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2-opt. In optimization, 2-opt is a simple local search algorithm for solving the traveling salesman problem. The 2-opt algorithm was first proposed by Croes in 1958, [ 1 ] although the basic move had already been suggested by Flood. [ 2 ] The main idea behind it is to take a route that crosses over itself and reorder it so that it does not.
Combination. In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple ...
Stars and bars (combinatorics) In the context of combinatorial mathematics, stars and bars (also called "sticks and stones", [ 1 ] "balls and bars", [ 2 ] and "dots and dividers" [ 3 ]) is a graphical aid for deriving certain combinatorial theorems. It can be used to solve many simple counting problems, such as how many ways there are to put n ...
Inclusion–exclusion principle. In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as. where A and B are two finite sets and | S | indicates the cardinality of a ...
The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. It is a special case of the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem, a general case of ...
A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form. where. is a function, where X is a set to which the elements of a sequence must belong.
Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula. which using factorial notation can be compactly expressed as.
Gale–Shapley algorithm. In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm, [1] propose-and-reject algorithm, [2] or Boston Pool algorithm[1]) is an algorithm for finding a solution to the stable matching problem. It is named for David Gale and Lloyd Shapley, who ...