Bottleneck traveling salesman problem - Bottleneck traveling salesman problem The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in discrete or combinatorial optimization. It is stated as follows: Find the Hamiltonian cycle in a weighted graph with the minimal length of the longest edge. The problem is known to be NP-hard. Related topics Travelling salesman problem.
Traveling salesman problem - Traveling salesman problem The traveling salesman problem (TSP), also known as the traveling salesperson problem, is a problem in discrete or combinatorial optimization. It is a prominent illustration of a class of problems in computational complexity theory. The problem can be stated as: Given a number of cities and the costs of travelling from one to the other, what is the cheapest roundtrip route that visits each city and then returns to the starting city? An equivalent formulation in terms of graph theory is: Find the shortest Hamiltonian cycle in a weighted graph. A related problem is the Bottleneck traveling salesman problem (bottleneck TSP): Find the Hamiltonian cycle in a weighted graph with the minimal length of the longest edge. Table of contents showTocToggle("show","hide") 1 Computational.
Shortest path problem - Shortest path problem In graph theory, the shortest path problem is the following: Given a weighted graph, (that is a set N of nodes, a set E of edges and a real-valued function f : E -> R), and given further two elements n, n' of N, find a path P from n to n', so that is minimal among all paths connecting n to n'. A solution to the shortest path problem is sometimes called a "pathing algorithm". The most important algorithm for solving this problem in case all edge weights are greater than or equal to zero is Dijkstra's algorithm. Without worsening the run time, this algorithm can in fact compute the shortest paths from a given start point s to all other nodes. Another algorithm.
Graph theory - article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to handle graphs is therefore of major interest in computer science. Table of contents showTocToggle("show","hide") 1 History 2 Graph problems 3 Important algorithms 4 Generalizations 5 Related areas of mathematics 6 See also History Leonhard Euler's paper on Seven Bridges of Königsberg is considered to be the first result in graph theory. It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. This illustrates the deep connection between graph theory and topology. Graph problems Graph coloring: the four color theorem Route problems: Seven bridges of Königsberg Minimum spanning tree Shortest path problem Route.
Greedy algorithm - Greedy algorithms are algorithms which follow the problem solving meta-heuristic of making the locally optimum choice at each stage with the hope of finding the global optimum. For instance, applying the greedy strategy to the traveling salesman problem yields the following algorithm: "At each stage visit the nearest unvisited city to the current city". Greedy algorithms rarely find the globally optimal solution consistently, since they usually don't operate exhaustively on all the data. Nevertheless they are useful because they are quick to think up and often give good approximations to the optimum. If a greedy algorithm can be proven to yield the global optimum for a given problem class, it typically becomes the method of choice. Examples of such greedy algorithms are Kruskal's algorithm and Prim's algorithm. The theory of matroids.
Dijkstra's algorithm - scientist Edsger Dijkstra, solves a shortest path problem for a directed and connected graph G(V,E) which has nonnegative (>=0) edge weights. The set V is the set of all vertices in the graph G. The set E is the set of ordered pairs which represent connected vertexes in the graph (if (u,v) belongs to E then there is a connection from vertex u to vertex v). Assume that the function w: V x V -> [0, ∞] describes the cost w(x,y) of moving from vertex x to vertex y (non-negative cost). (We can define the cost to be infinite for pairs of vertices that are not connected by an edge.) The cost of a path between two vertices is the sum of costs of the edges in that path. The cost.
Ant colony algorithm - path from the colony to a food source, other ants are more likely to follow that path; positive feedback eventually leaves all the ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve. Ant colony algorithms have been used to produce good near-solutions to the traveling salesman problem. They have an advantage over simulated annealing and genetic algorithm approaches when the graph may change dynamically; the ant colony algorithm can be run continuously and adapt to changes in real time. This is of interest in network routing. External Links An Introduction to Ant Colony Algorithms.
Brute-force search - of systematically enumerating every possible solution of a problem until a solution is found, or all possible solutions have been exhausted. For example, an anagram problem can be solved by enumerating all possible combinations of words with the same number of letters as the desired phrase, and checking one by one whether the words make a valid anagram. Table of contents showTocToggle("show","hide") 1 Combinatorial explosion 2 Speeding up brute-force searches 3 Brute force attacks in cryptography Combinatorial explosion Many real-world problems have exponential time or space complexity, so that only very small problems are amenable to this approach. For example, searching the space of six-digit numbers takes only a million iterations: assuming a thousand instructions are needed to check the solution, a single modern computer can solve the problem in a.
Combinatorial optimization - be somewhat different. The domain of combinatorial optimization is optimization problems where the set of feasible solutionss is discrete, and the goal is to find the best possible solution. Examples are Traveling salesman problem, minimum spanning tree problem, linear programming problem. References William J. Cook, William H. Cunningham, William R. Pulleyblank, Alexander Schrijver; Combinatorial Optimization; John Wiley & Sons; 1 edition (November 12, 1997); ISBN: 047155894X. Christos H. Papadimitriou, and Kenneth Steiglitz; Combinatorial Optimization : Algorithms and Complexity; Dover Pubns; (paperback, Unabridged edition, July 1998) ISBN: 0486402584. Journals Journal of Combinatorial Optimization.
Sharp-P - theory. It is the set of counting problems associated with the decision problems in the set NP. An NP problem is often of the form: Are there any solutions that satisfy certain constraints? For example: Are there any subsets of a list of integers that add up to zero? (subset sum problem) Are there any Hamiltonian cycles in a given graph with cost less than 100? (traveling salesman problem) Are there any variable assignments that satisfy a given DNF formula? (Boolean satisfiability problem) The corresponding #P problems ask "how many" rather than "are there any". For example: How many subsets of a list of integers add up to zero? How many Hamiltonian cycles in a given graph have cost less than 100? How many variable assignments satisfy a given DNF formula?.
Simulated annealing - energy associated with it, which indicates how good it is. The goal is to find the point with the minimum energy. The algorithm starts off at an arbitrary point; at each step chooses some neighbor of the current point and moves to that point with a certain probability. Neighbors are points that are "close" to each other in a problem-dependent fashion (For example, in the traveling salesman problem, we may define two tours to be neighbors if and only if one can be converted to the other by interchanging a pair of adjacent cities). The probability of transition is a function of the energy difference between the two points and a global time-dependent parameter called the temperature. Let δE be the difference in energy, and T the temperature. If δE is.
Nearest neighbour algorithm - was one of the first algorithms used to determine a solution to the traveling salesman problem, and usually comes to within twenty percent of the optimal route. It is also a lot faster than testing every route and some other algorithms. These are the steps of the algorithm: Make two sets of nodes, set A and set B, and put all nodes into set B Put your starting node into set A Pick the node which is closest to the last node which was placed in set A and is not in set A; put this closest neighbouring node into set A Repeat step 3 until all nodes are in set A and B is empty. The nearest neighbour algorithm is easy to implement and can be done quickly, but it.
NP-complete - ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use that algorithm to solve all NP problems quickly. (A more formal definition is given below. See also Complexity classes P and NP). One example of an NP-complete problem is the subset sum problem which is: given a finite set of integers, determine whether any non-empty subset of them adds up to zero. A supposed answer is very easy to verify for correctness, but no-one knows a faster way to solve the problem than to try every single possible subset, which is very slow. At present, all known algorithms for NP-complete problems require time which is exponential in the problem size. It is unknown.
Monte Carlo method - A refinement of this method is to somehow make the points random, but more likely to come from regions of high contribution to the integral than from regions of low contribution. In other words, the points should be drawn from a distribution similar in form to the integrand. Unfortunately, doing this precisely is just as hard as solving the integral in the first place, but there are approximate methods available: from simply making up an integrable function thought to be similar, to one of the adaptive routines discussed in the topics listed below. Integration methods Direct sampling methods Importance sampling Stratified sampling Recursive stratified sampling VEGAS algorithm Random walk Monte Carlo including Markov chains Metropolis-Hastings Markov Chain Monte Carlo Sampling Optimisation Another powerful and very popular application for random numbers in.
Leonard Adleman - Bachelor's degree in 1968 and his Ph.D in 1976. In 1994, his paper Molecular Computation of Solutions To Combinatorial Problems described the experimental use of DNA as a computational system. In it, he solved a seven-node instance of the Hamiltonian Graph problem, an NP-Complete problem similar to the traveling salesman problem. While the solution to a seven-node instance is trivial, this paper is the first known instance of the successful use of DNA to compute an algorithm. DNA computing has been shown to have potential as a means to solve several other large-scale combinatorial search problems. For his contribution to the invention of the RSA cryptosystem, Adleman was a recipient along with Ron Rivest and Adi Shamir of the 2002 ACM Turing Award, often called the Nobel Prize of Computer Science..
List of mathematical topics (S-U) - Separated sets -- Separation axiom -- Separation of variables -- Sequence -- Series -- Serre -- Serre, Jean-Pierre -- Set -- Set-builder notation -- Set-theoretic intersection -- see: Intersection (set theory) -- Set-theoretic union -- see: Union (set theory) -- Set theory -- Seven bridges of Königsberg -- Sexagesimal -- Sextant -- Sexy prime -- SHA-1 -- Shanks, Daniel -- Shannon, Claude E -- Shannon capacity -- Shannon-Fano coding -- Shannon-Hartley law -- Shannon switching game -- Shannon's theorem -- Shape -- Shapley value -- Sharp-P -- Sharp-P-complete -- Sheaf -- Sheaf space -- Shear -- Sheffer sequence -- Sheffer stroke -- Shelah, Saharon -- Shell integration -- Shellsort -- Shijie -- Shijie, Zhu -- Shimura, Goro -- Shor's algorithm -- Short five lemma -- Shortest path problem -- Sierpinski carpet.
List of graph theory topics - Graph coloring Bipartite graphs Disperser Expander Extractor Four color theorem Tait's conjecture Ramsey's theorem Paths and cycles Seven bridges of Königsberg Shortest path problem Dijkstra's algorithm Open shortest path first Flooding algorithm Route inspection problem Hamiltonian cycle problem, Hamiltonian path Knight's Tour How to solve the knight's tour Traveling salesman problem Nearest neighbour algorithm Trees Tree (graph theory) Terminology Root node, root (computing) Leaf node Child node, parent node Ply Tree structure Tree data structure AVL tree B-tree Binary tree Binary search tree Heap Binary heap Binomial heap Fibonacci heap Red-black tree Splay tree Tree rotation Trie Examples Abstract syntax tree Decision tree Evolutionary tree Family tree Game tree Parse tree Phylogenetic tree Technology tree Spanning tree Minimum spanning tree Boruvka's algorithm Kruskal's algorithm Prim's algorithm Tree traversal Inorder traversal Backward.
List of computability and complexity topics - automaton Pushdown automaton Context-free grammar Chomsky hierarchy Context-sensitive language Recursively enumerable language Markov algorithm String rewriting system L-system Cellular automaton Conway's Game of Life Turing machine Turing-complete Turing tarpit Oracle machine Lambda calculus Combinatory logic Combinator Parallel computing Flynn's taxonomy Quantum computer Church-Turing thesis Recursive function Decision problems Entscheidungsproblem Halting problem Correctness Post correspondence problem Decidable language Undecidable language Word problem for groups Wang tile Definability questions Computable number Definable number Halting probability Algorithmic information theory Data compression Complexity theory Polynomial time Exponential time Time hierarchy theorem Polynomial-time many-one reduction Polynomial-time Turing reduction Complexity classes Complexity classes P and NP P-complete NP, NP-complete, NP-hard, NP-equivalent Co-NP, Co-NP-complete Sharp-P, Sharp-P-complete PSPACE, PSPACE-complete EXPSPACE, EXPTIME, EXPTIME-complete BPP BQP Class NC RP UP (complexity) ZPP Named problems Clique problem Hamiltonian cycle problem Integer factorization.
Local search - algorithmic method that is commonly used for solving computationally hard problem such as the traveling salesman problem (TSP). Given a (typically very large) space of candidate solutions of a given problem instance and a neighbourhood relation on this space, the basic principle underlying local search is to start from an initial candidate solution and then to iteratively moves from one candidate solution to one of the candidate solution from its direct neighbourhood, based on local information, until a termination condition is met. A typical example of a local search method is the 2-opt algorithm for the TSP. Local search algorithms are widely applied to numerous hard computational problems, including problems from computer science (particularly artificial intelligence), mathematics, operations research, engineering, and bioinformatics..
Bottleneck - Bottleneck A bottleneck is literally the neck of a glass or pottery bottle. An hourglass has a bottleneck at its mid-point whose diameter governs the time that granular contents of a given mass will take to pass through. Metaphorically a bottleneck is a section of a route with a carrying capacity substantially below that characterising other sections of the same route. This is often a narrow part of a road, perhaps also with a smaller number of lanes, or a reduction of the number of tracks of a railway line. It may be due to a narrow bridge or tunnel, a deep cutting or narrow embankment, or work in progress on part of the road or railway. More generally, a bottleneck is one process in a.
