Questions tagged [dynamic-programming]

Questions about problems that can be solved by combining recursively obtained solutions of subproblems.

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Version of knap sack problem

The are cuisenaire rods with N differnt lengthes $x_1,x_2,...,x_n$ (each length is a natural number), the number of the Cuisenaire rods is unlimited. Given a natural number B. you should tell if you ...
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2answers
3k views

Shortest path between 2 vertices using at most K edges using Bellman-Ford

I'm a bit confused about stopping at Kth iteration on the Bellman-Ford algorithm to find the shortest path of at most length k from s to t. Let me show you a graph and explain you what I understand: ...
5
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1answer
480 views

A variant of the vertex cover problem on trees

Consider the following variation on the vertex cover problem: given a tree on $n$ vertices, we are asked to calculate minimum size of a multiset $S$ such for each edge $(u,v)$ in the tree at ...
5
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1answer
657 views

Find equidistant triplets in a tree

Given a tree $T$ with $n$ vertices, we want to find the number of triplets of vertices $(a,b,c)$ such $d(a,b) = d(b,c) = d(c,a)$ where $d$ is the distance function (length of the shortest path between ...
5
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2answers
141 views

Given a list of integers, how to find the smallest positive integer such that I can get all the integers in the process of dividing it by 2?

The title could be a little bit confusing, and it is not easy to summarize it within a sentence, therefore I will explain it in detail below. If you have any thoughts on optimizing and rephrasing the ...
5
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2answers
513 views

Can counting problems have optimal substructure?

I understand that for a problem to be solvable using dynamic programming, it needs to have the following properties: optimal substructure overlapping subproblems I stumbled upon an article which ...
5
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1answer
120 views

Finding a partition with minimum “maximal length”

We're given $n^2$ different points in $(0,1)$ : $x_1< x_2 < \dots < x_{n^2}$. We are required to choose $n$ points $x_{i_1}<\dots<x_{i_n}$ such that the value of $\max\,\{x_{i_1}, x_{...
5
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1answer
4k views

Proving optimality of a dynamic programming algorithm

We have a string $s$ containing $n \leq 100$ bits. The move we can make on it is erasing from $s$ some substring $x$, but only if $x$ is directly preceded by $x^R$, where $x^R$ means string $x$ ...
5
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1answer
299 views

Maximize product of sum of two subset

Given two sets $A = \{a_1, a_2, \dots, a_n\}$ and $B = \{b_1, b_2, \dots, b_n\}$, both consist of positive numbers, this problem is to find a subset $S$ in $\{1, 2, \dots, n\}$ to maximize $$ \left(\...
5
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1answer
396 views

Semi-local Levenshtein distance

If you have a long string of length $n$ and a shorter string of length $m$, what is a suitable recurrence to let you compute all $n-m+1$ Levevenshtein distances between the shorter string and all ...
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0answers
91 views

Find the 'best' longest common subsequence

I am writing a program that computes and displays diffs. I implemented Meyers algorithm that computes the LCS between 2 subsequences (seq1 and ...
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0answers
150 views

Count Wildcard Parenthesizations of a String

Let $\Sigma = \{ (, ), ? \}$ be an alphabet. For a given string $s \in \Sigma^*$, we denote by $f(s)$ the number of ways to replace each symbol $?$ either with $($ or with $)$ such that $s$ is ...
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4answers
285 views

Algorithms for 2-colouring a 2 x N matrix

Our task is to color a given $2 \times N$ matrix with two colours red (R) and blue (B) such that no two adjacent cells are blue. For red, there are no restrictions. An example of all possible ...
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4answers
597 views

Do all recursive problems have optimal substructure?

I am reading about dynamic programming and I understand the overlapping subproblem requirement but not sure why optimal substructure is explicitly stated. Are there problems that can be solved ...
4
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2answers
213 views

How can you compute the expected edit distance in $O(2^{3n/2})$ time?

In a coding challenge an answer claimed to be able to compute the expected edit distance between two binary strings of length $n$ in $O(2^{3n/2})$ edit distance calculations by dynamic programming. A ...
4
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1answer
268 views

Is dynamic programming restricted to optimization problems?

The usual criteria used to decide if a problem can be solved using dynamic programming is (1) if it has optimal sub-problems and (2) if it has overlapping sub-problems. Does the word "optimal" mean ...
4
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1answer
81 views

Calculating all products of $n-1$ factors when given $n$ factors

Let's assume we have an operator $$ \times: E^2\to E$$ of which we merely know that it is associative. Let's say a multiplication $e\times f$ always takes up a time of $M$ for all $e, f\in E$. We're ...
4
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1answer
1k views

Find maximum distance between elements given constraints on some

I have a list of numbered elements 1 to N that fit into positions on a number line starting with 1. I also have constraints for these elements: The element 1 is in position 1, and element N must be ...
4
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2answers
706 views

Problem contest with matrix and DP

I found this problem while I was reading an ACM problem and it is about dynamic programming. The problem says that you have a square matrix $n\times n$ filled with 1's or 0's, like this: $$\begin{...
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1answer
1k views

Is there a more efficient algorithm than backtracking/dynamic programming?

Consider the following game: One day a castle is attacked at sunrise (by surprise) by n soldiers. Each soldier carries a canon and a rifle. The castle has strength s. On the first day each ...
4
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2answers
159 views

Discrete assignment problem with penalties

I came across a problem were you have to plan an optimal assignment pattern. Let's say you have $j=1,\ldots,n$ tasks during $i=1,\ldots,m$ time periods. It's an single agent problem where we have to ...
4
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2answers
201 views

Two recurrences for the change-making problem with repetition

The change-making problem with unbounded repetition is: Input: Unlimited quantities of coins with values $x_1, \ldots, x_n$ and an amount $v$. Output: Can the given $v$ amount of money be made ...
4
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1answer
3k views

Using dynamic programming to maximize work done

Say that there are $n$ days and there is $x_1, x_2, ...,x_n$ amount of data to process on each day. Your computer can process $s_1$ amount of work on the first day since rebooting your computer, $s_2$ ...
4
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1answer
942 views

Adjacent house , dynamic programming problem

I have to be honest this is a homework problem, but I just need to discuss this with some one. The problem is there is a row of n houses, with different profit e.g profit1 for house 1, it can be ...
4
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2answers
4k views

Efficiently calculating minimum edit distance of a smaller string at each position in a larger one

Given two strings, $r$ and $s$, where $n = |r|$, $m = |s|$ and $m \ll n$, find the minimum edit distance between $s$ for each beginning position in $r$ efficiently. That is, for each suffix of $r$ ...
4
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1answer
2k views

Binomial coefficient to approach multi-way choices DP problem?

I'm trying to understand this dynamic programming related problem, adapted from Kleinberg's Algorithm Design book. Not homework: i've already a solution, just considering if i'm ok with the theory. ...
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3answers
4k views

Finding all solutions to subset sum for integers with bounded weights

Suppose I have the set of weights $W = \{w_1,w_2,\ldots,w_{50}\}$ where each $1 \le w_i \le 60$ is an integer. I am interested in determining all subsets (not just one, and not just the number of ...
4
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1answer
260 views

Smallest string length to contain all types of beads

I read this question somewhere, and could not come up with an efficient answer. A string of some length has beads fixed on it at some given arbitrary distances from each other. There are $k$ ...
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2answers
2k views

Maximum sum subset of an array with an extra condition

We are given numbers $n \leq 200$, $k \leq 10$ and an array of $3n$ positive integers not greater than $10^6$. Find the maximum possible sum of a subset of elements of this array, such that in every ...
4
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2answers
981 views

Fuzzy string matching algorithm with allowed events?

I want to be able to locate a substring in a string allowing for a specified number of mismatches, insertions and deletions - and at the same time know how many mismatches, insertions and deletions ...
4
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1answer
3k views

Maximum Schedulable Set Zero-Lateness Deadline Scheduling

This is a homework problem for my introduction to algorithms course. Recall the scheduling problem from Section 4.2 in which we sought to minimize the maximum lateness. There are $n$ jobs, each ...
4
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1answer
344 views

Why does Banded Needleman-Wunsch give alignments with no more than d base pairs of indels?

A common modification to the Needleman-Wunsch to reduce running time is to only fill in the cells along a diagonal band of the matrix (slides 27/28). Let 2d + 1 represent the width of the band. Then ...
4
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1answer
104 views

Which matrix of Q values is being used here?

This question refers to this paper: Using Free Energies to Represent Q-values in a Multiagent Reinforcement Learning Task In section 2.1, equations (5) and (6), I am wondering which Q values are ...
4
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1answer
545 views

Issues with using greedy algorithm (Interval scheduling variant)

I am trying to solve a problem of finding incompatible jobs set using greedy algorithm. However, I am not sure if greedy algorithm can solve this problem or I need to perform another approach. I have ...
4
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1answer
310 views

Find the longest subsequence of two strings

I want to know which is the best way to find the longest common subsequence of two strings
4
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1answer
268 views

How to extend Bellman-Ford to solve the $k$ shortest path routing?

Browsing the wikipedia I got to this page where it is said: Finding k shortest paths is possible by extending Dijkstra algorithm or Bellman-Ford algorithm and extend them to find more than one ...
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0answers
83 views

Assign n people to m rooms of different sizes, such that noone is alone

I'm looking for an efficient way to assign n people to m rooms in a very specific way. INPUT: The program receives two sets of people (set of males and set of females), as well as a set of available ...
4
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0answers
117 views

Maximum weight independent set in a King's graph

I would like to find a maximum weight independent set in a finite section of a King's graph. For an $m\times n$ King's graph where $n \ll m$, we can use an $O(2^{2n} m)$ bitmask dynamic programming ...
4
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0answers
140 views

Sequence Alignment with Skips

In my thesis I am working on a problem connected with sequence alignment, in particular, I deal with the Dynamic Time Warping (DTW) algorithm (see this for more), which is used to evaluate the ...
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0answers
78 views

What is the relationship between the Markov property and optimal substructure?

Dynamic programming can only be applied to problems with optimal substructure. The Markov property (e.g. in Markov Decision Processes, MDPs) means that the distribution of one state $x_{k+1}$ only ...
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0answers
1k views

can we solve dynamic knapsack problem using Memoization approach?

As we know Dynamic programming has two techniques. Bottom up dynamic programming approach. Top down memoization approach Normally dynamic knapsack problem is solved using Bottom up dynamic ...
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0answers
386 views

Does the Longest Common Subsequence problem reduce to its binary version?

I am working on a problem regarding the Longest Common Subsequence (LCS) of two strings, and I was wondering if there is any reduction from the general case of LCS to its binary version, i.e. by ...
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0answers
611 views

2D version of LeetCode house robber problem

The house robber problem of leetcode can be described as followed : A robber enters a colony of houses numbered from 1 to n. Every house has a number printed on the top of it. That number is the ...
4
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1answer
94 views

How to cluster similar objects into fixed size groups?

I have $n$ people each of which can meet on certain days of the week. I want to group them into $\frac{n}{k}$ groups of size $k$ such that all people in a group can meet on a day. eg - Suppose there ...
3
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2answers
4k views

Why do these recurrences determine the number of ways of tiling a 3xN rectangle with 2x1 dominoes?

http://www.algorithmist.com/index.php/UVa_10918 The above link is a solution to UVa 10918 Problem. The problem is based on Dynamic Programming. I am not able to understand this approach to the ...
3
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3answers
172 views

Optimize a linear recurrence

$$\begin{align*} T[1] &= 1 \\ T[2] &= 2 \\ T[i] &= T[i-1] + T[i-3] + T[i-4] & \text{for \(i \gt 2\)} \\ \end{align*}$$ I have to calculate $T[N]$, but $N$ is too big ($\approx ...
3
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3answers
2k views

Can't understand why the DP Subset Sum algorithm is not polynomial

I can not understand why the dynamic programming algorithm for the Subset Sum, is not polynomial. Even though the sum to find 'T' is greater than the total sum of the 'n' elements of the set , the ...
3
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2answers
2k views

Dynamic programming table for finding similar substrings is too large

Substring Diff Given two strings of length $n$, $P = p_1\dots p_n$ and $Q = q_1 \dots q_n$, we define $M(i, j, L)$ as the number of mismatches between $p_i \dots p_{i+L-1}$ and $q_j \dots q_{j+L-1}...
3
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1answer
4k views

Dynamic Programming Solution to 0,1 KnapSack Problem

I am trying to understand the DP solution to the basic knapsack problem.However even after reading through a variety of tutorials ,its still beyond my comprehension.I am taking an algorithmics course ...
3
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1answer
4k views

Recurrence relation of the coin change problem

I'm trying to wrap my head around the coin change problem, where you try to find the total number of ways $N$ cents can be exchanged using $M$ coins $\{C_1, C_2, ..., C_m\}$. The recurrence relation ...

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