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The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset of integers and a target-sum , and the question is to decide whether any subset of the integers sum to precisely . [1] The problem is known to be NP-complete.
The goal is to construct m triplets, each of which contains one element from A, one from B and one from C, such that the sum of each triplet is T. [ 2 ] The 4-partition problem is a variant in which S contains n = 4 m integers, the sum of all integers is m T {\displaystyle mT} , and the goal is to partition it into m quadruplets, all ...
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.
In computational complexity theory, the 3SUM problem asks if a given set of real numbers contains three elements that sum to zero. A generalized version, k-SUM, asks the same question on k elements, rather than simply 3. 3SUM can be easily solved in () time, and matching (⌈ / ⌉) lower bounds are known in some specialized models of computation (Erickson 1999).
The loss function is defined using triplets of training points of the form (,,).In each triplet, (called an "anchor point") denotes a reference point of a particular identity, (called a "positive point") denotes another point of the same identity in point , and (called a "negative point") denotes an point of an identity different from the identity in point and .
Prefix sums are trivial to compute in sequential models of computation, by using the formula y i = y i − 1 + x i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.
A slightly different generalization allows the sum of (k + 1) n th powers to equal the sum of (n − k) n th powers. For example: For example: ( n = 3 ): 1 3 + 12 3 = 9 3 + 10 3 , made famous by Hardy's recollection of a conversation with Ramanujan about the number 1729 being the smallest number that can be expressed as a sum of two cubes in ...
If any of the above matrices, say A, is applied to a triple (a, b, c) T having the Pythagorean property a 2 + b 2 = c 2 to obtain a new triple (d, e, f) T = A(a, b, c) T, this new triple is also Pythagorean.