# Questions tagged [lower-bounds]

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### guillotine cuts versus general cuts

Cutting problems are problems where a certain large object should be cut to several small objects. For example, imagine you have a factory that works with large sheets of raw glass, of width $W$ and ...
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### How to analyze the lower bound of the horse racing problem using adversary argument?

Consider the following problem: Give an algorithm to find the $1^{st}, 2^{nd}, 3^{th}$ fastest horses from 25 horses. In each round, at most 5 horses can race and you can get the exact position of ...
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### Can you do an in-place reversal of a string on a vanilla turing machine in time $o(n^2)$?

By a vanilla Turing machine, I mean a Turing machine with one tape (no special input or output tapes). The problem is as follows: the tape is initially empty, other than a string of $n$ $1$s and $0$s ...
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### Lower-bound complexities for finding common elements between two unsorted arrays

I'm facing some problems that deal with finding common elements between unsorted arrays and I'd like to know whether there are well-known lower-bounds for the worst-case and, eventually, what are ...
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### Time complexity in Big O notation for Harmonic series with first k terms missing

Firstly, let's suppose there exists an algorithm where $i$ iterates from $1$ to $n$, spending $\frac{n^2}{i}$ time in each iteration. Thanks to the well known $O(\log n)$ upper bound for the Harmonic ...
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### A Problem on Time Complexity of Algorithms

For every integer $t$, is there a problem whose solutions can be verified in $O(n^{s})$ time but cannot be found in $O(n^{st})$ time? By verifying, I mean that given a candidate solution $y$, we can ...
857 views

### Lower bound for sorting n arrays of size k each

Given $n$ arrays of size $k$ each, we want to show that at least $\Omega(nk \log k)$ comparisons are needed to sort all arrays (indepentent of each other). My proof is a simple modification of the ...
532 views

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### Search spaces and computation time

This question follows on previous questions (1), (2), where we define an initial space of possibilities and reason about how a solution is chosen from that. Consider a problem P where we are given: ...
789 views

### Input that causes an operation on a binomial heap to run in $\Omega(\log n)$ time?

I was studying binomial heaps and its time analysis. Are there any inputs that cause DELETE-MIN, DECREASE-KEY, and DELETE to run in $\Omega(\log n)$ time for a binomial heap rather than $O(\log n)$?
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### Progress of algorithms in problem spaces

Continuing in the vein of two prior questions (1) and (2), we started with sorting, where we had a set of $n!$ input possibilities a goal space of only one element consisting of the one correct ...
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### Lower bound on size of proof that a list of integers is sorted

Suppose we have a list of unbounded integers, written in binary, and we want to write a (formal) proof that the list is sorted in ascending order. Such a proof might look (informally) like: "2 < 3,...
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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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### Proofs based on narrowing down sets of possibilities

Consider the argument made in this question based on the comparison sorting lower-bounds proof, which runs as follows. First, the comparison sorting lower-bounds proof was recited: For $n$ distinct ...
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### Is there any nontrivial problem in the theory of serial algorithms with a nontrivial polynomial lower bound of $\Omega(n^2)$?

In the theory of distributed algorithms, there are problems with lower bounds, as $\Omega(n^2)$, that are "big" (I mean, bigger than $\Omega(n\log n)$), and nontrivial. I wonder if are there problems ...
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### 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 ...