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For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.
The exceptional graph is a regular hexagon with one diagonal and a vertex at the center added; only 1 / 6 of its permutations can be attained, which gives an instance of the exotic embedding of S 5 into S 6. For larger versions of the n puzzle, finding a solution is easy. But, the problem of finding the shortest solution is NP-hard.
A[-1, *] % The last row of A A[[1:5], [2:7]] % 2d array using rows 1-5 and columns 2-7 A[[5:1:-1], [2:7]] % Same as above except the rows are reversed Array indices can also be arrays of integers. For example, suppose that I = [0:9] is an array of 10 integers.
Conversely, suppose there exists a solution S′′ to the Partition instance. Then, S′′ must contain either z 1 or z 2, but not both, since their sum is more than sum(S) + T. If S'' contains z 1, then it must contain elements from S with a sum of exactly T, so S'' minus z 1 is a solution to the SubsetSum
The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort , merge sort ), multiplying large numbers (e.g., the Karatsuba algorithm ), finding the closest pair of points , syntactic ...
In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times.
The earliest description of the Bubble sort algorithm was in a 1956 paper by mathematician and actuary Edward Harry Friend, [4] Sorting on electronic computer systems, [5] published in the third issue of the third volume of the Journal of the Association for Computing Machinery (ACM), as a "Sorting exchange algorithm".
In computer science, an in-place algorithm is an algorithm that operates directly on the input data structure without requiring extra space proportional to the input size. In other words, it modifies the input in place, without creating a separate copy of the data structure.