Sayre equation
In graph theory, the level ancestor problem is the problem of preprocessing a given rooted tree T in order to answer level ancestor queriesTemplate:Dn.Template:Huh Let T be a rooted tree and let v be a node of T, the level ancestor query LA(v,d) requests the depth d ancestor of node v. The depth of a node v in a tree is the number of edges on the shortest path from the root of the tree to node v.
Level ancestor query
One of the fundamental algorithmic problems on trees is to find level ancestors of nodes. The simplest way to find a level ancestor of a node is to climb up the tree towards the root of the tree. On the path to the root of the tree, every ancestor of a node can be visited and therefore reported. In this case, the tree does not need to be preprocessed and the time to answer a query is O(h), where "h" is the height of the tree. This approach is not feasible in situations where a large number of a queries are required to be processed. Alternatively, all the possible solutions can be precomputed and stored in a table. In this case, the queries can be answered in O(1) but the space and the preprocessing time is O(n2). However, a tree can be preprocessed in O(n) such that every Level Ancestor Query can be answered in O(1). There is a number of known algorithms [1][2] that can optimally pre-process a given tree and answer such queries in constant time. The simplest queries that can be answered in constant time without any preprocessing are LA(v, 0) and LA(v, depth(v)). In the former case, the answer is always the root of the tree and in the latter case, the answer is the node v itself. Each of these results takes O(1). By encoding the paths using skew binary random access lists instead, you can obtain results in O(log h), [3] while retaining the ability to extend the tree with new nodes in constant time in exchange for a constant factor of storage space.
Jump pointer algorithm
The jump pointer algorithm [1] is a simple algorithm that pre-processes a tree in O(n log n) time and answers level ancestor queries in O(log n) time. The jump pointer algorithm associates up to log n pointers to each vertex of the tree. These pointers are called jump pointers because they jump up the tree towards the root of the tree. For a given node v of a tree, the algorithm stores an array of length jumpers where . The ith element of this array points to the 2ith ancestor of v. Using such data structure helps us jump half way up the tree from any given node. When the algorithm is asked to process a query, we repeatedly jump up the tree using these pointers. Observe that the number of jumps will be at most log n and therefore queries can be answered in log n time.
Ladder algorithm
Ladder algorithm [1] is based on the idea of simplifying a tree into a bunch paths. The reason for this is the fact that paths are easier to be queried when it comes to level ancestor queries. Consider a path P consisting of n nodes rooted at a node r. We can store the path into an array of size n called Ladder and we can quickly answer a level ancestor query of LA(v, d) by returning Ladder[d] if depth(v)≤d. This will take O(1). However, this will only work if the given tree is a path. Otherwise, we need to decompose it into a bunch of paths. This is done in two stages: long-path decomposition and extending the long paths into ladders.
Stage 1: long-path decomposition
This is a recursive method that decomposes a given tree into a bunch of paths. This stages starts off by finding the longest root-to-leaf path in the tree. It then removes this path by breaking its ties from the tree which will break the remaining of the tree into a bunch of sub-trees and then it recursively processes each sub-tree. Every time a path is decomposed, an array is created in association with the path that contains the elements on the path from the root to the leaf. The base case of this recursion is when the tree is a path in which case its removal leaves an empty graph. Each vertex v has a unique ladder which is the ladder containing it and we call it the "v's ladder". However, after this pre-processing stage, the queries cannot be answered quickly. In fact in order to answer a level ancestor query, the algorithm needs to jump from a path to another until it reaches the root and there could be Θ(√n) of such paths on a leaf-to-root path. This leads us to an algorithm that can pre-process the tree in O(n) time and answers queries in O(√n). In order to reach the optimal query time, we need to process the results in a second stage described below.
Stage 2: extending the long paths into ladders
The first stage of the algorithm decomposes the tree into a number of disjoint paths. In the second stage of the algorithm, each path is extended and therefore the resulting paths will not be mutually exclusive. In the first stage of the algorithm, each path is associated with an array of size h' . We extend this path by adding the h' immediate ancestors at the top of the path in the same array. This will extend each array twice its original size at most which will result in 2n total number of nodes in all the ladders. Notice that the number of ladders is not changed and each node's ladder remains the same. Although a node v can be listed in multiple paths now but its ladder is the one that was associated to v in the first stage of the algorithm. These two stages can be processes in O(n) but the query time is not constant yet. Consider a level ancestor query on a node u of height h. By travelling to the top of u's ladder, a vertex of height at least 2h will be reached. Observe that all the nodes have a height of at least 1 and therefore after doing this i times, we reach a node of height at least 2i and therefore we need to do this at most log n times. This gives us a query time of O(log n).
It turns out that the Ladder algorithm does not do the trick on its own. In fact the Jump pointer algorithm and the Ladder algorithm complement one another. Notice that the two algorithms work in opposite directions. The Jump pointer algorithm makes exponentially increasing hops and the Ladder algorithm makes exponentially decreasing hops and that's the magic. We combine the two algorithms and we can answer queries in O(1) time. We first do a single jump pointer which will take us at least halfway up the tree and then we only need to climb up one ladder in order to answer the query. This results in O(n log n) pre-processing time and O(1) query time.
See also
References
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Template:Cite web