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| == Vibram Soldes 000 == | | The '''r-to-1 collision problem''' is an important theoretical problem in [[Computational complexity theory|complexity theory]], [[quantum computing]], and [[computational mathematics]]. The collision problem most often refers to the 2-to-1 version:<ref>{{cite paper |author = Scott Aaronson |authorlink = Scott Aaronson|title = Limits on Efficient Computation in the Physical World |year = 2004|url = http://www.scottaaronson.com/thesis.pdf|format=[[PDF]]}}</ref> given <math>n</math> even and a function <math>f:\,\{1,\ldots,n\}\rightarrow\{1,\ldots,n\}</math>, we are promised that f is either [[1-to-1]] or 2-to-1. We are only allowed to make queries about the value of <math>f(i)</math> for any <math>i\in\{1,\ldots,n\}</math>. The problem then asks how many such queries we need to make to determine with certainty whether f is 1-to-1 or 2-to-1. |
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Classical Solutions
Deterministic
Solving the 2-to-1 version deterministically requires queries, and in general distinguishing r-to-1 functions from 1-to-1 functions requires queries.
This is a straightforward application of the pigeonhole principle: if a function is r-to-1, then after queries we are guaranteed to have found a collision. If a function is 1-to-1, then no collision exists. Thus, queries suffice. If we are unlucky, then the first queries could return distinct answers, so queries is also necessary.
Randomized
If we allow randomness, the problem is easier. By the birthday paradox, if we choose (distinct) queries at random, then with high probability we find a collision in any fixed 2-to-1 function after queries.
Quantum Solution
The BHT algorithm, which uses Grover's algorithm, solves this problem optimally using by only making queries to f.
References
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The majority of owner-occupied homes will have lower tax charges in Singapore Finances 2013. be allowed to launch units on the market 15 months from the date of award of the sites or SINGAPORE The Housing and Development Board (HDB) has launched one other government condominium (EC) website for tender. The site is expected to yield about 620 residential units. More detail The Laurels is a freehold condominium growth at Cairnhill Street, Singapore (District 9). Developers could conduct property launches in Singapore by an in-house advertising staff, or more doubtless (to minimise staffing), by appointing exterior commercial real estate for sale estate agencies to do the advertising. The Singapore real estate market has simply undergone its seventh cooling measure in four years. New Apartment Q Bay Residences @ Tampines
The Master Plan revealed that there might be more plots of land used for dwelling ownership and property launch in anticipation of the increase in inhabitants. Many analyst has revealed that they can discover an increase within the number of plots of land set aside for home ownership. I really feel that these cooling measures actually help to preserve the property prices and be certain that they will not crash dramatically by 40% prefer it did in late nineteen nineties within the aftermath of the Asian monetary disaster. Property investment becomes much more attractive if the value is stable and will increase regularly over the long term. Components that determine Singapore's property market Property tax charges for non-residential properties stay unchanged at a flat 10 per cent. Singapore District