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If the longest path problem could be solved in polynomial time, it could be used to solve this decision problem, by finding a longest path and then comparing its length to the number k. Therefore, the longest path problem is NP-hard. The question "does there exist a simple path in a given graph with at least k edges" is NP-complete. [2]
For dimensions two through seven the lengths of the longest possible doubled coils are 4, 6, 8, 14, 26, 46. Beyond that, the best lengths found so far for dimensions eight through thirteen are 94, 186, 362, 662, 1222, 2354. For both the snake and the coil in the box problems, it is known that the maximum length is proportional to 2 n for an n ...
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.
In the first stage of the algorithm, each path is associated with an array of size h' . We extend this path by adding the h' immediate ancestors at the top of the path in the same array. This will extend each array twice its original size at most which will result in 2n total number of nodes in all the ladders. Notice that the number of ladders ...
Seidel's algorithm is an algorithm designed by Raimund Seidel in 1992 for the all-pairs-shortest-path problem for undirected, unweighted, connected graphs. [1] It solves the problem in () expected time for a graph with vertices, where < is the exponent in the complexity () of matrix multiplication.
This enables the simple 2-approximation algorithm for TSP with triangle inequality above to operate more quickly. In general, for any c > 0, where d is the number of dimensions in the Euclidean space, there is a polynomial-time algorithm that finds a tour of length at most (1 + 1/c) times the optimal for geometric instances of TSP in
A Fenwick tree or binary indexed tree (BIT) is a data structure that stores an array of values and can efficiently compute prefix sums of the values and update the values. It also supports an efficient rank-search operation for finding the longest prefix whose sum is no more than a specified value.
An adjacency list representation for a graph associates each vertex in the graph with the collection of its neighbouring vertices or edges. There are many variations of this basic idea, differing in the details of how they implement the association between vertices and collections, in how they implement the collections, in whether they include both vertices and edges or only vertices as first ...