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If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array. If the array contains all non-positive numbers, then a solution is any subarray of size 1 containing the maximal value of the array (or the empty subarray, if it is permitted).
The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., = (+ +). This special case of SSP is known as the partition problem . SSP can also be regarded as an optimization problem : find a subset whose sum is at most T , and subject to that, as close as possible to T .
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
All the values of x begin at the 15 th decimal, so Excel must take them into account. Before calculating the sum 1 + x , Excel first approximates x as a binary number. If this binary version of x is a simple power of 2, the 15 digit decimal approximation to x is stored in the sum, and the top two examples of the figure indicate recovery of x ...
The set of all size-k subsets contains: (1) all size-k subsets that do contain the distinguished element, and (2) all size-k subsets that do not contain the distinguished element. If a size- k subset of a size-( n + 1) set does contain the distinguished element, then its other k − 1 elements are chosen from among the other n elements of our ...
The best case for the algorithm now occurs when all elements are equal (or are chosen from a small set of k ≪ n elements). In the case of all equal elements, the modified quicksort will perform only two recursive calls on empty subarrays and thus finish in linear time (assuming the partition subroutine takes no longer than linear time).
The following decision problem related to maximum cuts has been studied widely in theoretical computer science: Given a graph G and an integer k, determine whether there is a cut of size at least k in G. This problem is known to be NP-complete. It is easy to see that the problem is in NP: a yes answer is easy to prove by presenting a large ...
MAX-SAT, and the corresponded weighted version Weighted MAX-SAT; MAX-kSAT, where each clause has exactly k variables: MAX-2SAT; MAX-3SAT; MAXEkSAT; The partial maximum satisfiability problem (PMAX-SAT) asks for the maximum number of clauses which can be satisfied by any assignment of a given subset of clauses. The rest of the clauses must be ...