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Questions tagged [subsequences]

Questions about algorithms related to subsequences, or about properties of subsequences.

A subsequence of another sequence S is a sequence whose elements belong to S and keep the same order as in S. A sequence is a subsequence of itself.

See also: substrings.

105 questions
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Longest Repeated (Scattered) Subsequence in a String

Informal Problem Statement: Given a string, e.g. $ACCABBAB$, we want to colour some letters red and some letters blue (and some not at all), such that reading only the red letters from left to right yields the same result as reading only the blue…
Sekti
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How to use a greedy algorithm to find the non-decreasing sequence closest to the given one?

You are given n integers $a_1, \ldots, a_n$ all between $0$ and $l$. Under each integer $a_i$ you should write an integer $b_i$ between $0$ and $l$ with the requirement that the $b_i$'s form a non-decreasing sequence. Define the deviation of such a…
Aden Dong
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Is there an algorithm which finds sorted subsequences of size three in $O(n)$ time?

I want to prove or disprove the existence of an algorithm which, given an array $A$ of integers, finds three indices $i, j$ and $k$ such that $i < j < k$ and $A[i] < A[j] < A[k]$ (or finds that there is no such triple) in linear time. This is not a…
Christopher Done
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12
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Finding the longest repeating subsequence

Given a string $s$, I would like to find the longest repeating (at least twice) subsequence. That is, I would like to find a string $w$ which is a subsequence (doesn't have to be a contiguous) of $s$ such that $w=w' \cdot w' $. That is, $w$ is a…
Dan D-man
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10
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1 answer

Find the longest repeated pattern in a string

I'm looking for an efficient algorithm to find the longest repeated pattern in a string. For example, consider the following string of numbers: 5431428571428571428571428571427623874534. As you can see, 142857142857 is the longest pattern which is…
MBZ
9
votes
2 answers

Count of distinct substrings in string inside range

Having string $S$ of length $n$, finding the count of distinct substrings can be done in linear time using LCP array. Instead of asking for unique substrings count in whole string $S$, query $q$ containing indexing $(i,j)$ where $0 \le i \le j < n$…
mechsoul
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9
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Polynomial time algorithm for finding a maximal monotone subset

Input: Some fixed $k>1$, vectors $x_i,y_i\in\mathbb R^k$ for $1\le i\le n$. Output: A subset $I\subset\{1,\dots,n\}$ of maximal size such that $(x_i-x_j)^T(y_i-y_j) \ge 0$ for all $i,j\in I$. Question: Can this be computed in polynomial time in…
Klaas
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9
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Test if there exists an integer k to add to one sequence to make it a subsequence of another sequence

Suppose that sequence $A$ contains $n$ integers $a_1,a_2,a_3,\ldots,a_n$ and sequence $B$ contains $m$ integers $b_1,b_2,b_3,\ldots,b_m$. We know that $m \geq n$. We assume without loss of generality that both sequences $A$ and $B$ are sorted in…
iouvxz
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8
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4 answers

Fastest algorithm for finding the longest palindrome subsequence

First of all we must read a word, and a desired size. Then we need to find the longest palindrome created by characters in this word used in order. For example for size = 7 and word = "abcababac" the answer is 7 ("abababa"). Postscript: the size…
Lin Yon Xong
8
votes
1 answer

Find subsequence of maximal length simultaneously satisfying two ordering constraints

We are given a set $F=\{f_1, f_2, f_3, …, f_N\}$ of $N$ Fruits. Each Fruit has price $P_i$ and vitamin content $V_i$; we associated fruit $f_i$ with the ordered pair $(P_i, V_i)$. Now we have to arrange these fruits in such a way that the sorted…
Jack
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7
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Subsequence of one string but not of others

Let $\Sigma$ be an alphabet, and let $x^+,x^-_1,\dots,x^-_n \in \Sigma^*$ be strings over that alphabet. Call a string $s \in \Sigma^*$ good if $s$ is a subsequence of $x^+$ but not a subsequence of any of $x^-_1,\dots,x^-_n$. Given…
D.W.
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6
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Count unique increasing subsequences of length 3 in $O(n\log n)$

Problem: Given an array of $n$ integers, $A[1 \dots n]$, such that any integer occurs at most 2 times in the array, we have to find the number of unique increasing subsequences of length 3 (duplicate subsequences must be counted only once). In other…
mayank
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6
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Computing the mode of XOR subsequences

I was confronted with this problem in an online programming challenge and it has been bugging me since: In the problem, you are given a list of 16-bit numbers, say $a_0, a_1, ..., a_n$. An "XOR subsequence" is defined as the exclusive-or combination…
chbaker0
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6
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2 answers

Longest subsequence such that A[i].x < A[i+1].y

I have an issue for which I am looking for an algorithm (if it exists) What I have: An array of items which have certain properties, e.g. item $A$ has properties $x$ and $y$. Example: $[ A(x,y), B(x,y), C(x,y), D(x,y), E(x,y) ]$ What I want: A…
Dutchy
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How can I efficiently generate the shortest list of arguments for the range() function that will generate a given list of integers?

I ran into an interesting problem at work when trying to generate the inputs for an API given the expected output. I've tried to formalize and anonymize the problem below. I've been trying to design a fast algorithm that works here but I'm a little…
Daniel Ong
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