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# Questions tagged [lower-bounds]

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### Is it really possible to prove lower bounds?

Given any computational problem, is the task of finding lower bounds for such computation really possible? I suppose it boils down to how a single computational step is defined and what model we use ...
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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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### How to prove that matrix multiplication of two 2x2 matrices can't be done in less than 7 multiplications?

In Strassen's matrix multiplication, we state one strange ( at least to me) fact that matrix multiplication of two 2 x 2 takes 7 multiplication. Question : How to prove that it is impossible to ...
645 views

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### How to prove that matrix inversion is at least as hard as matrix multiplication?

Suppose we are given a matrix $A$ over real numbers and we want to computer the inverse of matrix $A$. There are various algorithms to do so and it also turn out that we can use matrix multiplication ...
2k views

### Lower bounds: queues that return their min elements in $O(1)$ time

First, consider this simple problem --- design a data structure of comparable elements that behaves just like a stack (in particular, push(), pop() and top() take constant time), but can also return ...
277 views

### Is there an intuitive proof for the existence of hard functions?

I am referring to the theorem on page 115 of the book by Arora and Barak, which states that, For every $n>1$, there exists a function $f:\{0,1\}^n \rightarrow \{0,1\}$ that cannot be computed by ...
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### Space complexity below $\log\log$

Show that for $l(n) = \log \log n$, it holds that $\text{DSPACE}(o(l)) = \text{DSPACE}(O(1))$. It's well known fact in Space Complexity, but how to show it explicitly?
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### Does finding a cycle with $\log n$ length in $\text{P}$?

Let $G$ be an arbitrary graph with $n$ vertices and we want to find a simple cycle with $\log n$ length. Is there exists a known polynomial algorithm for this problem?
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### (Nontrivial) Algorithms for finding the third largest element of a set

According to the lecture note by Jeff Erickson, the lower bound for finding the third largest element of a set of $n$ distinct elements is open. See the related post: What is the lower bound for ...
180 views

### Are there any known lower-bounds for complexity on Non-determinsitic machines

For some problems, like sorting, we know that on a deterministic RAM Machine, any comparison sort must take at least $\Omega(n\log n)$ time. Are they any problems where we have known lower bounds for ...
871 views

### Lower bound for finding majority element in a sorted array

Suppose $A$ is a sorted array with $n$ elements. I want to know whether we can determine if there are majority elements in $A$ with time complexity $O(1)$. Recall that a majority element of $A$ is ...
943 views

### Generalizing the Comparison Sorting Lower Bound Proof

Let's start with the comparison sorting lower bound proof, which I'll summarize as follows: For $n$ distinct numbers, there are $n!$ possible orderings. There is only one correct sorted sequence of ...
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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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### TM recognizing $0^n1^n$ requires Ω(log n) space

I am trying to prove that any deterministic 1-tape Turing Machine which recognizes the language $L = \lbrace{0^n1^n | n \geq 0 \rbrace}$ requires $\Omega(\text{log }n)$ space. I believe this can be ... 119 views

### Determining if $G$ contains $K_4$ as a minor in polynomial time

I am trying to devise an algorithm for determining if an undirected graph $G$ contains $K_4$ as a minor. I was able to show in a previous problem how to test for $K_{2,3}$ by looking at all pairs of ... 202 views

### 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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### Problems with Θ(n³) complexity on TMs with lower bounds by communication complexity arguments

One of the most used simple examples of application of Communication Complexity is the $\Omega(n^2)$ lower bound for recognizing palindromes of length $2n$ on a single tape Turing machine. Is there a ...
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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 ...
863 views

### Is Green's the best 16-input sorting network so far?

Every paper says that Green's construction is the best 16-input sorting network as for now. But why does Wikipedia says: "Size, lower bound: 53"? I thought "lower bound" meant:"If there exists at ...
664 views

### space complexity of DFA intersection problem

the DFA-intersection computation problem, given two DFAs specified on the input, compute the intersection DFA, or the decision problem to determine its emptiness, turns out to have wider/ deeper ...
363 views

### Linearithmic lower bound for 1-D "distinct" closest pair of points problem

The 1-D distinct closest pair of points problem is as follows: Given a set of n distinct integer points on real line, find a pair of points with the smallest distance between them, here the distance ...