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A family of hash functions is said to be '''<math>k</math>-independent''' or '''<math>k</math>-universal'''<ref name=CLRS> | |||
{{cite book | |||
| first = Thomas H. | |||
| last = Cormen | |||
| coauthors = Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford | |||
| title = Introduction to Algorithms | |||
| publisher = MIT Press | |||
| year = 2009 | |||
| isbn = 0-262-03384-4 | |||
| edition = 3rd | |||
}} | |||
</ref> if selecting a [[hash function]] at random from the family guarantees that the hash codes of any designated <math>k</math> keys are [[Independence (probability theory)|independent random variables]] (see precise mathematical definitions below). Such families allow good average case performance in randomized algorithms or data structures, even if the input data is chosen by an adversary. The trade-offs between the degree of independence and the efficiency of evaluating the hash function are well studied, and many <math>k</math>-independent families have been proposed. | |||
== Introduction == | |||
{{see also|Hash function}} | |||
The goal of hashing is usually to map keys from some large domain (universe) <math>U</math> into a smaller range, such as <math>m</math> bins (labelled <math>[m] = \{0, \dots, m-1\}</math>). In the analysis of randomized algorithms and data structures, it is often desirable for the hash codes of various keys to "behave randomly". For instance, if the hash code of each key were an independent random choice in <math>[m]</math>, the number of keys per bin could be analyzed using the [[Chernoff bound]]. A deterministic hash function cannot offer any such guarantee in an adversarial setting, as the adversary may choose the keys to be the precisely the [[Image (mathematics)|preimage]] of a bin. Furthermore, a deterministic hash function does not allow for ''rehashing'': sometimes the input data turns out to be bad for the hash function (e.g. there are too many collisions), so one would like to change the hash function. | |||
The solution to these problems is to pick a function ''randomly'' from a large family of hash functions. The randomness in choosing the hash function can be used to guarantee some desired random behavior of the hash codes of any keys of interest. The first definition along these lines was [[universal hashing]], which guarantees a low collision probability for any two designated keys. The concept of <math>k</math>-independent hashing, introduced by Wegman and Carter in 1981,<ref name=WC81> | |||
{{cite journal | |||
| last1 = Wegman | |||
| first1 = Mark N. | author1-link = Mark N. Wegman | |||
| last2 = Carter | |||
| first2 = J. Lawrence | |||
| title = New hash functions and their use in authentication and set equality | |||
| journal = Journal of Computer and System Sciences | |||
| volume = 22 | |||
| issue = 3 | |||
| pages = 265–279 | |||
| year = 1981 | |||
| doi = 10.1016/0022-0000(81)90033-7 | |||
| id = Conference version in FOCS'79 | |||
| url = http://www.fi.muni.cz/~xbouda1/teaching/2009/IV111/Wegman_Carter_1981_New_hash_functions.pdf | |||
| accessdate = 9 February 2011 | |||
}}</ref> strengthens the guarantees of random behavior to families of <math>k</math> designated keys, and adds a guarantee on the uniform distribution of hash codes. | |||
=== Mathematical Definitions === | |||
The strictest definition, introduced by Wegman and Carter<ref name=WC81 /> under the name "strongly universal<math>_k</math> hash family", is the following. A family of hash functions <math>H=\{ h:U \to [m] \}</math> is <math>k</math>-independent if for any <math>k</math> distinct keys <math>(x_1, \dots, x_k) \in U^k</math> and any <math>k</math> hash codes (not necessarily distinct) <math>(y_1, \dots, y_k) \in [m]^k</math>, we have: | |||
: <math>\Pr_{h \in H} \left[ h(x_1)=y_1 \land \cdots \land h(x_k)=y_k \right] = m^{-k}</math> | |||
This definition is equivalent to the following two conditions: | |||
# for any fixed <math>x\in U</math>, as <math>h</math> is drawn randomly from <math>H</math>, <math>h(x)</math> is uniformly distributed in <math>[m]</math>. | |||
# for any fixed, distinct keys <math>x_1, \dots, x_k \in U</math>, as <math>h</math> is drawn randomly from <math>H</math>, <math>h(x_1), \dots, h(x_k)</math> are independent random variables. | |||
Often it is inconvenient to achieve the perfect joint probability of <math>m^{-k}</math> due to rounding issues. Following,<ref name=Siegel> | |||
{{cite journal | |||
| last1 = Siegel | |||
| first1 = Alan | |||
| title = On universal classes of extremely random constant-time hash functions and their time-space tradeoff | |||
| journal = SIAM Journal on Computing | |||
| volume = 33 | |||
| issue = 3 | |||
| pages = 505–543 | |||
| year = 2004 | |||
| id = Conference version in FOCS'89 | |||
| url = http://www.cs.nyu.edu/faculty/siegel/FASTH.pdf | |||
| doi=10.1137/S0097539701386216}} | |||
</ref> one may define a <math>(\mu, k)</math>-independent family to satisfy: | |||
: <math>\forall</math> distinct <math>(x_1, \dots, x_k) \in U^k</math> and <math>\forall (y_1, \dots, y_k) \in [m]^k</math>, <math>~~\Pr_{h \in H} \left[ h(x_1)=y_1 \land \cdots \land h(x_k)=y_k \right] \le \mu / m^k</math> | |||
Observe that, even if <math>\mu</math> is close to 1, <math>h(x_i)</math> are no longer independent random variables, which is often a problem in the analysis of randomized algorithms. Therefore, a more common alternative to dealing with rounding issues is to prove that the hash family is close in [[statistical distance]] to a <math>k</math>-independent family, which allows black-box use of the independence properties. | |||
==See also== | |||
* [[Universal hashing]] | |||
* [[Tabulation hashing]], a technique for generating 3-independent hash functions | |||
== References == | |||
<references /> | |||
== Further reading == | |||
* {{cite book | |||
| last1 = Motwani | |||
| first1 = Rajeev | |||
| last2 = Raghavan | |||
| first2 = Prabhakar | |||
| title = Randomized Algorithms | |||
| publisher = Cambridge University Press | |||
| year = 1995 | |||
| isbn = 0-521-47465-5 | |||
| page = 221 | |||
}} | |||
[[Category:Hash functions]] | |||
[[Category:Search algorithms]] | |||
[[Category:Error detection and correction]] | |||
Revision as of 13:44, 3 October 2013
A family of hash functions is said to be -independent or -universal[1] if selecting a hash function at random from the family guarantees that the hash codes of any designated keys are independent random variables (see precise mathematical definitions below). Such families allow good average case performance in randomized algorithms or data structures, even if the input data is chosen by an adversary. The trade-offs between the degree of independence and the efficiency of evaluating the hash function are well studied, and many -independent families have been proposed.
Introduction
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The goal of hashing is usually to map keys from some large domain (universe) into a smaller range, such as bins (labelled ). In the analysis of randomized algorithms and data structures, it is often desirable for the hash codes of various keys to "behave randomly". For instance, if the hash code of each key were an independent random choice in , the number of keys per bin could be analyzed using the Chernoff bound. A deterministic hash function cannot offer any such guarantee in an adversarial setting, as the adversary may choose the keys to be the precisely the preimage of a bin. Furthermore, a deterministic hash function does not allow for rehashing: sometimes the input data turns out to be bad for the hash function (e.g. there are too many collisions), so one would like to change the hash function.
![{\displaystyle [m]=\{0,\dots ,m-1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9353ecda11480fd9431a5a15de494b7827f0020)
![{\displaystyle [m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b649f5418f2ac84d66063f41319368b0926ccec4)
The solution to these problems is to pick a function randomly from a large family of hash functions. The randomness in choosing the hash function can be used to guarantee some desired random behavior of the hash codes of any keys of interest. The first definition along these lines was universal hashing, which guarantees a low collision probability for any two designated keys. The concept of -independent hashing, introduced by Wegman and Carter in 1981,[2] strengthens the guarantees of random behavior to families of designated keys, and adds a guarantee on the uniform distribution of hash codes.
Mathematical Definitions
The strictest definition, introduced by Wegman and Carter[2] under the name "strongly universal hash family", is the following. A family of hash functions is -independent if for any distinct keys and any hash codes (not necessarily distinct) , we have:
![{\displaystyle H=\{h:U\to [m]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07dbe0dc6d02fc97f45867f07178c7000d6343b9)
![{\displaystyle (y_{1},\dots ,y_{k})\in [m]^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c433de4188bd9bedbb3f582b748b1d60640d904)
![{\displaystyle \Pr _{h\in H}\left[h(x_{1})=y_{1}\land \cdots \land h(x_{k})=y_{k}\right]=m^{-k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a0259a6436aaaf651ede961e7cbb8f4079fb5d)
This definition is equivalent to the following two conditions:
- for any fixed , as is drawn randomly from , is uniformly distributed in .
- for any fixed, distinct keys , as is drawn randomly from , are independent random variables.
Often it is inconvenient to achieve the perfect joint probability of due to rounding issues. Following,[3] one may define a -independent family to satisfy:
- distinct and ,
![{\displaystyle \forall (y_{1},\dots ,y_{k})\in [m]^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25ea7aa2578de6be638cd97492f09890b62c1b84)
![{\displaystyle ~~\Pr _{h\in H}\left[h(x_{1})=y_{1}\land \cdots \land h(x_{k})=y_{k}\right]\leq \mu /m^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e58f823c6bd14ab7fd9123f0edc3099c264989)
Observe that, even if is close to 1, are no longer independent random variables, which is often a problem in the analysis of randomized algorithms. Therefore, a more common alternative to dealing with rounding issues is to prove that the hash family is close in statistical distance to a -independent family, which allows black-box use of the independence properties.
See also
- Universal hashing
- Tabulation hashing, a technique for generating 3-independent hash functions
References
- ↑
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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 - ↑
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
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
Further reading
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534