Questions tagged [lower-bounds]

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CLRS Question 8-6 Lower bound on merging sorted lists

I'm doing the CLRS Problems and there's a part I'm having trouble following. The question is: Part a) Given 2n numbers, compute the number of possible ways to divide them into two sorted lists, each ...
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Lower bound on algorithm solving certain recurrence

I have to find the lower bound of the following recursion: $A_1 = C_1 = p_1$, $B_1 = D_1 = 1-p_1$, $F_k = A_k + B_k$. Evaluate $F_n$. \begin{align} A_{k+1} &= (A_k + 2C_k) p_{k+1} + (1-p_k) p_{k+1}...
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How many strings for CLOSEST STRING lower bound to apply

In the CLOSEST STRING problem, one is given (bit-)strings $s_1, \dots, s_t$, each of length $L$ and an integer $d$. The question to be answered is whether there exists a string $s$ that has hamming ...
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Hardness of boolean functions

For a boolean function $f:\{0,1\}^n\longrightarrow\{0,1\}$, $H_{avg}(f)$ is a function from $\mathbb{N}\longrightarrow \mathbb{N}$, termed as the average case hardness, if $\forall$ circuit $C_n$ of ...
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How to Show that an Algorithm Depends on all N inputs (and thus has a lower bound of Omega(n)) [duplicate]

In the algorithm detailed on this page: Fastest Algorithm for Computing Expected Value, I was wondering how to prove that the algorithm depends on all $n$ of the inputs it is fed (and thus has $\Omega(...
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Connection between Pseudo random generators and hardness

For a boolean function $f:\{0,1\}^n\longrightarrow\{0,1\}$ $H_{avg}(f)$ is defined as the largest $S(n)$ s.t. for all circuit $C_n$ of size $S(n)$, $\Pr_{x\in U_n}[C_n(x)=f(x)]<1/2+1/S(n)$. Here $...
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Is there a super-linear lower bound on the time complexity of all solutions of NP complete problems?

$P \ne NP$ would imply that any polynomial is a lower bound on the time complexity of any NP complete problem. Is some non-trivial lower bound known at all?
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Existence of boolean function with exponential average case hardness

Show that for every large enough $n$, there is a boolean function $f\colon \{0,1\}^n\longrightarrow\{0,1\}$, whose average case hardness is exponential. The question is taken from Arora Barak ...
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Why there is no polynomially large sequence of polynomial large weights that derandomize the isolation lemma?

I was studying the paper Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size by Arvind and Mukhopadhyay and came across the following claim (Observation 1.2 on page 3): "More ...
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Lower bound for $k$-sorting an array

This is exercise 2 of the lecture note by Jeff Erickson on decision tree lower bounds. We say that an array $A[1 \ldots n]$ is $k$-sorted if it can be divided into $k$ blocks, each of size $n/k$ (we ...
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What is the difference between saying there is no ϵ>0 such that a problem can be solved in $O(n^{2-\epsilon})$ time and $n^{2-o(1)}$ or $\Omega(n^2)$?

I have seen the formulations there is no ϵ>0 such that a problem can be solved in $O(n^{2-\epsilon})$ time a problem requires time $n^{2-o(1)}$ a problem requires time $\Omega(n^2)$ being used ...
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Information-theoretic limits for a weighing puzzle

Consider the following problem: You are given $n$ coins with labels $1, \ldots, n$. You know that coins have weights $1, \ldots, n$, but you don't know whether the labels are correct (i.e. they can ...
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Why decision tree method for lower bound on finding a minimum doesn't work

(Motivated by this question. Also I suspect that my question is a bit too broad) We know $\Omega(n \log n)$ lower bound for sorting: we can build a decision tree where each inner node is a comparison ...
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Counting circuits with constraints

Please forgive me if this question is trivial, I couldn’t come up with an answer (nor finding one). In order to show that there are boolean functions $f : \{0,1\}^n \rightarrow \{0,1\}$ which can be ...
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Lower bound on comparison-based sorting

I have a question from one of the exercises in CLRS. Show that there is no comparison sort whose running time is linear for at least half of the $n!$ inputs of length $n$. What about a fraction of $1/...
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Lower bounds for orthogonal matrix multiplication

Is it possible, according to the current state of knowledge, that orthogonal matrices can be multiplied faster than arbitrary matrices? More precisely, let $T(N)$ denote the worst-case time of the ...
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Communication Complexity for Product Distributions

In general for the (two-party) set disjointness problem for inputs of length n, we know that the parties need to communicate $\Omega(n)$. Surprisingly, today I discovered (if I understood correctly) ...
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Algebra for min/max bounds

I am trying to model some set operations which are only well-defined if one is a subset of the other. The way the sets are constructed, I'll have a series of constraints of the form $x \subseteq y$, ...
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Lower bound time complexity for obtaining an arbitrary entry in a hashtable

I just answered this question on StackOverflow, which asks for an efficient algorithm such that given a nonempty hashtable, the algorithm should return a pointer to an arbitrary nonempty entry in the ...
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Decision tree lower bound for finding two array elements summing to zero

I have to solve this exercise: Given an unordered array $A[1], \ldots, A[n]$ of positive and negative integers, determine if there are two indices $i \neq j$ such that $A[i] + A[j] = 0$. ...
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Resolution exponential lower bound… alternative proofs?

I am reading the Resolution proof system exponential lower bound via Haken's bottleneck method for the Pigeonhole Principle as presented in Arora and Barak's Computational Complexity: A Modern ...
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Least number of comparisons needed to sort (order) 5 elements

Find the least number of comparisons needed to sort (order) five elements and devise an algorithm that sorts these elements using this number of comparisons. Solution: There are 5! = 120 possible ...
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A Question related to the method of find lower bound : Trivial lower bounds

In Trivial lower bounds we just need to count the number of items in the input that needs to be processed and the number of items that need to be generated and the trivial lower bound time is then the ...
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Finding the lower bound through decision trees

One way to find the lower bound of a comparison based algorithm is to use the decision tree. U have two questions regarding this method : 1) We know that the height of the tree is path that connects ...
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Lower bound and worst case scenario

We know that the lower bound is the minimum amount of work needed to solve a problem. So for a given problem say x it has the best algorithm ( the most efficient algorithm to solve this problem ) say ...
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What is the min # of moves to sort an array from 1 to n?

Problem: You are required to sort an array with numbers from 1 to n. You can do a "move", which means choosing one element and moving it to any place you want (insert to any place, not swap). Prove ...
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How to use adversary arguments for selection and insertion sort?

I was asked to find the adversary arguments necessary for finding the lower bounds for selection and insertion sort. I could not find a reference to it anywhere. I have some doubts regarding this. I ...
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What is an $O(n \log(n))$ binary sorting algorithm with a guaranteed low scaling constant on the run-time?

Let $O_c(f(n))$ denote that $c$ is the scaling constant for the run-time (e.g. $\text{run time} \leq c\cdot f(n) + B$ if $n$ is large enough) The absolute lower limit on the run-time for a binary ...
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Efficiently computing minimal elements over partially ordered sets

I have a list of sets that I would like to sort into a partial order based on the subset relation. In fact, I do not require the complete ordering, only the minimal elements. If I am not mistaken, ...
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Conditional lower bounds on the running time of the single source shortest path problem

Just out of curiosity, I was wondering whether there is a conditional lower-bound on the running time of an algorithm for the Single Source Shortest Path Problem (on directed or undirected graphs). I ...
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Standard information-theoretic lower bound?

There should be a simple argument, but I'm struggling to see it. Suppose Alice has a string $x \in \{0, 1\}^n$ and sends a message $s = s(x)$ to Bob. And suppose that given $s$, Bob can reconstruct ...
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An example where the algorithm of Hopcroft and Karp performs poorly?

I have been trying to construct an example, where Hopcroft and Karp's algorithm for the maximum matching problem performs poorly (say at least $\Omega(\log n)$ rounds). However, all the examples I ...
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Why is the lower bound for sorting strings Ω(d + nlogn)?

I'm taking an Advanced Algorithms course. I'm currently studying efficient algorithms for sorting strings. In this chapter, it is provided a lower bound for the time complexity of $\Omega(d + n\log{n})...
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Dealing with test condition '=' for a while loop when determining a bound function/loop variant

The following is the definition of what a bound function for a while loop must satisfy: The bound function is an integer-valued, total function of some of the inputs, variables and global data that ...
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Find both lower and upper asymptotic bounds for $T(n) = 2T(\frac{n}{2})+n^4$

So far we have learned Recursion Tree, Substitution Method, and Master's Theorem. I'm not sure how we can find lower AND upper bounds. I know that using Master's Theorem, we get $T(n) = \Theta(n^4)$, ...
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Decision Tree for searching an element in an n*n matrix

I just learnt decision tree concept in class. I have a question for homework. It says to prove that for searching an element in n*n matrix the lower bound is logn and prove it using decision tree. My ...
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Minimum amount of rectangles to create a 2-dimensional matrix

From this codegolf question. Consider an $r$ by $c$ matrix of nonnegative integers, called $M$. You also have a zero matrix of the same dimensions, called $N$. A "move" consists of replacing a ...
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Lower bound for merging $m$ sorted arrays (decision tree leaves count - permutations)

I need some help understanding how to calculate the lower bound on the time complexity of merging $m$ sorted arrays of length $n$. The bound should be $nm \lg(m)$. I need to prove this using a ...
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Useful conditions for proving super polynomial lower bound for some kind of recurrences

Given a recurrence of the form $\forall n,m.\ \ T(n,m)=\begin{cases}1,&,m=1\\\sum_i{T(n_i,m_i)}&,\text{else}\end{cases}$ Note: both $n_i$ and $m_i$ are dependent on $n,m$ so they should have ...
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Problems that provably require quadratic time

I'm looking for examples of problem which has a lower bound of $\Omega(|x|^2$) for input $x$. The problem needs to have the following properties: $\Omega(n^2)$ runtime proof for any algorithm - ...
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Number of comparisons in array where each element appears n/k times [duplicate]

Given an array of $n$ elements with $k$ distinct elements, each appearing $n/k$ times, how can I show that the number of comparisons to the sort the array in the worst case is in $\Omega(n \log k)$?
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Lower Bound for Time Complexity of Pairing Problem

Given an array X and array Y both of length n, the pairing algorithm will return the elements of the arrays matched so that the smallest element in X will be matched with the smallest element of Y, ...
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1answer
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Pebble game lower bound?

This paper says pebble games have super linear lower bound for every fixed $k$ https://dl.acm.org/citation.cfm?doid=62.322433. Why is it not considered proof of constructive example for a function in ...
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Asymptotics of a sinusoid

Consider the function $$ f(n) = 2n^2 |\sin(\pi \cdot n/2)|. $$ Which of the following classes does $f(n)$ belong to? $$ O(n^2), \Omega(n^2), \Theta(n^2), \omega(n^2), o(n^2). $$ I'm working in this ...
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Comparison-based lower-bound for finding duplicates in an array of $n$ numbers

Decision Problem: Given $n$ real numbers, give an algorithm that outputs "1" iff there are at least two numbers that are identical and outputs "0" otherwise. (Assume that comparison between any two ...
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Lower bound of disjointness by discrepancy?

I need to show that $Disc_\mu(Disj) \geq \frac{1}{2n+1}$ for any distribution $\mu: \{0,1\}^n \times \{0,1\}^n \to [0,1]$. Disjointness is defined as $Disj(X,Y)=\left\{ \begin{array}[ll]+1 & \...
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Why is finding minimum number of comparisons to sort $n$ elements so difficult?

In The Art of Computer Programming 2nd Ed, Vol 3, Section 5.3.1 then discuss a function $S(n)$ which is define as: $S(n)$ : The minimum number of comparisons that suffice to sort $n$ elements. ...
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Lower bound for maxima on 2D plane

Given $n$ points $(x_1, y_1), \ldots, (x_n, y_n)$ on a 2-dimensional plane. A point $(x_1, y_1)$ dominates $(x_2, y_2)$ if $x_1 > x_2 \land y_1 > y_2$. A point is called a maxima if no other ...
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Finding maximum takes at least $\lceil n/2 \rceil$ comparisons

We are given an array $A$ with $n$ elements, $n \in \mathbb{N}$ and all elements are in the set $\{1,2,3, \cdots, n \}$. I want to prove that finding the maximum in $A$ (that is, outputting the index ...

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