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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 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-dimensional box, asymptotically as n grows large, and bounded above by 2 n − 1. However the constant of proportionality is not known, but is observed to be in the range 0.3 - 0.4 for current best known values.
Conversely, if H has an induced path or cycle of length k, any maximal set of nonadjacent vertices in G from this path or cycle forms an independent set in G of size at least k/3. Thus, the size of the maximum independent set in G is within a constant factor of the size of the longest induced path and the longest induced cycle in H.
Two primary problems of pathfinding are (1) to find a path between two nodes in a graph; and (2) the shortest path problem—to find the optimal shortest path. Basic algorithms such as breadth-first and depth-first search address the first problem by exhausting all possibilities; starting from the given node, they iterate over all potential ...
In computer programming and computer science, "maximal munch" or "longest match" is the principle that when creating some construct, as much of the available input as possible should be consumed. The earliest known use of this term is by R.G.G. Cattell in his PhD thesis [ 1 ] on automatic derivation of code generators for compilers .
In fact in order to answer a level ancestor query, the algorithm needs to jump from a path to another until it reaches the root and there could be Θ(√ n) of such paths on a leaf-to-root path. This leads us to an algorithm that can pre-process the tree in O( n ) time and answers queries in O( √ n ).
The NAG Library [1] can be accessed from a variety of languages and environments such as C/C++, [2] Fortran, [3] Python, [4] AD, [5] MATLAB, [6] Java [7] and .NET. [8] The main supported systems are currently Windows, Linux and macOS running on x86-64 architectures; 32-bit Windows support is being phased out. Some NAG mathematical optimization ...
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [5] [6] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [7] or they can be used to build a pointer based quadtree.