Top 10 similar words or synonyms for combinatorial
libraries 0.687787
library 0.631238
rational 0.620648
combinatory 0.578670
shuffling 0.562406
combinatorially 0.551942
silico 0.544807
screening 0.541470
biocombinatorial 0.537748
mutagenesis 0.533501
Top 30 analogous words or synonyms for combinatorial
Article | Example |
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Combinatorial search | A study of computational complexity theory helps to motivate combinatorial search. Combinatorial search algorithms are typically concerned with problems that are NP-hard. Such problems are not believed to be efficiently solvable in general. However, the various approximations of complexity theory suggest that some instances (e.g. "small" instances) of these problems could be efficiently solved. This is indeed the case, and such instances often have important practical ramifications. |
Combinatorial search | Common algorithms for solving combinatorial search problems include: |
Combinatorial search | More sophisticated search techniques such as alpha-beta pruning are able to eliminate entire subtrees of the search tree from consideration. When these techniques are used, lookahead is not a precisely defined quantity, but instead either the maximum depth searched or some type of average. |
Combinatorial principles | The rule of product is another intuitive principle stating that if there are "a" ways to do something and "b" ways to do another thing, then there are "a" · "b" ways to do both things. |
Combinatorial principles | Generating functions can be thought of as polynomials with infinitely many terms whose coefficients correspond to terms of a sequence. This new representation of the sequence opens up new methods for finding identities and closed forms pertaining to certain sequences. The (ordinary) generating function of a sequence "a" is |
Combinatorial proof | An archetypal double counting proof is for the well known formula for the number formula_1 of "k"-combinations (i.e., subsets of size "k") of an "n"-element set: |
Combinatorial proof | and after division by "k"! this leads to the stated formula for formula_1. In general, if the counting formula involves a division, a similar double counting argument (if it exists) gives the most straightforward combinatorial proof of the identity, but double counting arguments are not limited to situations where the formula is of this form. |
Combinatorial proof | gives an example of a combinatorial enumeration problem (counting the number of sequences of "k" subsets "S", "S", ... "S", that can be formed from a set of "n" items such that the subsets have an empty common intersection) with two different proofs for its solution. The first proof, which is not combinatorial, uses mathematical induction and generating functions to find that the number of sequences of this type is (2 −1). The second proof is based on the observation that there are 2 −1 proper subsets of the set {1, 2, ..., "k"}, and (2 −1) functions from the set {1, 2, ..., "n"} to the family of proper subsets of {1, 2, ..., "k"}. The sequences to be counted can be placed in one-to-one correspondence with these functions, where the function formed from a given sequence of subsets maps each element "i" to the set {"j" | "i" ∈ "S"}. |
Combinatorial proof | Stanley does not clearly distinguish between bijective and double counting proofs, and gives examples of both kinds, but the difference between the two types of combinatorial proof can be seen in an example provided by , of proofs for Cayley's formula stating that there are "n" different trees that can be formed from a given set of "n" nodes. Aigner and Ziegler list four proofs of this theorem, the first of which is bijective and the last of which is a double counting argument. They also mention but do not describe the details of a fifth bijective proof. |
Combinatorial auction | A combinatorial auction is a type of smart market in which participants can place bids on combinations of discrete items, or “packages”, rather than individual items or continuous quantities. |