆实现的一种队列)
- Trie: 字典树 (一种用于快速检索的树形数据结构)
- Big O notation: 大O符号 (一种表示算法时间复杂度的记号) - Sorting algorithm: 排序算法
- Searching algorithm: 算法
- Abstract data type (ADT): 抽象数据类型
- Hashing: 哈希函数的计算过程
- Collision: 哈希冲突 (发生在两个不同的键值被映射到相同的哈希桶时)。
RTree空间索引详解
2
Spatial Database (Ia)
• Consider: Given a city map, ‘index’ all university buildings in an efficient structure for quick topological search.
3
Spatial Database (Ib)
• Where the child-pointer points to the child of this node, and I is the MBR that encompasses all the regions in the child-node’s pointer’s entries.
I(A) I(B) … I(M) I(a) I(b) I(c) I(d)
• An entry E in a leaf node is defined as (Guttman, 1984):
E = (I, tuple-identifier)
• Where I refers to the smallest binding n-dimensional region (MBR) that encompasses the spatial data pointed to by its tuple-identifier.
cs3460_ch1 Data Structures 美国高校《数据结构》(Data Structure)ppt课件,共9章,英文版
Series - 2
• The top three examples are arithmetic series
• Proofs are typically by induction, which we will see in a minute
• The series for HN is a harmonic series; the eo Euler’s constant 0.57721..
• Then show p1 * p2 * p3 * … * pk + 1 is a prime and it is clearly larger than pk
Recursion
• Some mathematical functions are naturally recursive
– factorial: n! =
• Two steps –
– Prove a base case, for example, if you want to prove a theorem for any integer value n >= 1, you would first prove it for n = 1
– Assuming the theorem is true for values 1..k for some arbitrary k, prove it is true for k + 1
Proof by Induction - 2
• We first prove the base case with n = 1, which results in 1 = 1 (1 + 1) (2 * 1 + 1) / 6
• To prove the general case, we let n = k and prove the formula for k+1 using algebraic manipulation
小学下册J卷英语第1单元期末试卷(有答案)
小学下册英语第1单元期末试卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1. A _____ (89) is a small stream.2.We go _____ (shopping) on Saturdays.3.The Sun's core is extremely ______ and hot.4.I want to ________ (run) a marathon.5. A ______ (植物的生态监测) can provide valuable data.6.My friend loves __________ (参与新项目).7.My mom enjoys __________ (和朋友聚会) on weekends.8.Chemical reactions can be affected by surface area and ______.9. A molecule with a high affinity for electrons is called ______.10.The Earth's atmosphere contains various ______ gases.11.I have a toy _____ that can transform.12.We go _____ (hiking) every spring.13.What is the name of the dog in "The Wizard of Oz"?A. TotoB. SnoopyC. PlutoD. Brian答案:A14.What do we call the lines that run north to south on a map?A. LongitudeB. LatitudeC. EquatorD. Meridian答案:a15.ers are __________ (香味) and attract bees. Some flo16.Incorporating native plants into your garden can support local ______. (在花园中加入本地植物可以支持当地生态。
雅思剑12阅读真题Test5Passage1真题精讲
雅思剑12阅读真题Test5Passage1真题精讲雅思剑12阅读真题Test5Passage1原文及答案!剑桥雅思真题练习是权威的复习资料，建议每一位考生都应该做相关练习，目前剑桥雅思已经更新到12了，下面小编为大家带来雅思剑12阅读真题Test5Passage1原文及答案，供大家复习。
雅思剑12阅读真题Test5Passage1原文及题目READING PASSAGE 1You should spend about 20 minutes on Questions 1-13, which are based on Reading Passage 1 below.CorkCork - the thick bark of the cork oak tree (Quercus suber) - is a remarkable material. It is tough, elastic, buoyant, and fire-resistant, and suitable for a wide range of purposes. It has also been used for millennia: the ancient Egyptians sealed their sarcophagi (stone coffins) with cork, while the ancient Greeks and Romans used it for anything from beehives to sandals.And the cork oak itself is an extraordinary tree. Its bark grows up to 20 cm in thickness, insulating the tree like a coat wrapped around the trunk and branches and keeping the inside at a constant 20℃ all year round. Developed most probably as a defence against forest fires, the bark of the cork oak has a particular cellular structure - with about 40 million cells per cubic centimetre - that technology has never succeeded in replicating. The ceils are filled with air, which is why cork is so buoyant.It also has an elasticity that means you can squash it and watch it spring back to its original size and shape when you release the pressure.Cork oaks grow in a number of Mediterranean countries, including Portugal, Spain, Italy, Greece and Morocco. Theyflourish in warm, sunny climates where there is a minimum of 400 millimetres of rain per year, and not more than 800 millimetres. Like grape vines, the trees thrive in poor soil, putting down deep roots in search of moisture and nutrients. Southern Portugal’s Alentejo region meets all of these requirements, which explains why, by the early 20th century, this region had become the world’s largest producer of cork, and why today it accounts for roughly half of all cork production around the world.Most cork forests are family-owned. Many of these family businesses, and indeed many of the trees themselves, are around 200 years old. Cork production is, above all, an exercise in patience. From the planting of a cork sapling to the first harvest takes 25 years, and a gap of approximately a decade must separate harvests from an individual tree. And for top-quality cork, it’s necessary to wait a further 15 or 20 years. You even have to wait for the right kind of summer’s day to harvest cork. If the bark is stripped on a day when it’s too cold - or when the air is damp - the tree will be damaged.Cork harvesting is a very specialised profession. No mechanical means of stripping cork bark has been invented, so the job is done by teams of highly skilled workers. First, they make vertical cuts down the bark using small sharp axes, then lever it away in pieces as large as they can manage. The most skilful cork- strippers prise away a semi-circular husk that runs the length of the trunk from just above ground level to the first branches. It is then dried on the ground for about four months, before being taken to factories, where it is boiled to kill any insects that might remain in the cork. Over 60% of cork then goes on to be made into traditional bottle stoppers, with most of the remainder being used in the construction trade. Corkboard andcork tiles are ideal for thermal and acoustic insulation, while granules of cork are used in the manufacture of concrete.Recent years have seen the end of the virtual monopoly of cork as the material for bottle stoppers, due to concerns about the effect it may have on the contents of the bottle. This is caused by a chemical compound called 2,4,6-trichloroanisole (TCA), which forms through the interaction of plant phenols, chlorine and mould. The tiniest concentrations - as little as three or four parts to a trillion - can spoil the taste of the product contained in the bottle. The result has been a gradual yet steady move first towards plastic stoppers and, more recently, to aluminium screw caps. These substitutes are cheaper to manufacture and, in the case of screw caps, more convenient for the user.The classic cork stopper does have several advantages, however. Firstly, its traditional image is more in keeping with that of the type of high quality goods with which it has long been associated. Secondly - and very importantly - cork is a sustainable product that can be recycled without difficulty. Moreover, cork forests are a resource which support local biodiversity, and prevent desertification in the regions where they are planted. So, given the current concerns about environmental issues, the future of this ancient material once again looks promising.following statements agree with the information given in Reading Passage 1?In boxes 1-5 on your answer sheet, writeTRUE if the statement agrees with the informationFALSE if the statement contradicts the informationNOT GIVEN if there is no information on this1 The cork oak has the thickest bark of any living tree.2 Scientists have developed a synthetic cork with the same cellular structure as natural cork.3 Individual cork oak trees must be left for 25 years between the first and second harvest.4 Cork bark should be stripped in dry atmospheric conditions.5 The only way to remove the bark from cork oak trees is by hand.Questions 6-13Complete the notes below.Choose ONE WORD ONLY from the passage for each answer.Write your answers in boxes 6-13 on your answer sheet.Comparison of aluminium screw caps and cork bottle stoppersAdvantages of aluminium screw capsdo not affect the 6 ______ of the bottle contentsare 7 ______ to produceare 8 ______ to useAdvantages of cork bottle stopperssuit the 9 ______ of quality productsmade from a 10 ______ materialeasily 11 ______cork forests aid 12 ______cork forests stop 13 ______ happening雅思剑12阅读Test5 Passage1答案解析Test 5 Passage 1Question 1答案：NOT GIVEN关键词：cork oak; thickest bark定位原文：第二段第二句“Its bark grows up to 20cm in thickness, insulating the tree….”解题思路：原文只提到软木橡树的树皮能长到20厘米的厚度，并未提到它的树皮是否是所有树中最厚的。
树种调查英文专业术语
树种调查英文专业术语Conducting a Tree Species Investigation: A Comprehensive ExplorationThe world around us is teeming with diverse and fascinating flora, each species uniquely adapted to its environment. Among the most captivating of these natural wonders are the myriad tree species that grace our landscapes, providing not only aesthetic beauty but also vital ecological functions. As professionals in the field of environmental science, it is our responsibility to delve deeper into the intricacies of these arboreal marvels, uncovering their taxonomic classifications, physiological characteristics, and roles within their respective ecosystems.At the heart of this endeavor lies the tree species investigation, a meticulous process of identifying, categorizing, and studying the trees that populate a given region. This endeavor requires a comprehensive understanding of botanical terminology, as well as a keen eye for the subtle differences that distinguish one species from another. By mastering the English language lexicon associated with tree taxonomy, we can unlock a wealth of knowledge and unlock the doors to a deeper appreciation of the natural world.To begin our investigation, we must first familiarize ourselves with the fundamental taxonomic hierarchy that governs the classification of trees. At the broadest level, we have the kingdom Plantae, which encompasses all photosynthetic organisms, including both flowering and non-flowering plants. Within this kingdom, trees are classified under the division Tracheophyta, also known as vascular plants, which possess specialized tissues for the transport of water and nutrients.Delving deeper, we find that trees belong to the class Magnoliopsida, or dicotyledons, characterized by the presence of two seed leaves, or cotyledons, upon germination. This class is further divided into numerous orders, each with its own distinctive features and adaptations. For instance, the order Fagales includes the familiar oak, beech, and chestnut trees, while the order Pinales encompasses the coniferous species, such as pines, firs, and spruces.At the species level, the specific epithet, or scientific name, serves as the primary means of identification. This binomial nomenclature system, pioneered by the renowned Swedish naturalist Carl Linnaeus, assigns each plant a unique two-part name, consisting of the genus and species. For example, the common oak tree is known scientifically as Quercus rubra, where Quercus represents the genus and rubra denotes the specific species.In addition to taxonomic classification, the tree species investigation also delves into the morphological characteristics that distinguish one species from another. These include the shape and arrangement of leaves, the texture and pattern of bark, the structure of the branching pattern, and the overall form and size of the tree. By carefully observing and documenting these features, we can build a comprehensive understanding of the trees within a given ecosystem.Beyond the external features, the physiological attributes of trees also play a crucial role in their identification and classification. Factors such as the presence or absence of needle-like leaves, the arrangement and structure of the vascular system, and the unique chemical compounds produced by the tree can all provide valuable insights into its identity and evolutionary relationships.In the context of a tree species investigation, the study of leaf morphology is particularly important. Characteristics such as leaf shape, venation patterns, and the presence of trichomes (hair-like structures) can be used to differentiate between species. For instance, the maple tree (Acer) is known for its distinctive palmate leaves, while the oak tree (Quercus) typically displays lobed or deeply indented leaf blades.Bark texture and pattern also serve as important identificationmarkers, as the outer layer of a tree's trunk can exhibit a wide range of characteristics, from the smooth, silvery bark of the birch (Betula) to the deeply furrowed, scaly bark of the shagbark hickory (Carya ovata). By familiarizing ourselves with the diverse range of bark textures and patterns, we can enhance our ability to accurately identify tree species in the field.Another crucial aspect of the tree species investigation is the study of the overall tree form, including the branching pattern, the shape of the crown, and the overall height and diameter of the trunk. These characteristics can provide valuable insights into the tree's adaptation to its environment, as well as its stage of growth and developmental maturity. For example, the towering, columnar shape of the bald cypress (Taxodium distichum) is a testament to its ability to thrive in wetland environments, while the broad, spreading canopy of the American sycamore (Platanus occidentalis) is an adaptation to life in open, sunny landscapes.As we delve deeper into the world of tree species, we must also consider the unique chemical compounds and secondary metabolites produced by these remarkable organisms. From the fragrant terpenes of the pine (Pinus) to the medicinal alkaloids found in the bark of the willow (Salix), these chemical signatures can serve as valuable tools in the identification and classification of trees. By understanding the role of these compounds in the tree's overallphysiology and ecological interactions, we can gain a more holistic appreciation for the complex and interconnected nature of the natural world.In the course of our tree species investigation, we may also encounter instances of hybridization, where two distinct species interbreed to produce offspring with characteristics intermediate between the parent plants. This phenomenon can present a unique challenge, as the resulting trees may exhibit a blend of features that do not neatly fit into established taxonomic categories. In such cases, a thorough understanding of genetic analysis and molecular techniques may be necessary to accurately identify the hybrid and its parent species.Throughout the tree species investigation, it is essential to maintain meticulous records and documentation. This includes the creation of detailed field notes, the collection of herbarium specimens, and the compilation of comprehensive photographic documentation. By building a robust database of information, we can not only aid in the identification and classification of tree species, but also contribute to the broader scientific understanding of the natural world.As we embark on this journey of tree species investigation, we must remember that our work extends far beyond the mere cataloging of these remarkable organisms. By deepening our knowledge andappreciation of the trees that surround us, we can unlock insights into the complex web of ecological relationships that sustain our planet. From the vital role of trees in the carbon cycle to their importance as habitats for countless other species, our understanding of these arboreal wonders can inform and shape our efforts to protect and preserve the natural environments that we call home.In conclusion, the tree species investigation is a multifaceted and captivating endeavor that requires a deep understanding of botanical terminology, morphological characteristics, and physiological adaptations. By mastering the English language lexicon associated with tree taxonomy, we can unlock a wealth of knowledge and become more effective stewards of the natural world. Through our continued efforts to identify, classify, and study the diverse tree species that grace our landscapes, we can contribute to the greater scientific understanding of the living world and inspire others to appreciate the beauty and complexity of the natural environment.。
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1 Introduction and motivation
The successful realization of any data model in a large-scale database requires supporting its language features with e cient secondary storage manipulation. Consider the relational data model of 8]. While the declarative programming features (relational calculus and algebra) of the model are important, it is crucial to support these features with data structures for searching and updating that make optimal use of secondary storage. B-trees and their variants B+ -trees 1,9] are examples of such data structures. They have been an unquali ed success in supporting external dynamic 1-dimensional range searching in relational database systems. The general data structure problem underlying e cient secondary storage manipulation for many data models is external d-dimensional range searching. In particular, 2-dimensional range searching (in various forms) is crucial in the implementation of object-oriented, hierarchical, temporal, spatial, and constraint databases 18,19,27,34]. B-trees, that perform 1-dimensional range searching well, are ine cient at handling more general problems like two and higher dimensional range searching. The problem of 2-dimensional range searching in both main memory and secondary memory has been the subject of much research. Many elegant data structures like the range tree 3], priority search tree 22], segment tree 2], and interval tree 12,13] have been proposed for use in main memory for 2-dimensional range searching and its special cases (see 7] for a detailed survey). Most of these algorithms are not e cient when mapped to secondary storage. However, the practical need for good I/O support has led to the development of a large number of empirical external data structures 15,16,21,23,24,28,29,30,32] which do not have good theoretical worst-case bounds but have good average-case behavior for common spatial database problems. The worst-case performance of these data structures is much worse than the optimal bounds achievable for dynamic external 1-dimensional range searching using B+ -trees. (See 19] for a more complete reference on the eld.) In this paper, we study the problem of 2-dimensional range searching in secondary memory. We present a secondary storage data structure for this problem that is space-optimal and answers queries in time that is within a very small additive term of the optimum. We also present secondary memory data structures for important special cases of 2-dimensional range searching. These data structures occupy optimal linear storage and answer queries in nearly optimal time. Our data structures also allow updates to be made in amortized logarithmic time. These data structures are based upon a new 2-dimensional searching structure called the p-range tree that we show has applications for range searching in main memory as well. Using p-range trees, we give e cient static data structures for d-dimensional range searching in main memory. The space-time product of our main memory data structure is better than any previously known data structure for range searching in dimension 3. We also prove a lower bound on the number of disk blocks needed for constructing an e cient general 2-dimensional range searching data structure in secondary memory. Our lower bound shows that it is not possible to e ciently map a main memory data structure for general 2-dimensional range-searching on to secondary memory without incurring a space overhead. Preliminaries: We now introduce our model for secondary storage algorithms. We make the standard assumption that each secondary memory access transmits one disk block or B units of data, and we count this as one I/O. Function symbols and conventions: The symbol IL will denote the iterated log function. IL (n) is the number of times we must apply the log function to n before the result becomes 2. It is also the 3rd row inverse of the Ackerman function. IL (n) is the inverse of A(3; n) 26] where A denotes the ackerman function. Unless otherwise stated all logarithms will be with respect to base 2. Also, throughout this article we will use the terms in-core and main memory to mean the same thing. The e ciency of our algorithms will be measured in terms of the number of I/O operations they perform. For example, consider the problem of 1-dimensional range searching. Given a set P of points on the real line, a 1-dimensional query q is of the form a x b and the answer to q consists of all the points x 2 P 1