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In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
The algorithm can be made much more effective by first sorting the list of items into decreasing order (sometimes known as the first-fit decreasing algorithm), although this still does not guarantee an optimal solution and for longer lists may increase the running time of the algorithm. It is known, however, that there always exists at least ...
NC = P problem The P vs NP problem is a major unsolved question in computer science that asks whether every problem whose solution can be quickly verified by a computer (NP) can also be quickly solved by a computer (P). This question has profound implications for fields such as cryptography, algorithm design, and computational theory. [1]
Specific applications of search algorithms include: Problems in combinatorial optimization, such as: . The vehicle routing problem, a form of shortest path problem; The knapsack problem: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as ...
Motion planning, also path planning (also known as the navigation problem or the piano mover's problem) is a computational problem to find a sequence of valid configurations that moves the object from the source to destination.
Tarjan and Van Leeuwen also developed one-pass Find algorithms that retain the same worst-case complexity but are more efficient in practice. [4] These are called path splitting and path halving. Both of these update the parent pointers of nodes on the path between the query node and the root.
They are also called Krom formulas, after the work of UC Davis mathematician Melven R. Krom, whose 1967 paper was one of the earliest works on the 2-satisfiability problem. [ 2 ] Each clause in a 2-CNF formula is logically equivalent to an implication from one variable or negated variable to the other.
A 1999 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems related to the field of combinatorial algorithms and algorithm engineering, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem.