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Mathematical binary relation
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing
Subsequence
Algorithmic problem on pairs of sequences
A longest common subsequence (LCS) is the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from
Longest_common_subsequence
Computer science problem
science, the longest increasing subsequence problem aims to find a subsequence of a given sequence in which the subsequence's elements are sorted in an ascending
Longest increasing subsequence
Longest_increasing_subsequence
Sorting algorithm
the algorithm efficiently computes the length of a longest increasing subsequence in a given array. The algorithm's name derives from a simplified variant
Patience_sorting
Combinatorial problem
and computer science, in the longest alternating subsequence problem, one wants to find a subsequence of a given sequence in which the elements are in
Longest alternating subsequence
Longest_alternating_subsequence
Sufficiently long sequences of numbers have long monotonic subsequences
(r-1)(s-1)+1} contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The proof appeared in the same
Erdős–Szekeres_theorem
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. An equivalent formulation is that a subset of R n {\displaystyle \mathbb
Bolzano–Weierstrass_theorem
Probabilistic data structure
made possible by maintaining a linked hierarchy of subsequences, with each successive subsequence skipping over fewer elements than the previous one (see
Skip_list
On when a family of real, continuous functions has a uniformly convergent subsequence
defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions
Arzelà–Ascoli_theorem
Theorem
analysis about the Cesàro convergence of a subsequence of random variables (or functions) and their subsequences to an integrable random variable (or function)
Komlós'_theorem
Generalization of the concept of subsequence to the case of nets
subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition
Subnet_(mathematics)
Topological space where every sequence has a convergent subsequence
if every sequence of points in X {\displaystyle X} has a convergent subsequence converging to a point in X {\displaystyle X} . Every metric space is
Sequentially_compact_space
Contiguous part of a sequence of symbols
substring of "It was the best of times". In contrast, "Itwastimes" is a subsequence of "It was the best of times", but not a substring. Prefixes and suffixes
Substring
Metric used for testing NLP models
reference summaries. ROUGE-L: Longest Common Subsequence (LCS) based statistics. Longest common subsequence problem takes into account sentence-level structure
ROUGE_(metric)
known as Hunt–McIlroy algorithm, is a solution to the longest common subsequence problem. It was one of the first non-heuristic algorithms used in diff
Hunt–Szymanski_algorithm
theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are
Minimal prime (recreational mathematics)
Minimal_prime_(recreational_mathematics)
Data mining technique
repeats, finding tandem repeats, and finding unique subsequences and missing (un-spelled) subsequences. Alignment problems: that deal with comparison between
Sequential_pattern_mining
Type of comparison sorting algorithm
into a subsequence of S {\displaystyle S} of length at most three. First, y 4 {\displaystyle y_{4}} is inserted into the three-element subsequence ( x 1
Merge-insertion_sort
Mathematical concept for comparing objects
starting point of an infinite increasing subsequence. The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering
Well-quasi-ordering
Sequence that contains itself as a subsequence
mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3,
Fractal_sequence
On convergent subsequences of functions that are locally of bounded total variation
uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space
Helly's_selection_theorem
Prime numbers that occupy prime-numbered positions
known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence
Super-prime
Type of mathematical space
set is compact if and only if every infinite sequence in the set has a subsequence that converges to a point of the set. Likewise, whereas every real-valued
Compact_space
Limit of some subsequence
mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In
Subsequential_limit
Subsequence of video frames
Subsequence of video frames
Group_of_pictures
Computer science problem
data deduplication and plagiarism detection. Unlike the longest common subsequence problem, which finds insertions or deletions within the common text,
Longest_common_substring
Computer science metric of string similarity
distance are obtained by restricting the set of operations. Longest common subsequence (LCS) distance is edit distance with insertion and deletion as the only
Edit_distance
Subpermutation of a longer permutation
to the number pi), then π is said to contain σ as a pattern if some subsequence of the entries of π has the same relative order as all of the entries
Permutation_pattern
that every sequence of elements of that length contains a non-empty subsequence adding up to 0. In symbols, this is D ( G ) = min { N : ∀ ( { g n } n
Davenport_constant
Online database of integer sequences
representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Sequence of data points over time
cluster) subsequence time series clustering (single timeseries, split into chunks using sliding windows) time point clustering Subsequence time series
Time_series
shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subsequence problem. Given two sequences X = <
Shortest_common_supersequence
theorem is a result from probabilistic combinatorics. It deals with the subsequences of a randomly uniformly drawn permutation from the set { 1 , 2 , … ,
Baik–Deift–Johansson_theorem
Mathematical formula for the number of Young tableaux
and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula gives the number of semi-standard Young tableaux,
Hook_length_formula
finite alphabet Σ {\displaystyle \Sigma } , as partially ordered by the subsequence relation, is a well partial order. That is, if w 1 , w 2 , … ∈ Σ ∗ {\displaystyle
Higman's_lemma
Hybrid sorting algorithm based on insertion sort and merge sort
2002 for use in the Python programming language. The algorithm finds subsequences of the data that are already ordered (runs) and uses them to sort the
Timsort
Conjecture in number theory
for every initial subsequence of 2 and odd numbers, and every non-constant growth rate, there is a continuation of the subsequence by odd numbers whose
Gilbreath's_conjecture
number of increasing subsequences of the sequence. This is a measure of presortedness, and in particular measures how many subsequences must be merged to
Run_of_a_sequence
Compact embedding theorem concerning Sobolev spaces
theorem implies that any uniformly bounded sequence in W1,p(Ω; R) has a subsequence that converges in Lq(Ω; R). Stated in this form, in the past the result
Rellich–Kondrachov_theorem
Family of graphs based on the Fibonacci sequence
contiguous subsequences. Within these two subsequences, the path can be constructed recursively by the same rule, linking the two subsequences at the ends
Fibonacci_cube
Line-breaking algorithm used in the TeX typesetting package
optimum can be shown to be a special case of the convex least-weight subsequence problem, which can be solved in O ( n ) {\displaystyle O(n)} time. Methods
Knuth–Plass line-breaking algorithm
Knuth–Plass_line-breaking_algorithm
Mathematics concept
are mathematical constants that describe the lengths of longest common subsequences of random strings. Although the existence of these constants has been
Chvátal–Sankoff_constants
Iterative Viterbi decoding is an algorithm that spots the subsequence S of an observation O = {o1, ..., on} having the highest average probability (i
Iterative_Viterbi_decoding
algorithm for finding all non-overlapping, contiguous, maximal scoring subsequences in a sequence of real numbers. The Ruzzo–Tompa algorithm was proposed
Ruzzo–Tompa_algorithm
Probability distribution
appears in the distribution of the length of the longest increasing subsequence of random permutations, as large-scale statistics in the Kardar-Parisi-Zhang
Tracy–Widom_distribution
Algorithm for solving the partition problem
is exactly 2 − 1 k {\displaystyle 2-{\frac {1}{k}}} . In the min-max subsequence problem, the input is a multiset of n numbers and an integer parameter
Largest_differencing_method
Binary tree derived from a sequence of numbers
sequence, and recursively construct its left and right subtrees from the subsequences before and after this number. It is uniquely defined as a min-heap whose
Cartesian_tree
Space-filling curve
{\displaystyle c} is replaced by a contiguous subsequence of the centers of these nine smaller squares. This subsequence is formed by grouping the nine smaller
Peano_curve
Guidance and navigation computer used in Apollo spacecraft
subsequence. Simple instructions, such as TC, executed in a single subsequence of 12 pulses. More complex instructions required several subsequences.
Apollo_Guidance_Computer
Type of continuous linear operator
usually not compact, and bounded sequences need not have convergent subsequences. Compact operators partly restore this finite-dimensional behavior by
Compact_operator
Parallel sorting algorithm
of the (green) subsequence with the element of the other (orange) subsequence at the respective index produces two bitonic subsequences. These two bitonic
Bitonic_sorter
Shell command for comparing file content
z From a longest common subsequence, it is only a small step to get diff-like output: if an item is absent in the subsequence but present in the first
Diff
Self-balancing binary search tree data structure
Since the length of the subsequences in S is ∈ O ( | I | ) {\displaystyle \in O(|I|)} and in every stage the subsequences are being cut in half, the
Red–black_tree
Sequence with limited alternation of symbols
values occurring in the sequence, then the sequence does not contain a subsequence ... x, ... y, ..., x, ..., y, ... consisting of s + 2 values alternating
Davenport–Schinzel_sequence
applications to percolations and longest increasing subsequence. To study the longest increasing subsequence of a random permutation π {\displaystyle \pi }
Kingman's subadditive ergodic theorem
Kingman's_subadditive_ergodic_theorem
Topics referred to by the same term
provides location information Longest increasing subsequence, algorithm to find the longest increasing subsequence in an array of numbers Laser Isotope Separation
LIS
Algorithm that generates an approximation of a random number sequence
next bit a one (or zero) with probability one-half; and any selected subsequence contains no information about the next element(s) in the sequence. K3
Pseudorandom_number_generator
Theorem in measure theory
{\displaystyle m} -dimensional Euclidean space), then there exist a subsequence ( μ n k ) {\displaystyle (\mu _{n_{k}})} and a probability measure μ
Prokhorov's_theorem
Relates three different kinds of weak compactness in a Banach space
following statements are equivalent: each sequence of elements of A has a subsequence that is weakly convergent in X each sequence of elements of A has a weak
Eberlein–Šmulian_theorem
Sequences of convex sets in a bounded set have convergent subsequences
contained in a bounded set, the theorem guarantees the existence of a subsequence { K n m } {\displaystyle \{K_{n_{m}}\}} and a convex set K {\displaystyle
Blaschke_selection_theorem
Resampling method
a subsequence, and there are M such subsequences (phases) multiplexed together. The dot product is the sum of the dot products of each subsequence with
Downsampling (signal processing)
Downsampling_(signal_processing)
On distances between points on a circle
1 {\displaystyle n+1} distinct consecutive subsequences of length n {\displaystyle n} . Each subsequence occurs infinitely often with a certain frequency
Three-gap_theorem
Sequence in which every finite string appears as a subsequence
Sequence in which every finite string appears as a subsequence
Disjunctive_sequence
Topics referred to by the same term
for factoring an integer into its prime factors Factor, a substring, a subsequence of consecutive symbols in a string Authentication factor, a piece of
Factor
similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim, and, soon
Delta-convergence
Series of clashes in Sagaing Region, Myanmar in (2021-present)
The Kalay clashes are a series of clashes between the Tatmadaw and armed civilians in the town of Kalay and surrounding villages in Kale Township during
Kalay_clashes
Finite or infinite ordered list of elements
above and bounded from below, then the sequence is said to be bounded. A subsequence of a given sequence is a sequence formed from the given sequence by deleting
Sequence
String distance measure
and the transposition of two adjacent characters; the longest common subsequence (LCS) distance allows only insertion and deletion, not substitution;
Jaro–Winkler_distance
Property of certain normed spaces
spaces stating that every bounded sequence of points in the space has a subsequence that is convergent in the mean (also known as Cesàro summation or limesable)
Banach-Saks_property
Notion of convergence in mathematics
structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the same subsequential limit, the sequence itself
Pointwise_convergence
Measure on group representations
probabilistic problems, especially in the study of longest increasing subsequence of a random permutation σ {\displaystyle \sigma } . As a result of its
Plancherel_measure
Monoid of all words in the alphabet of positive integers modulo Knuth equivalence
given by Craige Schensted (1961) in his study of the longest increasing subsequence of a permutation. It was named the "monoïde plaxique" by Lascoux & Schützenberger
Plactic_monoid
Probability measure in thermodynamics
randomness) subsequence of finite-volume Gibbs distributions. It was proved for Euclidean lattices that there always exists a deterministic subsequence along
Metastate
pattern is order-isomorphic to the subsequence. For instance, if π is the permutation 25314, then it has ten subsequences of length three, forming the following
Superpattern
Mathematical term in complex analysis
called a normal family if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets of X to a continuous function
Normal_family
Mathematical property of subsets in order theory
and nets, where “cofinal subnet” is the appropriate generalization of "subsequence". They are also important in order theory, including the theory of cardinal
Cofinal_(mathematics)
Property of a sequence or series
and Cauchy convergence, together with the existence of a convergent subsequence implies convergence. The concept of completeness of metric spaces, and
Modes_of_convergence
Type of mathematical sequence
sequence with the property that for all values of N {\displaystyle N} , its subsequence x 1 , … , x N {\displaystyle x_{1},\ldots ,x_{N}} has a low discrepancy
Low-discrepancy_sequence
Product of any collection of compact topological spaces is compact
characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence. They also include statements less obviously related
Tychonoff's_theorem
Given a cover of a compact metric space, all small subsets are subset of some cover set
. Since X {\displaystyle X} is sequentially compact, there exists a subsequence { x n k } {\displaystyle \{x_{n_{k}}\}} (with k ∈ Z > 0 {\displaystyle
Lebesgue's_number_lemma
Sequence generating game between two players
until either player A's or player B's sequence appears as a consecutive subsequence of the coin toss outcomes. The player whose sequence appears first wins
Penney's_game
Natural number
of six elements, exactly 238 of them have a unique longest increasing subsequence. There are 238 compact and paracompact hyperbolic groups of ranks 3 through
238_(number)
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Cluster point in a topological space
x_{\bullet }} if and only if x {\displaystyle x} is a limit of some subsequence of x ∙ . {\displaystyle x_{\bullet }.} The set of all cluster points
Accumulation_point
On chains and antichains in partial orders
width of this partial order is n. The Erdős–Szekeres theorem on monotone subsequences can be interpreted as an application of Dilworth's theorem to partial
Dilworth's_theorem
embedded in Y: i.e. X ⊆ Y and every ||⋅||X-bounded sequence in X has a subsequence that is ||⋅||Y-convergent; and Y is continuously embedded in Z: i.e.
Ehrling's_lemma
Number, sum of distinct powers of 4
in their binary representations, the Moser–de Bruijn sequence forms a subsequence of the fibbinary numbers. It follows from either the binary or base-4
Moser–de_Bruijn_sequence
Russian mathematician (1933–2024)
infinite symmetric groups and applications to the longest increasing subsequences. Vershik studied at Leningrad State University (later renamed to Saint
Anatoly_Vershik
Intersection graph for intervals on the real number line
non-overlapping times. Other applications include assembling contiguous subsequences in DNA mapping, and temporal reasoning. An interval graph is an undirected
Interval_graph
Set of natural numbers
of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni). A set A of natural numbers is an IP set if there exists an infinite
IP_set
Data structure
Parsing Pattern matching Compressed pattern matching Longest common subsequence Longest common substring Sequential pattern mining Sorting String rewriting
Ternary_search_tree
Product of numbers from 1 to n
i} numbers by splitting it into two subsequences of i / 2 {\displaystyle i/2} numbers, multiplies each subsequence, and combines the results with one last
Factorial
Computer science metric for string similarity
characters alongside insertion, deletion, substitution; the longest common subsequence (LCS) distance allows only insertion and deletion, not substitution;
Levenshtein_distance
Elementary cellular automaton
contiguous subsequence of values in one row of the triangle are all 0 or 2, then Rule 90 can be used to determine the corresponding subsequence in the next
Rule_90
Mathematics lemma in functional analysis
x 2 , … {\displaystyle x_{1},x_{2},\ldots } contains no convergent subsequence, which implies that the closed unit ball is not compact. Thus, if X {\displaystyle
Riesz's_lemma
Theorem in symplectic topology
in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or (a
Gromov's compactness theorem (topology)
Gromov's_compactness_theorem_(topology)
Continuous real function on a closed interval has a maximum and a minimum
that there exists a subsequence that converges to a point in the domain. Use continuity to show that the image of the subsequence converges to the supremum
Extreme_value_theorem
eventually, or equivalently, that the property is satisfied by one of its subsequences ( a n ) n ≥ N {\displaystyle (a_{n})_{n\geq N}} , for some N ∈ N {\displaystyle
Eventually_(mathematics)
Binary sequence
and thus fail to pick out an infinite subsequence. We only consider those that do pick an infinite subsequence. Stated in another way, each infinite binary
Algorithmically random sequence
Algorithmically_random_sequence
Contiguous sequence of errors occurring in a communications channel
the first and last symbols are in error and there exists no contiguous subsequence of m correctly received symbols within the error burst. The integer parameter
Burst_error
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