Questions tagged [greedy-algorithms]

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

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Parition a multiset of numbers into two subsets, how to maximize the sum of their medians?

Given a multiset $S$ of numbers, partition it into two subsets $S_1$ and $S_2$. How to maximize the sum of their medians? For example, the median of {1,2} is ...
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Graph in which greedy algorithm for maximum matching is a 2-approximation

Here is a greedy algorithm for maximum bipartite matching: Iteratively select an edge that is not incident to previously selected edges. This algorithm returns a 2-approximation, and runs in linear ...
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Find set of vertices of max size under some restrictions [duplicate]

I've faced a problem and I don't know what approach I must follow, dynamic programming or greedy method, so here is the question. Question: Given a directed tree $T=(V,\ E)$. We're required to find a ...
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Find most vertices in a directed tree where no path of length less than 3 connects any pair

Given a directed tree $T = (V, E)$, we need to find a set of vertices $A \subseteq V$ such that for every two vertices $v,u \in A$ either there is no path between them or the path between them is of ...
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Is the optimal substructure property necessary to guarantee the existence of a greedy algorithm that always finds the optimal result?

I'm trying to understand what is necessary to guarantee that a problem has a greedy algorithm that always produces optimal solutions. In multiple places I find that a greedy algorithm can be ...
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Where does each part of the $1 - (1 - 1/k)^k$ approximation for the Maximum Coverage problem come from?

A solution to an instance of the Maximum Coverage problem with a budget of k subsets can be approximated with a greedy algorithm that, at each iteration, picks one of the subsets that adds the most ...
113 views

Greedy Stays Ahead proof of "Jump Game"

(Source: https://leetcode.com/problems/jump-game-ii/) Consider the following problem: Given an array of non-negative integers nums, you are initially positioned at the first index of the array. Each ...
33 views

Optimal order to minimise time until first success

Say you are a salesperson and you want to make a sale; there are $N$ customers on your list; customer $i$ takes $t_i$ time to talk to and there is $p_i$ probability to make the sale to that customer - ...
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discrete optimization problem with a matrix inverse

I'm trying to solve this discrete optimization problem:$\newcommand{\I}{\mathcal{I}}\newcommand{\R}{\mathbb{R}}$ \max_{|\I| \le k} f(\I) \qquad\text{where}\; f(\I) :=x_{\I}^{\top} (\Sigma_{\I})^{-1} ...
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Maximum number of teams that can be formed with given age such that the sum of their age is more than 18

An array of students age is given : $A = \{ A_1,A_2, ..., A_n \}$. Form maximum number of teams such that the sum of age for each team is more than 18. Note that each team consists of only 2 people. ...
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Given n positive integers, pick two elements and subtract each by one with one operation. Find max operations

Problem Description: we have an array of n positive integers and in one operation we have to choose two elements in the array and decrease them by 1. (Elements on which we are performing this ...
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Why can we solve this problem by a greedy algorithm?

https://atcoder.jp/contests/past202004-open/tasks/past202004_f I am trying to solve the following problem: We have $N$ tasks. Let today be Day $1$. You can do the $i$-th task on or after Day $A_i$, ...
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Algorithm that finds a matrix with a specific number of 1s in its rows & columns

Question: Given integer $n \geq 2$ and two lists of size $n$, $A$ and $B$, of non-negative integers, determine if there exists an $n \times n$ matrix whose $i$-th row has $A[i]$ $1$s and whose $j$-th ...
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Determines the smallest set of unit-length closed intervals that contains all of the given points

Question: Describe an efficient algorithm that, given a set $\{x1, \cdots, x_n\}$ of points on the real line, determines the smallest set of unit-length closed intervals that contains all of the given ...
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The preliminary of the Bandit Gradient Algorithm

In the papers introducing The Bandit Gradient Algorithm as Stochastic Gradient Ascent, the following relationship: is always considered as a preliminary and lacks proof for it. Does anyone know how ...
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Optimal Word Guessing Algorithm in $O(n \log n)$

Say that your friend picks a word $(w_1, w_2,\dots,w_n)$ according to a known probability distribution $(p_1,p_2,\dots,p_n)$. You ask yes or no questions until you are certain which word has been ...
28 views

Social Networking Disease Transmission

You are given an undirected graph (social network) in which each edge $e = (v, v')$ has an interval $I_e = [l_e, u_e]$ on it. The meaning is that you know that $v$ and $v'$ met at some point during ...
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Bin packing problem and optimality proof

Let $W$ be an array of weights. Store all the weights of $W$ in bins such that in each bin a heavier weight always go before a lighter weight (if $w_i\in W$ is stored before $w_j\in W$ then ...
107 views

Time-varying edge cost Minimum Spaning Tree

I am having a hard time wrapping my head around the time-varying edge cost of this question : Suppose we have a connected graph $G = (V, E)$. Each edge e now has a time-varying edge cost given by a ...
52 views

Why is this greedy strategy always correct for this problem?

I'm trying to solve the following problem: https://cses.fi/problemset/task/1084/ My first idea to solve this was to sort the applicants and apartments in increasing order. Then, I iterate through all ...
48 views

Greedy approach behind SPOJ Aggressive Cows problem

My doubt is related to the given SPOJ problem: Farmer John has built a new long barn, with N (2 <= N <= 100,000) stalls. The stalls are located along a straight line at positions x1,...,xN (0 &...
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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 ...
306 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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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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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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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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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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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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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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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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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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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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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: ...