Questions tagged [efficiency]

Using as few resources (e.g. time, space) as possible while solving a problem. Use this tag if your question is specifically about resource usage, not for generic algorithm questions that happen to mention running times.

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What the algorith should return and how to implement

I was studying algorithms, and that problem below came up. Write an efficient algorith that calculates the biggest single positive forward change in value given the values. There is a second image ...
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Why use Welford's Method over a more naive approach?

I came across this pair of blog posts explaining the problem with calculating variance on floating point data: https://jvns.ca/blog/2023/01/13/examples-of-floating-point-problems/#example-3-a-variance-...
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Shortcut for multiple highly-redundant dot-products?

In statistical computing, we often have linear models that involve computing multiple vector dot products, but there's a high amount of redundancy in the operations when you break them down. As an ...
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Chien search complexity

I am working on Reed-Solomon decoding, where Chien search is used to determine the roots of the so-called locator polynomial. From my understanding, Chien search consists in trying every possible ...
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Finding number of combinations of numbers from multiple arrays that add up to a given value

Let $ A $ be an array of $ n $ integer arrays with unknown lengths and $ s \in \mathbb{Z} $ a given number. I want to find the number of combinations of numbers from each array, such that their sum ...
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Efficiently sorting a list of integers depending on the index of said integers in a second list

Introduction Hi everyone, I am currently pondering how to efficiently design an algorithm for a sorting problem and hope that someone here may help me out a bit. The problem I have two lists of ...
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Are there adaptive algorithms/data structures for sparse and non sparse vectors?

A sparse 1D array of integers is commonly encoded as pairs of [index, value], which consumes 2 memory spots per value. A dense 1D array is commonly encoded as a linear array of values [value1, value2, ...
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$O(n \log n)$ Algorithm for first Train Problem

On $n$ parallel rails there are $n$ trains with constant speeds $v_1, ..., v_n$. At time $0$ the trains are at positions $k_1, ..., k_n$. Find an $O(n \log n)$ algorithm that determines which trains ...
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There is a problem-name and/or algorithm for calculating functions which are constant on different intervals?

I frequently have to calculate this kind of function Those functions have constant values on arbitrary (predefined) intervals I wonder if there is an algorithm for fast calculation of it, so I can ...
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What is the most computationally efficient Turing complete system?

I'm making a program that involves making models of and working with arbitrary systems (or programs). What is the most computationally efficient Turing complete system to model these in? By "...
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How much faster would a semantic segmentation model be with just 2 classes compared to 100 classes?

Let's say I have a semantic segmentation model that distinguishes between 100 classes of objects, and the speed of running the model is 1 image per second. Now let's say I take the same model ...
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Retrieving the cheapest path of a graph with time-dependent edge weights

There are many efficient algorithms for finding the shortest path in a network, like dijkstra's or bellman-ford's. But what if the weights of edges are time-dependent? I'm trying to find an efficient ...
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The interpretation of expected time bound for searches in a hash table

As CLRS book,page 260 stated, Thus, the total time required for a successful search is $\Theta{\left(2+\alpha/2-\alpha/2n\right)}=\Theta{(1+\alpha)}$ I wouldn't have any problem if the author says ...
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Number of operands to take when evaluating postfix notation?

All sources I have seen list taking two operands from the stack once we encounter an operator. Why? What if not all of my operators are binary (taking two arguments)? What if I have custom unary and ...
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Merging Tuples of Intervals

Suppose I have a list of tuples. Each tuple contains 2 intervals. The intervals in each tuple have nothing to do with each other. I would like to find a smaller list of tupels that covers all elements ...
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How to design fast/efficient algorithms?

Let's say, if you could travel to the past, what would you teach your younger self? Which key points define efficient algorithms?
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Fast computing of a matrix power for large integer values in C++

I'm working with squared matrices that can be quite large, for instance, a M = 50 x 50 matrix. My objective is to compute the power of the squared matrix ...
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Efficient way to find key points on spline to approximate it with line strip

Given a spline, what is an efficient way to find (approximately) the least amount (and position) of key-points to approximate the spline with a line strip, so that the largest distance between the ...
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How can we solve halting problem efficiently?

I was doing exercises regarding the halting problem and there is this question where I am stuck Ques: it goes like suppose if you can decide the halting problem with a query "Is <tm,s> ...
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Efficient method for finding 1's in binary representations

Say I have a binary number of N bits, and I need to find every combination that have M 1's. For example, if N = 3 and M = 1, then 100, 010, 001 are the allowed combinations. I have read that a more ...
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Which of the query approaches are more efficient?

There are two relations, registered(participant,topic) and fee(participant,amount). The primary key for registered is (participant, topic) and the primary key for fee is participant. The premise is ...
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About the connection of pipelined execution and latency

Let's consider we want to calculate a[i]=a[i]*c for a vector the size of N=12 on some random processor. We do assume that ...
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What is the name of this type of program optimization where two loops operating over common data are combined into a single loop?

On an imperative programming language, let us consider the following program: ...
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Can we make at most 3 comparisons in the closest points algorithm instead of 7?

Let's say I am using the divide and conquer algorithm outlined here, but I only want to return the minimum distance. I understand why that algorithm puts an upper-bound at 7 but I think that can be ...
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Efficient triangular decomposition of an integer

Euclidean division is an iterative process that has been made super-efficient at the CPU level, right? Its specification is that if I do (q, r) = f(n, d), I get ...
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Floating-point oblivious way to compute multiset numbers

I have to compute $R = \left(\!\!{n + 1\choose k}\!\!\right)$, which happens to be: $$ R = \left(\!\!{n+1\choose k }\!\!\right) = \binom{n+k}{k} = \frac{(n + k)!}{n!k!} = \frac{(n+1)(n+2)\cdots(n+k)}{...
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Making an algorithm that picks a unique random number in fixed set more efficient

I have been working on a project that simulates an online bank. At this point, I'm implementing the code used to create user accounts. Each account will have a sortcode and account number, I have ...
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Convex hull on set of squares

Imagine a set of two to six squares within 3D-space. The goal is to generate a convex hull around these squares as efficiently as possible. The following constraints are known: Each of the two to six ...
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1 answer
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Overhead cost of spawning child processes

I am curious as to the overhead cost of spawning child processes using fork in a Linux environment. Suppose I have a C program such as ...
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3 answers
162 views

Faster computation of $ke^{-(x - h)^2}$

The question is quite simple; almost every computer language today provides the $\exp(x)$ function in their standard library to compute expressions like $ke^{-(x - h)^2}.$ However, I would like to ...
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1 answer
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Finding most efficient sorting algorithm

Arr is an array that contains $ n$ numbers. Suggest the most efficient algorithm for each case and analyze the runtime. Explain why the algorithm you chose is the best one. Arr contains exactly $\...
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Big O Proof , f(n) = 2n + 1 and I have to prove f(n) is O n^2

If I have $f(n) = 2n + 1$ and I have to prove $f(n) \in O(n^2)$, by proving there exists positive constants $c$ and $n_0$ such that $f(n)<cn^2, \forall n\ge n_0$, can I do this all in one step by ...
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How to prove Big-O when $f(n)$ is defined sectionwise

I'm given a function which is defined based on a condition, for example $$ f(n) = \begin{cases} 4n+1, \ \text{n is even}\\ 3n^2+2, \ \text{n is odd} \end{cases}. $$ I have to prove or ...
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Calculating efficiency?

A program must construct and then use a set $S$ of 1000 integers. ...
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Efficient algorithm to compare arrays problem

I was submitted an interesting problem, but I wasn't able to find a solution. Define a function p(x, y) that takes int x and y, with ...
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1 vote
1 answer
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Understanding recursion tree for withdrawal formula

$$ T(n) = T(n-a) + T(a) + cn $$ Now the solution says that the height of the tree $(h)$ is: $$ h = \left \lfloor n/a \right \rfloor $$ And I don't understand why. Maybe I didn't understand the ...
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Why substituting the search part in INSERTION SORT doesnt yield a running time of $\Theta(nlgn)$

$$ \Theta - Tight \ asymptotic \ bound $$ If we change lines $5-7$ in Insertion sort With BINARY-SEARCH(A,p,r,v) Why don't we get a running time of $\Theta(n\lg n)$ as we go through the array $\...
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What's the most time efficient way to find the number of nodes reachable from each root and no other root?

Suppose there are n nodes. These nodes are connected by m unique directed edges. Sets of these edges may form cycles. Each node ...
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Parent Node in BST

Is it a good idea to store a pointer to the parent node in a binary search rather than to find the parent node several times? ...
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Comparing the efficiecy of 2 run times

I have a heap with $n$ elements. $k$ represent a number that is the height of one of the elements in the tree. I need to compare two run times and prove what i claim. The 2 run times are: $$ (1)O(...
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Findind The n/lgn intermidiate values in an unsorted array with asymptotic run time of $\Theta(n)$ SELECT algorithm

Let $A[1..n]$ be an unsorted array, we want to find the $n/lgn$ intermidiate numbers in the array. Namely the $(n/2)+1$ biggest number and the $(n/2) + 2$ biggest number and so on... until the $(n/2)...
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reordering a max heap that the values of one of her subtrees has been changed

Let $H$ be a max binary heap with $n$ elements (vertexes). Pick a vertex $z$ in the heap with height of $k$ ($0<k<\lg n$) To every element in the sub heap of $z$ we add the constant value $c &...
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Heaps and Heapsort - Find the 7'th biggest value in a min heap by $O(1)$

I have a min heap. I need to find the 7'th biggest value in the heap with $O(1)$. I need to build the algorithm. I dont realy have an idea how to get to this efficiency. Help? Thanks.
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What's the best bucket fill algorithm in terms of efficiency? [duplicate]

I am looking for an algorithm that fills a given region of connected particular nodes in minimum time. I have tried using flood flow algorithm but it's too slow and inefficient for large array, it ...
3 votes
1 answer
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Dynamic Program to solve an NP-complete partitioning problem

I have this problem for which I am struggling to find an efficient dynamic programming algorithm. Would be thankful for some help!! Let $A = \{ a_1, a_2, ..., a_n \}$ be a set where $a_i \in \mathbb{...
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For $T(n) = 16T(n/4) + n^2\lg^3n$ prove: $T(n) = \Theta(n^2\lg^3n)$

Define: $ \lg x = \log_2x $. Let $ f(n), g(n) $ be some non-negative functions. Define $ f(n) = \Theta (g(n)) $ if $$ \exists c_1,c_2 \in R\colon 0 < c_1g(n) \leq f(n) \leq c_2g(n) $$ I want ...
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Does the word "efficient" usually refer to polynomial time or polylogarithmic time?

This question is strictly about terminology. I'm not an expert in CS, but I've almost always seen the word "efficient" applied to an algorithm to mean "of polynomial runtime". E.g. this question and ...
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Regex vs line by line read efficiency - general concept question

Say, I have a file that is somewhere between 10-20 pages long. I need to extract 10-20 pieces of information from it that I can fairly easily extract using a series of regex-es or by running code that ...
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To calculate how many times a certain year repeats itself in the calendar within a given year range

Let's say I have a year $Y$. I wanna know how can I calculate the number of times the calendar configuration of the year $Y$ repeats itself in the year range $[A, B]$. Is there any method to it ...
3 votes
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Add edges to undirected graph to make connected and minimize longest path

I am trying to find an efficient algorithm to solve to following problem: Given an undirected disconnected graph, I want to add as few as possible edges to make to graph connected while minimizing the ...

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