# Questions tagged [lower-bounds]

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### Problems/properties of dynamic graphs with strong lower bounds

I know from  that the lower bound for the maximum hitting time of simple random walk on a dynamic graph is $\Omega(2^n)$. Smoothed analysis has been applied to the maximum hitting time  and ...
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### Can smoothed analysis be studied on the batch dynamic model?

I was reading about the batch dynamic CONGEST model for distributed computing. This model assumes a fixed communication network, but where local inputs to nodes may change over time. It aims to study ...
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### Counting number of swaps to make two strings equal in linear time

The input to our problem is a pair of strings, say $x$ and $y$. We treat our alphabet size as a constant, i.e., our input is effectively a pair of arrays with the values therein bounded by a constant. ...
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### Coming up with an adversary strategy for a clique of maximum size

I’m having trouble coming up with a good adversary strategy for this problem: Input: a graph G Output: the maximum size of any clique in G Where the algorithm asks each time, “are vertices x and y ...
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### Why does IP = PSPACE

Can anyone give an intuitive explanation to why IP = PSPACE, or at least one direction of it? I looked at many research papers but its very hard to understand the formalism unless you have a solid ...
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### Lower bound on worst-case time complexity of all sorting algorithms neglecting reading input and accessing elements time

We know that the worst-case time complexity of any comparison sorting algorithm is $\Omega(n\log n)$. Is there a lower bound on the worst-case running time of sorting algorithms of any type? Not just ...
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### Prove a lower bound

Prove: $n^{5}-3n^{4}+\log\left(n^{10}\right)∈\ Ω\left(n^{5}\right)$. I always get stuck in these types of questions, where there is a $"-(xy^{z})"$ in the expression. Whenever I see the solutions for ...
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### Prove that the following algorithm is $\Theta(n^3)$ by induction

I have the following algorithm runtime: $T(1) = b$ for some positive constant. Otherwise, $T(n)=8T(\frac n 2) + 100n^2$ I am trying to prove that it is $\Theta(n^3)$ by induction. I proved that it is ...
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### $\log n$ lower bound for space complexity

I am currently reading Arora and Barak's Computational complexity. In Chapter 4 (Space complexity), they say the following: Since the TM's work tapes are separated from its input tape, it makes sense ...
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### Communication complexity of equality gap problem

I'm interested to know what is the biggest known $0\le \epsilon\le 1$ such that the $gap-EQUALITY$ problem that is defined by: f_\text{GEQ}(x,y)=\cases{1&$x=y$\\0 & $x$ and $y$ differ in at ...
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### Is it possible to prove that this algorithm is big Omega $n^2logn$ time complexity?

Considering the following recursive algorithm: $T(n)= T(\frac{n}{2})+c_1(\frac {n}{2})^2+c_2n$. I was able to prove that this algorithm is $O(n^2 logn)$ I was trying to understand whether it is a ...
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### How to prove that the lower bound of the Huffman coding problem is $\Omega(n \log n)$?

how to prove that the lower bound of the Huffman coding problem is $\Omega(n \log n)$? Here Huffman coding problem is Huffman encoding. For example, ...
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### Information-theoretic lower bound for succinct string dictionary of the Unicode Name property

Background The literature on succinct data structures refers often to the “information-theoretic lower bound” of encoding data, i.e., the minimum number of bits needed to store the data – a concept ...
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### Finding upper bound on number of I/Os needed to generate all permutations of some input in external memory

The general approach outlined in this paper in its proof of the lower bound on the average number of I/Os needed to obtain a given permutation of some input in external memory is as follows. Note that ...
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### Techniques to prove lower bounds on randomized algorithms

I am interested in proving lower bounds for AM-like languages. The usual techniques for lower bounds in non-probabilistic machines don't work for probabilistic machines. Intuitively, when I think ...
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### "Equality" problem in distributed computation

I recently started learning about distributed computation on graphs (not to be confused with parallel computation with threads). I have seen as a side note in a few lower bound proofs, a reference ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 $T(n) = \Theta(n^4)$, ...