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Consider a path P consisting of n nodes rooted at a node r. We can store the path into an array of size n called Ladder and we can quickly answer a level ancestor query of LA(v, d) by returning Ladder[d] if depth(v)≤d. This will take O(1). However, this will only work if the given tree is a path. Otherwise, we need to decompose it into paths.
The k shortest path routing problem is a generalization of the shortest path routing problem in a given network. It asks not only about a shortest path but also about next k−1 shortest paths (which may be longer than the shortest path). A variation of the problem is the loopless k shortest paths.
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] In weighted complete graphs with non-negative edge weights, the weighted longest path problem is the same as the Travelling salesman path problem, because the longest path always includes all ...
Problem 2. Find the path of minimum total length between two given nodes P and Q. We use the fact that, if R is a node on the minimal path from P to Q, knowledge of the latter implies the knowledge of the minimal path from P to R. is a paraphrasing of Bellman's Principle of Optimality in the context of the shortest path problem.
Maximum lengths of snakes (L s) and coils (L c) in the snakes-in-the-box problem for dimensions n from 1 to 4. The problem of finding the longest path or cycle that is an induced subgraph of a given hypercube graph is known as the snake-in-the-box problem. Szymanski's conjecture concerns the suitability of a hypercube as a network topology for ...
Path coloring Models a routing problem in graphs Radio coloring Sum of the distance between the vertices and the difference of their colors is greater than k + 1, where k is a positive integer. Rank coloring If two vertices have the same color i, then every path between them contain a vertex with color greater than i Subcoloring
Branch and bound (BB, B&B, or BnB) is a method for solving optimization problems by breaking them down into smaller sub-problems and using a bounding function to eliminate sub-problems that cannot contain the optimal solution. It is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical ...
Every three-digit sequence occurs exactly once if one visits every vertex exactly once (a Hamiltonian path). The de Bruijn sequences can be constructed by taking a Hamiltonian path of an n -dimensional de Bruijn graph over k symbols (or equivalently, an Eulerian cycle of an ( n − 1)-dimensional de Bruijn graph).