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Questions tagged [lower-bounds]

262 questions
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3 answers

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 for the proof, but given that, do we really prove…
hsalin
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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 outcomes. Therefore a binary tree for the sorting…
PleaseHelp
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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 multiply two 2 x 2 matrices in 6…
Complexity
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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 - first priority is to have as simple as possible lower…
R B
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1 answer

Is detecting "doubly" arithmetic progressions 3SUM-hard?

This is inspired by an interview question. We are given an array of integers $a_1, \dots, a_n$ and have to determine if there are distinct $i \lt j \lt k$ such that $a_k - a_j = a_j - a_i$ $k - j = j - i$ i.e, the sequences $\{a_i, a_j, a_k\}$ and…
Knoothe
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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 length $L$. There are several buyers, each of…
Erel Segal-Halevi
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15
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4 answers

Is every linear-time algorithm a streaming algorithm?

Over at this question about inversion counting, I found a paper that proves a lower bound on space complexity for all (exact) streaming algorithms. I have claimed that this bound extends to all linear time algorithms. This is a bit bold as in…
Raphael
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Bound on space for selection algorithm?

There is a well known worst case $O(n)$ selection algorithm to find the $k$'th largest element in an array of integers. It uses a median-of-medians approach to find a good enough pivot, partitions the input array in place and then recursively…
user834
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Is there a data-structure which is more efficient than both arrays and linked lists?

Background: In this question we care only about worst-case running-time. Array and (doubly) linked lists can be used to keep a list of items and implement the vector abstract data type. Consider the following three operations: $Location(i)$:…
Kaveh
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9
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$O(\frac{\log n}{\log \log n})$ algorithm for the prefix parity problem

The prefix parity problem can be defined as follows. You are given a string $S$ of length $n$ and initially every character is $0$. Then you want to build a data structure that can support updates such as follows. For a given $i$ change $S[i]$ to…
jsguy
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2 answers

Find the central point in a metric-space point set, in less than $O(n^2)$?

I have a set of $n$ points which are defined in a metric space – so I can measure a 'distance' between points but nothing else. I want to find the most central point within this set, which I define as the point with the minimum sum of distances to…
Open Door Logistics
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Is integer sorting possible in O(n) in the transdichotomous model?

To my knowledge there doesn't exist a $O(n)$ worst-case algorithm that solves the following problem: Given a sequence of length $n$ consisting of finite integers, find the permutation where every element is less than or equal to its successor. But…
orlp
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9
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4 answers

Can element uniqueness be solved in deterministic linear time?

Consider the following problem: Input: lists $X,Y$ of integers Goal: determine whether there exists an integer $x$ that is in both lists. Suppose both lists $X,Y$ are of size $n$. Is there a deterministic, linear-time algorithm for this problem? …
D.W.
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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 understand that adversary arguments are usually…
user5507
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Optimal upper bound on the number of states in the complement of an NFA

I have my own version of lex and I would like to add the complement operation. Derived from that I can then add the intersection and difference also. My version also supports the generation of NFAs (Non-deterministic Finite Automaton) and of course…
Bryan Olivier
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