The phrase "a sequence of duplicate" is not correct in standard English; it should be "a sequence of duplicates." You can use it when referring to a series of identical items or elements in a specific order.
Example: "In the dataset, there was a sequence of duplicates that needed to be removed to ensure data integrity."
Alternatives: "a series of duplicates" or "a chain of duplicates."
A duplication history is a sequence of duplicate events that first builds up a set of seed duplication blocks by duplicating and aggregating duplicons from their ancestral loci and then successively constructs the remaining duplication blocks by duplicating substrings of previously constructed blocks.
The proof of this claim is similar to that of Lemma 2. In that proof we showed how to transform a sequence of duplicate and delete operations into a sequence of duplicate-delete operations of at most the same cost.
Let OPT x, y) denote minimum length of a sequence of duplicate operations that generate y from x.
We show how a sequence of duplicate operations that generates a string can be described by a context-free grammar (CFG).
Consider a sequence of duplicate operations δ x (s1, t1, p1),..., δ x (s k, t k, p k ) that generates y from an initially empty target string.
We claim that every sequence of duplicate and delete operations can be transformed into a sequence of duplicate operations of the same length.
By duplicating y1 separately using x i we obtain a sequence of duplicates that generates y whose length is 1 + d x, y2,| y |).
Since duplicate operations are a special case of duplicate-delete operations, the cost of a minimal sequence of duplicate-delete operations and delete operations that generates y cannot be more than that of a sequence of just duplicate operations and delete operations.
A sequence of k duplicate operations that constructs Y from X uniquely defines a feasible generator Ψ X with length k whose elements correspond, respectively, to substrings of X that are duplicated conjointly in a single operation.
We show the (stronger) claim that an arbitrary sequence of duplicate-delete and delete operations that produces a string y with cost c can be transformed into a sequence of just duplicate-delete operations that generates y with cost at most c by induction on the number of delete operations.
Lemma 2. If the affine cost for duplications is non-decreasing and Φ is non-decreasing and obeys the triangle inequality then the cost of a minimum sequence of duplicate and delete operations that generates a target string y from a source string x is equal to the cost of a minimum sequence of duplicate-delete operations that generates y from x. Proof.
