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| {{About|the data structure|the type of metric space|Real tree}}
| | [http://morrowsmiles.com/singapore-takes-new-steps-cool-property-market-%E2%80%93-sharesinvcom morrowsmiles.com] Developers selling models in a completed housing venture with Certificate of Statutory Completion and particular person legal titles for the items issued will not be required to use the standard type of Choice to Buy and Sale & Buy Agreement prescribed under the Housing Developers Rules.<br><br>The weak spot available in the market is prone to weigh on the sales outlook of smaller listed builders of premium properties comparable to Wheelock Properties (Singapore) Ltd , Ho Bee Land Ltd and Wing Tai Holdings Ltd It feels like we're back in 2008," stated Christine Li, head of research at actual estate firm OrangeTee, referring to the property droop that affected Singapore through the global monetary disaster. An established property fund supervisor, Keppel Land has two property fund management autos, Keppel REIT , a pan-Asian commercial actual estate investment trust, and Alpha Funding Companions (Alpha), a personal fairness arm. When property is made accessible (PRIME is issued) to the client. BELLA CASITA PERSONAL CONDOMINIUM RESIDENCE IPOH LANE, SINGAPORE (DISTRICT 15)<br><br>SAINT PATRICK'S RESIDENCES NON-PUBLIC CONDOMINIUM CONDOMINIUM SAINT PATRICK'S STREET, SINGAPORE (DISTRICT 15) SEA ESTA PERSONAL CONDOMINIUM APARTMENT JALAN LOYANG BESAR, PASIR RIS RISE, SINGAPORE (DISTRICT 18) SEAHILL PRIVATE CONDOMINIUM RESIDENCE WEST COAST HYPERLINK, WEST COAST CRESCENT, SINGAPORE (DISTRICT 05) SIGNATURE AT LEWIS PRIVATE CONDOMINIUM RESIDENCE LEWIS STREET, SINGAPORE (DISTRICT eleven) SKIES MILTONIA NON-PUBLIC CONDOMINIUM RESIDENCE MILTONIA CLOSE, YISHUN AVENUE 1, SINGAPORE (DISTRICT 27) SKYLINE 360 @ ST THOMAS WALK PERSONAL CONDOMINIUM APARTMENT SAINT THOMAS STROLL, SINGAPORE (DISTRICT 09) SUMMER TIME SCENT PRIVATE CONDOMINIUM APARTMENT KANG CHOO BIN ROAD , SINGAPORE (DISTRICT 19) WHITE COVE PONGGOL SEVENTEENTH AVENUE, SINGAPORE (DISTRICT 19) CEL Growth Pte Ltd<br><br>You have to read this earlier than continuing, because it explains the use and nondisclosure terms and situations between Templeton Asset Management Ltd ("TAML") and you (the "Agreement") which apply to the release of portfolio holdings information, together with but not restricted to prime contributors and detractors to portfolio efficiency of one or more non-U.S. domiciled collective investment schemes which can be registered as recognised schemes with the Monetary Authority of Singapore and sponsored by Franklin Templeton Investments (every a "Fund" and together the "Funds").<br><br>You need to read this earlier than proceeding, because it explains the use and nondisclosure phrases and situations between Templeton Asset Administration Ltd ("TAML") and you (the "Agreement") which apply to the release of portfolio holdings data, including but not limited to top contributors and detractors to portfolio efficiency of a number of non-U.S. domiciled collective investment schemes which can be registered as recognised schemes with the Monetary Authority of Singapore and sponsored by Franklin Templeton Investments (each a "Fund" and together the "Funds").<br><br>Cheng additionally revealted that TIEC Holdings had obtained a number of gives however chose to work with Teho because they had a typical imaginative and prescient for the enterprise and believe the acquisition of an current property improvement enterprise akin to TIEC gives a jumpstart for Teho into the trade. 'We already have a staff with a collection of expertise in property development, and the business is running effectively. All we nee is powerful monetary backing, and to ensure compliance with listing regulations," says Cheng."<br><br>Exercise of the Choice/entry into a sale and buy agreement requires cost of the stability of a five to 10 per cent deposit of the acquisition price, plus the cash for stamp obligation. The customer's solicitors will lodge a caveat at the Singapore Land Authority to ensure discover is given of the customer's curiosity in the property and the priority of his interest as from the date of the caveat. The sale contract must be stamped within 14 days of acceptance.<br><br>Soilbuild is an integrated property developer with a development portfolio of mid to excessive-finish residential properties and enterprise house properties for multi-nationwide companies and SMEs. Where you might be buying property which is below development, the Singapore Academyof Legislation will hold a portion of the acquisition monies as stakeholders until the expiryof the defects legal responsibility interval provided in the Settlement/Contract. Check the indicative valuation for the property you propose to buy as valuation instantly affects the amount of loan you can get from the bank for the property. For a foreigner, main banks or monetary institutions in Singapore can only loan as much as 70% of the valuation or buy value (whichever is decrease) IP World (Intellectual Property) |
| {{Infobox data structure
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| |name=R-tree
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| |type=tree
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| |invented_by=[[Antonin Guttman]]
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| |invented_year=1984
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| |space_avg=
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| |space_worst=
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| |search_avg= O(''log<sub>M</sub>n'')
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| |search_worst=
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| |insert_avg=
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| |insert_worst= O(''n'')
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| |delete_avg=
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| |delete_worst=
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| }}
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| [[Image:R-tree.svg|thumb|400px|right|Simple example of an R-tree for 2D rectangles]]
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| [[Image:RTree-Visualization-3D.svg|thumb|400px|right|Visualization of an R*-tree for 3D cubes using [[Environment for DeveLoping KDD-Applications Supported by Index-Structures|ELKI]]]]
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| '''R-trees''' are [[tree data structure]]s used for [[spatial index|spatial access method]]s, i.e., for indexing multi-dimensional information such as [[Geographic coordinate system|geographical coordinates]], [[rectangle]]s or [[polygon]]s. The R-tree was proposed by Antonin Guttman in 1984<ref name="guttman">{{cite doi | 10.1145/602259.602266 }}</ref> and has found significant use in both theoretical and applied contexts.<ref name="rtree-book">{{cite book|author1=Y. Manolopoulos|author2=A. Nanopoulos|author3=Y. Theodoridis|title=R-Trees: Theory and Applications|url=http://books.google.com/books?id=1mu099DN9UwC&pg=PR5|accessdate=8 October 2011|year=2006|publisher=Springer|isbn=978-1-85233-977-7}}</ref> A common real-world usage for an R-tree might be to store spatial objects such as restaurant locations or the polygons that typical maps are made of: streets, buildings, outlines of lakes, coastlines, etc. and then find answers quickly to queries such as "Find all museums within 2 km of my current location", "retrieve all road segments within 2 km of my location" (to display them in a [[navigation system]]) or "find the nearest gas station" (although not taking roads into account).
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| == R-tree idea ==
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| The key idea of the data structure is to group nearby objects and represent them with their [[minimum bounding rectangle]] in the next higher level of the tree; the "R" in R-tree is for rectangle. Since all objects lie within this bounding rectangle, a query that does not intersect the bounding rectangle also cannot intersect any of the contained objects. At the leaf level, each rectangle describes a single object; at higher levels the aggregation of an increasing number of objects. This can also be seen as an increasingly coarse approximation of the data set.
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| Similar to the [[B-tree]], the R-tree is also a balanced search tree (so all leaf nodes are at the same height), organizes the data in pages, and is designed for storage on disk (as used in [[database]]s). Each page can contain a maximum number of entries, often denoted as <math>M</math>. It also guarantees a minimum fill (except for the root node), however best performance has been experienced with a minimum fill of 30%–40% of the maximum number of entries (B-trees guarantee 50% page fill, and [[B*-tree]]s even 66%). The reason for this is the more complex balancing required for spatial data as opposed to linear data stored in B-trees.
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| As with most trees, the searching algorithms (e.g., [[intersection (set theory)|intersection]], containment, [[nearest neighbor search]]) are rather simple. The key idea is to use the bounding boxes to decide whether or not to search inside a subtree. In this way, most of the nodes in the tree are never read during a search. Like B-trees, this makes R-trees suitable for large data sets and [[database]]s, where nodes can be paged to memory when needed, and the whole tree cannot be kept in main memory.
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| The key difficulty of R-trees is to build an efficient tree that on one hand is balanced (so the leaf nodes are at the same height) on the other hand the rectangles do not cover too much empty space and do not overlap too much (so that during search, fewer subtrees need to be processed). For example, the original idea for inserting elements to obtain an efficient tree is to always insert into the subtree that requires least enlargement of its bounding box. Once that page is full, the data is split into two sets that should cover the minimal area each. Most of the research and improvements for R-trees aims at improving the way the tree is built and can be grouped into two objectives: building an efficient tree from scratch (known as bulk-loading) and performing changes on an existing tree (insertion and deletion).
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| R-trees do not guarantee good [[worst-case performance]], but generally perform well with real-world data.<ref>{{cite doi | 10.1007/978-3-540-45072-6_2}}</ref> While more of theoretical interest, the (bulk-loaded) [[Priority R-tree]] variant of the R-tree is worst-case optimal,<ref name="prtree">{{cite doi | 10.1145/1007568.1007608 }}</ref> but due to the increased complexity, has not received much attention in practical applications so far.
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| When data is organized in an R-tree, the [[k nearest neighbors]] (for any [[Lp space|L<sup>p</sup>-Norm]]) of all points can efficiently be computed using a spatial join.<ref>{{cite doi | 10.1145/170036.170075}}</ref> This is beneficial for many algorithms based on the k nearest neighbors, for example the [[Local Outlier Factor]]. DeLi-Clu,<ref>{{cite doi | 10.1007/11731139_16}}</ref> Density-Link-Clustering is a [[cluster analysis]] algorithm that uses the R-tree structure for a similar kind of spatial join to efficiently compute an [[OPTICS algorithm|OPTICS]] clustering.
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| ==Variants==
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| *[[R* tree]]
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| *[[R+ tree]]
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| *[[Hilbert R-tree]]
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| *[[X-tree]]
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| ==Algorithm==
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| === Data layout ===
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| Data in R-trees is organized in pages, that can have a variable number of entries (up to some pre-defined maximum, and usually above a minimum fill). Each entry within a non-[[leaf node]] stores two pieces of data: a way of identifying a [[child node]], and the [[bounding box]] of all entries within this child node. Leaf nodes store the data required for each child, often a point or bounding box representing the child and an external identifier for the child. For point data, the leaf entries can be just the points themselves. For polygon data (that often requires the storage of large polygons) the common setup is to store only the MBR (minimum bounding rectangle) of the polygon along with a unique identifier in the tree.
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| === Search ===
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| The input is a search rectangle (Query box). Searching is quite similar to searching in a [[B+ tree]]. The search starts from the root node of the tree. Every internal node contains a set of rectangles and pointers to the corresponding child node and every leaf node contains the rectangles of spatial objects (the pointer to some spatial object can be there). For every rectangle in a node, it has to be decided if it overlaps the search rectangle or not. If yes, the corresponding child node has to be searched also. Searching is done like this in a recursive manner until all overlapping nodes have been traversed. When a leaf node is reached, the contained bounding boxes (rectangles) are tested against the search rectangle and their objects (if there are any) are put into the result set if they lie within the search rectangle.
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| === Insertion ===
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| To insert an object, the tree is traversed recursively from the root node. At each step, all rectangles in the current directory node are examined, and a candidate is chosen using a heuristic such as choosing the rectangle which requires least enlargement. The search then descends into this page, until reaching a leaf node. If the leaf node is full, it must be split before the insertion is made. Again, since an exhaustive search is too expensive, a heuristic is employed to split the node into two. Adding the newly created node to the previous level, this level can again overflow, and these overflows can propagate up to the root node; when this node also overflows, a new root node is created and the tree has increased in height.
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| ==== Choosing the insertion subtree ====
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| At each level, the algorithm needs to decide in which subtree to insert the new data object. When a data object is fully contained in a single rectangle, the choice is clear. When there are multiple options or rectangles in need of enlargement, the choice can have a significant impact on the performance of the tree.
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| In the classic R-tree, objects are inserted into the subtree that needs the least enlargement. In the more advanced [[R*-tree]], a mixed heuristic is employed. At leaf level, it tries to minimize the overlap (in case of ties, prefer least enlargement and then least area); at the higher levels, it behaves similar to the R-tree, but on ties again preferring the subtree with smaller area. The decreased overlap of rectangles in the R*-tree is one of the key benefits over the traditional R-tree (this is also a consequence of the other heuristics used, not only the subtree choosing).
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| ==== Splitting an overflowing node ====
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| Since redistributing all objects of a node into two nodes has an exponential number of options, a heuristic needs to be employed to find the best split. In the classic R-tree, Guttman proposed two such heuristics, called QuadraticSplit and LinearSplit. In quadratic split, the algorithm searches the pair of rectangles that is the worst combination to have in the same node, and puts them as initial objects into the two new groups. It then searches the entry which has the strongest preference for one of the groups (in terms of area increase) and assigns the object to this group until all objects are assigned (satisfying the minimum fill).
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| There are other splitting strategies such as Greene's Split,<ref name="greene">{{cite doi | 10.1109/ICDE.1989.47268 }}</ref> the [[R*-tree]] splitting heuristic<ref name="rstar">{{cite doi | 10.1145/93597.98741}}</ref> (which again tries to minimize overlap, but also prefers quadratic pages) or the linear split algorithm proposed by Ang and Tan<ref name="ang-tan">{{cite doi | 10.1007/3-540-63238-7_38}}</ref> (which however can produce very unregular rectangles, which are less performant for many real world range and window queries). In addition to having a more advanced splitting heuristic, the [[R*-tree]] also tries to avoid splitting a node by reinserting some of the node members, which is similar to the way a [[B-tree]] balances overflowing nodes. This was shown to also reduce overlap and thus increase tree performance.
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| Finally, the [[X-tree]]<ref name="xtree2">{{Cite journal| first1 = Stefan | last1 = Berchtold| first2 = Daniel A. | last2 = Keim| first3 = Hans-Peter | last3 = Kriegel| author3-link = Hans-Peter Kriegel| title = The X-Tree: An Index Structure for High-Dimensional Data| journal = Proceedings of the 22nd [[VLDB]] Conference| place = Mumbai, India| year = 1996| pages = 28–39| url = http://www.dbs.ifi.lmu.de/Publikationen/Papers/x-tree.ps}}</ref> can be seen as a R*-tree variant that can also decide to not split a node, but construct a so-called super-node containing all the extra entries, when it doesn't find a good split (in particular for high-dimensional data).
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| {{Gallery
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| |title=Effect of different splitting heuristics on a database with Germany postal districts
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| |width=300 | height=300 | align=center | lines=6
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| |File:Zipcodes-Germany-GuttmanRTree.svg|Guttman quadratic split.<ref name="guttman" /><br /> There are many pages that extend from east to west all over Germany, and pages overlap a lot. This is not beneficial for most applications as many only need a small rectangular area that intersects with many slices.
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| |File:Zipcodes-Germany-AngTanSplit.svg|Ang-Tan linear split.<ref name="ang-tan" /><br /> While the slices do not extend as far as with Guttman, the slicing problem affects almost every leaf page. Leaf pages overlap little, but directory pages do.
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| |File:Zipcodes-Germany-RStarTree.svg|[[R* tree]] topological split.<ref name="rstar" /><br /> The pages overlap very little since the R*-tree tries to minimize page overlap, and the reinsertions further optimized the tree. The split strategy also does not prefer slices, the resulting pages are much more useful for common map applications.
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| }}
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| === Deletion ===
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| Deleting an entry from a page may require updating the bounding rectangles of parent pages. However, when a page is underfull, it will not be balanced with its neighbors. Instead, the page will be dissolved and all its children (which may be subtrees, not only leaf objects) will be reinserted. If during this process the root node has a single element, the tree height can decrease.
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| {{Expand section|date=October 2011}}
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| === Bulk-loading ===
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| * Nearest-X - Objects are sorted by their first coordinate ("X") and then split into pages of the desired size.
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| * Packed [[Hilbert R-tree]] - variation of Nearest-X, but sorting using the Hilbert value of the center of a rectangle instead of using the X coordinate. There is no guarantee the pages will not overlap.
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| * Sort-Tile-Recursive (STR):<ref>{{cite paper | first1 = Scott T. | last1 = Leutenegger | first2 = Jeffrey M. | last2 = Edgington | first3 = Mario A. | last3 = Lopez | url = http://archive.org/details/nasa_techdoc_19970016975 | title = STR: A Simple and Efficient Algorithm for R-Tree Packing | year = 1997 | month = February}}</ref> Another variation of Nearest-X, that estimates the total number of leaves required as <math>l=\lceil \text{number of objects} / \text{capacity}\rceil</math>, the required split factor in each dimension to achieve this as <math>s=\lceil l^{1/d}\rceil</math>, then repeatedly splits each dimensions successively into <math>s</math> equal sized partitions using 1-dimensional sorting. The resulting pages, if they occupy more than one page, are again bulk-loaded using the same algorithm. For point data, the leaf nodes will not overlap, and "tile" the data space into approximately equal sized pages.
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| * [[Priority R-tree]]
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| {{Expand section|date=June 2008}}
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| ==See also==
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| * [[Segment tree]]
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| * [[Interval tree]] - A degenerate R-tree for one dimension (usually time).
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| * [[Bounding volume hierarchy]]
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| * [[Spatial index]]
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| * [[GiST]]
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| ==References==
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| <references/>
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| ==External links==
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| {{commons category|R-tree}}
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| * [http://www.rtreeportal.org/ R-tree portal]
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| * R-tree implementations: [http://superliminal.com/sources/sources.htm#C%20&%20C++%20Code C & C++], [http://gis.umb.no/gis/applets/rtree2/jdk1.1/ Java applet], [http://www.cliki.net/spatial-trees Common Lisp], [http://pypi.python.org/pypi/Rtree/ Python], [http://github.com/imbcmdth/RTree Javascript], [http://neirynck.us/rtree Javascript AMD module]
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| *[http://www.boost.org/doc/libs/release/libs/geometry/doc/html/index.html Boost.Geometry library containing R-tree implementation (various splitting algorithms)]
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| {{CS-Trees}}
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| {{Data structures}}
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| {{DEFAULTSORT:R-Tree}}
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| [[Category:R-tree| ]]
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morrowsmiles.com Developers selling models in a completed housing venture with Certificate of Statutory Completion and particular person legal titles for the items issued will not be required to use the standard type of Choice to Buy and Sale & Buy Agreement prescribed under the Housing Developers Rules.
The weak spot available in the market is prone to weigh on the sales outlook of smaller listed builders of premium properties comparable to Wheelock Properties (Singapore) Ltd , Ho Bee Land Ltd and Wing Tai Holdings Ltd It feels like we're back in 2008," stated Christine Li, head of research at actual estate firm OrangeTee, referring to the property droop that affected Singapore through the global monetary disaster. An established property fund supervisor, Keppel Land has two property fund management autos, Keppel REIT , a pan-Asian commercial actual estate investment trust, and Alpha Funding Companions (Alpha), a personal fairness arm. When property is made accessible (PRIME is issued) to the client. BELLA CASITA PERSONAL CONDOMINIUM RESIDENCE IPOH LANE, SINGAPORE (DISTRICT 15)
SAINT PATRICK'S RESIDENCES NON-PUBLIC CONDOMINIUM CONDOMINIUM SAINT PATRICK'S STREET, SINGAPORE (DISTRICT 15) SEA ESTA PERSONAL CONDOMINIUM APARTMENT JALAN LOYANG BESAR, PASIR RIS RISE, SINGAPORE (DISTRICT 18) SEAHILL PRIVATE CONDOMINIUM RESIDENCE WEST COAST HYPERLINK, WEST COAST CRESCENT, SINGAPORE (DISTRICT 05) SIGNATURE AT LEWIS PRIVATE CONDOMINIUM RESIDENCE LEWIS STREET, SINGAPORE (DISTRICT eleven) SKIES MILTONIA NON-PUBLIC CONDOMINIUM RESIDENCE MILTONIA CLOSE, YISHUN AVENUE 1, SINGAPORE (DISTRICT 27) SKYLINE 360 @ ST THOMAS WALK PERSONAL CONDOMINIUM APARTMENT SAINT THOMAS STROLL, SINGAPORE (DISTRICT 09) SUMMER TIME SCENT PRIVATE CONDOMINIUM APARTMENT KANG CHOO BIN ROAD , SINGAPORE (DISTRICT 19) WHITE COVE PONGGOL SEVENTEENTH AVENUE, SINGAPORE (DISTRICT 19) CEL Growth Pte Ltd
You have to read this earlier than continuing, because it explains the use and nondisclosure terms and situations between Templeton Asset Management Ltd ("TAML") and you (the "Agreement") which apply to the release of portfolio holdings information, together with but not restricted to prime contributors and detractors to portfolio efficiency of one or more non-U.S. domiciled collective investment schemes which can be registered as recognised schemes with the Monetary Authority of Singapore and sponsored by Franklin Templeton Investments (every a "Fund" and together the "Funds").
You need to read this earlier than proceeding, because it explains the use and nondisclosure phrases and situations between Templeton Asset Administration Ltd ("TAML") and you (the "Agreement") which apply to the release of portfolio holdings data, including but not limited to top contributors and detractors to portfolio efficiency of a number of non-U.S. domiciled collective investment schemes which can be registered as recognised schemes with the Monetary Authority of Singapore and sponsored by Franklin Templeton Investments (each a "Fund" and together the "Funds").
Cheng additionally revealted that TIEC Holdings had obtained a number of gives however chose to work with Teho because they had a typical imaginative and prescient for the enterprise and believe the acquisition of an current property improvement enterprise akin to TIEC gives a jumpstart for Teho into the trade. 'We already have a staff with a collection of expertise in property development, and the business is running effectively. All we nee is powerful monetary backing, and to ensure compliance with listing regulations," says Cheng."
Exercise of the Choice/entry into a sale and buy agreement requires cost of the stability of a five to 10 per cent deposit of the acquisition price, plus the cash for stamp obligation. The customer's solicitors will lodge a caveat at the Singapore Land Authority to ensure discover is given of the customer's curiosity in the property and the priority of his interest as from the date of the caveat. The sale contract must be stamped within 14 days of acceptance.
Soilbuild is an integrated property developer with a development portfolio of mid to excessive-finish residential properties and enterprise house properties for multi-nationwide companies and SMEs. Where you might be buying property which is below development, the Singapore Academyof Legislation will hold a portion of the acquisition monies as stakeholders until the expiryof the defects legal responsibility interval provided in the Settlement/Contract. Check the indicative valuation for the property you propose to buy as valuation instantly affects the amount of loan you can get from the bank for the property. For a foreigner, main banks or monetary institutions in Singapore can only loan as much as 70% of the valuation or buy value (whichever is decrease) IP World (Intellectual Property)