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The verifier must also have complete access to the contents of a function or predicate in order to unroll them as necessary. This has implications when used in conjunction with access modifiers . Specifically, hiding the contents of a function using the protected modifier can limit what properties can be established about it.
For example, AVERAGE=SUM/COUNT and RANGE=MAX−MIN. In the MapReduce framework, these steps are known as InitialReduce (value on individual record/singleton set), Combine (binary merge on two aggregations), and FinalReduce (final function on auxiliary values), [ 5 ] and moving decomposable aggregation before the Shuffle phase is known as an ...
Example: Let 픽 be a finite field and take A = 픽. Then since 픽 is closed under addition and multiplication, A + A = AA = 픽, and so | A + A | = | AA | = | 픽 |. This pathological example extends to taking A to be any sub-field of the field in question. Qualitatively, the sum-product problem has been solved over finite fields:
The first Project Euler problem is Multiples of 3 and 5. If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000. It is a 5% rated problem, indicating it is one of the easiest on the site.
Max-sum MSSP is a special case of MKP in which the value of each item equals its weight. The knapsack problem is a special case of MKP in which m=1. The subset-sum problem is a special case of MKP in which both the value of each item equals its weight, and m=1. The MKP has a Polynomial-time approximation scheme. [6]
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Given a function that accepts an array, a range query (,) on an array = [,..,] takes two indices and and returns the result of when applied to the subarray [, …,].For example, for a function that returns the sum of all values in an array, the range query (,) returns the sum of all values in the range [,].
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