Questions tagged [greedy-algorithms]

Questions about algorithms that make at each step the locally optimal choice.

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Finding an optimal solution for the following problem

This is an intriguing question given by my friend for which I want to find an optimal solution: Each week, you can choose to work ONLINE $(ON)$ or in person $(WR)$ in your department's office. If you ...
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23 views

Minimum Number of Refueling Stops with Dynamic Programming

This is a question from Leetcode and I wanted to solve it with Dynamic Programming. upon seeing the solution, I am not able to get the intuition behind the logic. I was able to understand the solution ...
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Time to form a complete graph from n vertices given that only k vertices can be used at a time [closed]

I know this problem is related to the greedy algorithm and max edges incomplete graph but can't come up with a solution. Problem You are given two numbers n and k: n >= k n is the total # of ...
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1answer
72 views

How to prove the my greedy algorithm for placing guards?

Given $n$ images placed in indexes $x_1 < x_2 < ... < x_n$ and an endless number of guards, where each guard if placed in index $y$ can protect $[y-0.5,y+1]$. I want to protect all images ...
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35 views

Filling Two Knapsacks with greedy algorithm

I've got the next problem, where I have two, instead of one, knapsacks. Formally, we have items $1, \ldots , n$ and each item $i$ has a positive integer weight $w_i \in \mathbb{N}$ and a positive ...
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3answers
33 views

Greedy Algorithm: Optimal Substructure

I don't have a CS degree but I have recently taken up studying algorithms very seriously. I have been studying greedy and dynamic programming for days and I come across the below definition a lot, ...
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21 views

Is correctness implied by an optimality proof?

New to proofs (in the context of analysis of algorithms). I'm wondering, if I were to prove a greedy algorithm is the optimal solution, does this imply its correctness as well? (partial correctness + ...
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25 views

Proof of Correctness Request for Greedy Algorithm that solves “The Weight Job Scheduling” problem

Today, in my self-lead studies, I found out about greedy algorithms, more specifically, a greedy approach to solve The Weighted Job Scheduling Problem. I understand how the solution is implemented but,...
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28 views

Which algorithms exhibits Greedy Choice Property but not Optimal Substructure Property

After a few courses using CLRS, I still have not been able to find a satisfying answer to the title question. This answer suggests Hoffman trees. But here the greedy choice is the two subtrees with ...
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1answer
76 views

Proof of Correctness : Arranging the sheep

I've come across a question in Codeforces contest 719(Div - 3). The problem goes like this : I was able to solve the problem by using another approach but had to use 4*n auxiliary space, where n is ...
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27 views

maximal independent set on grid-based graph proof of approximation ratio

We have a G = (V, E, w), in form of a grid graph with a single diagonal line in each grid in form of below. Where w is the V weight. We use a greedy algorithm that takes in each step maximum weighted ...
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1answer
24 views

Optimize stacking time series by offsetting start times (feels like a backpack problem?)

Given a time-series of data collected from a single running process that takes 8 hours to complete: Minute GB of Disk Space Used 0 0 1 8 2 15 3 22 ...Etc. It is sampled every minute, for 8 ...
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40 views

maximum sum of diameter of tree + leaf removal

Problem: A tree of n nodes is given. Have a sum variable. Choose the leaves furthest from one and other and add their distance(the number of edges between them) to the sum variable. Now remove one of ...
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1answer
36 views

Find the minimum cost of adding the elements of a set (greedy algorithm)

I'm VERY stuck with this problem: Given a set (with possible repeated elements), the cost of adding two elements $x, y$ is $x + y$. For example, the possible costs of the next set $\{1,2,5 \}$ are: ...
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1answer
44 views

Expected behavior of an algorithm to minimize rankings

Suppose $n$ students have preferences over $n$ different notebooks. Their preferences can be represented with a square matrix of size $n$ where each column is a different permutation of the vector $[1:...
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1answer
25 views

Greedy approach suggestions for assigning objects

Suppose there are three categories of people. Type X, Type Y, Type Z. In each type, there are two objects of subtype Type 'a' and type 'b'. For example. X: a1 , a2 , b1 , b2 Y: a3 , a4 , b3 , b4 Z: a5 ...
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1answer
43 views

k-center problem: proof for Gon algorithm gives a 2-approximation

The $k$-center problem is where we a given a graph $G(V,E)$, an integer $k$, a distance metric $d$ and we want to find a subset $C\subseteq V$ (such that $|C|\leq k$) which minimizes the following ...
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1answer
73 views

Find minimum number of points which intersect overlapping arcs

Say I have a circle of a fixed radius, with overlapping arc intervals along its edge. I want to return a minimum set of Points which intersects all arcs in $n^2$ time. I'm having some trouble proving ...
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1answer
31 views

Does any graph based optimization problem, which have a greedy algorithm, has guarantee that there exist an order which give the optimum

There exist greedy algorithms for vertex coloring like optimization problems. We know that for graph coloring, there is an order of vertices for which greedy coloring produces the optimum results. Is ...
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32 views

A problem about Huffman codeword

Under Huffman Encoding, what circumstances codeword length of each character be equal?(suppose number of character is power of 2) I think if all frequency of characters is same then codeword length ...
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1answer
60 views

Greedy algorithm for maintaining drugs

Given $n$ drugs such that each drug $d_i$ should maintain in interval $[c_i,h_i]$.We want to minimize number of containers to maintain medicines in compatible interval. My answer is as follows: I use ...
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1answer
12 views

Scheduling algorithm for a student's study times given assignments, expected time, due dates, and class times

I would like to create an algorithm that advises a student when they should work on certain assignments given the expected time each assignment will take, and the due date of each assignment. Say that ...
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1answer
114 views

Hamiltonian path greedy and anti-greedy algorithms

I had this question on a problem set recently, but I wasn't sure how to solve it. Given a complete weighted undirected graph $G$, here are two "algorithms" to find a Hamiltonian path: ...
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89 views

Finding name of a specific problem

Input : $x \in \mathbb Z$ and $a_1,a_2,\dots,a_n \in \mathbb N$. Output : $b_1,b_2,\dots,b_n \in \mathbb Z$ such that $\sum_{i} a_i b_i=x$ and $\sum_{i}|b_i|$ is minimized. Main question is, can we ...
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54 views

Need help with a very specific Greedy Algorithm. Are there fast solutions?

i need help for a specific problem i have at work. You have N number of rows in an array, each with some distribution of Numbers that range from 0 to N.Given an array of size 1000: Row 1 might look ...
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29 views

Minimum cost bin assignment

I've been trying to solve the below problem the entire day but couldn't come up with a solution. I have the suspicion that it could by solved by a graph algorithm (or maybe some greedy approach?) but ...
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1answer
32 views

Must an optimization problem with a greedy algorithm belong to P?

If it is known that for some optimization problem there is a greedy algorithm that solves it and the solution includes sorting of input at the preliminary stage, is it necessarily true that the ...
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2answers
69 views

Best approach to resource allocation problem

Problem The enemy army has taken $n$ of our cities. In each city $i$ the enemy has placed $e_i$ soldiers. We have $n$ teams, each team $j$ with $d_j$ soldiers. If we place more soldiers in a city ...
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1answer
26 views

Algorithm of split graph $G=(V,E)$ to 2 groups that at least half of the edges are between the groups [duplicate]

Can someone remind me the algorithm that split vertex of graph to 2 groups that at least half of the edges are external, I mean between the groups. As I remember it was a greedy algorithm, each time ...
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1answer
60 views

What Makes an Algorthim Greedy [Graph Coloring Algorithm]

I have a simple graph G = (V,E) and each vertex has a range [a,b].Every two vertices are connected only if [a_1,b_1] and [a_2,b_2] have a common subrange. Each range of vertex is rV1 = [0,5] rV2 = [1,...
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1answer
50 views

Is the following Greedy algorithm to generate Gray Codes always correct?

I recently solved the basic problem of generating a n-bit Gray Code. The solution I used involved building larger-bit Gray Codes from smaller ones recursively (the solution I've seen on most websites)....
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2answers
87 views

Maximize number of subsets

Given a list of subsets $S_1, \ldots, S_n$ of the universal set $U = \{e_1,\ldots, e_m\}$, find a subset $S \subset U$ of size $k$ that contains the maximum number of subsets $S_i$. In another words, $...
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1answer
47 views

How to (Efficiently) Sort a List of Items with Parent/Child Relationship

I have a list of items that have a parent/child/grandchild/etc. type of relationship. Each item has a list of descendants, and an _.isDescendentOf(other) member ...
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1answer
45 views

Maximum Independent Set of special Directed Graph

I was given this special type of Directed Graph and was asked to find it's Maximum Independent Set. Graph Properties : Graph has $N$ vertices and $N$ edges There can be no edge from a vertex $v$ to ...
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1answer
115 views

Greedy algorithm for subset sum on powers of 2

I have some $n$ numbers which are powers of $2$, say $a_1,a_2,a_3,\ldots,a_n$ which are not necessarily all distinct. I have option to give them any sign. I have to find if I can make their sum after ...
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23 views

Rearrange string k distance Apart

The question statement is as follows: Given a non-empty string s and an integer k, rearrange the string such that the same characters are at least distance k from each other. All input strings are ...
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19 views

Greedy Best-First Search Performance for Tree and Graph Space

I am currently reviewing the GBFS algorithm and when looking at its completeness I am confused between the difference of it being not optimal in Tree Search for Finite and Infinite Spaces that it is ...
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Spanning hypertree which connects the vertices as slowly as possible

I want to find a reference for the following problem or a similar problem for my paper. I found a greedy algorithm for this problem, but writing such an algorithm in a paper is not common in my area, ...
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2answers
52 views

Highest Product of X Items

🧩 How do you find the highest product of X items? This should optimize for runtime complexity and protect from overflows from large products. Inputs An array of both positive and negative Ints ...
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1answer
54 views

2-Approximation algorithm for for messages across a cyclic network

Question There are $n$ computers arranged in a cycle ($1,2,3..,n,1$), with undirected edges between adjacent computers. There are $m$ messages that need to be delivered. Message $i$ ($1 \le i \le m$) ...
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1answer
33 views

Modified topological sort

I recently asked a related question at the theoretical CS stack exchange, but I have modification to the problem that I think is a bit tougher. This seems like a better place anyways. Let's define a &...
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Does the existence of a matroid structure imply that the greedy algorithm is optimal?

I was going through the topic of matroid structures for the problems like Activity selection ,minimum spanning tree. I also came to know how to solve if a problem exhibits matroid structure. The ...
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30 views

Greedy Proof: For coin denomination systems, faster proof to ensure greedy solution yields the optimal solution

I have reasons to believe that there is a faster way to confirm, that for coin denomination systems if the greedy solution yields the optimal solution or not. I believe that if you check that for all ...
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1answer
642 views

How to understand the solution to Task Scheduler problem on LeetCode?

LeetCode Task Scheduler problem is the following: Given a characters array tasks, representing the tasks a CPU needs to do, where each letter represents a different task. Tasks could be done in any ...
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1answer
43 views

Serving $k$ customers with bounded time window

A person provides a service and he/she can serve $k$ clients each minute. Now, client number $i$ comes at the beginning of minute $a_{i}$ and waits $w_{i}$ minutes to receive the service and if they ...
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1answer
45 views

Ordering the tasks to minimize penalties

So I just started learning greedy algorithms and I have a problem that I want to solve. The statement is as follows: In your calendar you have an $L$ list of all the tasks you need to complete today. ...
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86 views

Correctness proof of a greedy approximation algorithm

How do I prove the correctness of this algorithm?
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46 views

Maximize the minimum gap while scheduling within intervals?

Problem There are N intervals in which a particular integer can be chosen. What is the maximum possible minimum gap between each integer if one integer is chosen for each of those intervals? For ...
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2answers
83 views

Why this problem has such a simple solution? How would you avoid looking for more complex solutions first?

There's a problem whose solution startles me because at first sigh, I wouldn't imagine that it could be solved so easily. The problem is: There are n tasks, each task ...
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1answer
40 views

Mathematical expression for the quantity that we are maximising in the stock buying and selling problem

Problem Statement: Say you have an array prices for which the $i^{th}$ element is the price of a given stock on day $i$. Design an algorithm to find the maximum profit. You may complete as many ...

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