All Questions
Tagged with ds.algorithms lower-bounds
36 questions
1
vote
1
answer
91
views
Is there a conditional lower bound for the k max subarray sum problem?
Consider an array $A$ of integers of length $n$. The $k$-max subarray sum asks us to find up to $k$ (contiguous) non-overlapping subarrays of $A$ with maximum sum. If $A$ is all negative then this ...
- 3,912
4
votes
1
answer
188
views
Is there a lower bound for the problem of finding the best straight line partition
I recently asked the following algorithms question on another site. The best answer so far is $O(n^4)$ time. The input is of size $O(n^2)$ and the output is just a number so I was wondering if there ...
- 3,912
6
votes
1
answer
590
views
Find odd-ranked numbers from a list
From a list of $n$ distinct numbers, I want to find the set consisting of all odd-ranked numbers (1st, 3rd, 5th, ...). How many comparison queries do I need?
I could sort the whole list using $O(n\log ...
- 133
12
votes
2
answers
699
views
Quadratic lower bound
Consider three arrays $A,B,C$ of size $N$ consisting of integers. I want to verify the following constraint: for any two indices $0 \leq i,j < N$, $A[i] < A[j] \land B[i] < B[j] \implies C[i] ...
- 1,187
5
votes
1
answer
327
views
Deciding if all matrix multiplication entries have at least two witnesses
Consider two square matrices $A(x,y)$ and $B(y,z)$ of dimensions $N×N$ containing boolean entries. Consider the output product matrix $C(x,z)$ where $C=AB$ (not boolean matrix multiplication but the ...
- 1,187
2
votes
1
answer
210
views
Divide and Conquer Algorithm for 1-Median Problem
Let $P_1$ and $P_2$ be two disjoint point sets in $\mathbb{R}^d$ and $n = \vert P_1\vert = \vert P_2\vert$ and $P = P_1\cup P_2$. Let $c^\star$ be the optimal 1-median for $P$ and $opt^\star$ is the ...
- 265
1
vote
0
answers
71
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Decision tree vs. pebble game lower bounds
This question concerns two types of lower bounds. In a pebbling lower bound, we are concerned with the complexity of constructing the output from the input. For example, if the only way we could ...
- 851
2
votes
0
answers
182
views
Triangle detection hardness in regular graphs
Consider a tripartite graph over $n^{1-\epsilon}$ vertices each in sets $I, J, K$. Suppose we impose a constraint that every vertex has degree $n^\epsilon/c$ for some constant $\epsilon > 0$ and ...
- 1,187
0
votes
0
answers
40
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Maximum resistor with sublinear number of measurements
Consider a set $X = \{x_1, \dots, x_n\}$ of positive real numbers (or natural numbers, if you like) to be a set of resistors. For any subset $S \subset X$, we can build resistive circuits and measure ...
- 111
6
votes
0
answers
168
views
Computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$
This question is about computing and maintaining the minimum of a set $S$ of integers while allowing updates on $S$.
The computation model we are considering is the unit-cost RAM machine with linear ...
- 820
1
vote
0
answers
118
views
Entropy bounds on solutions to problems in BPP and other complexity classes based on entropy demands
Has anyone studied the asymptotics of problems in complexity classes like $BPP$? The thought came to me that if a problem in $BPP$ only requires $O(log(n))$ bits of entropy to solve then, intuitively, ...
- 1,213
1
vote
1
answer
98
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The SQ argument in Balazs Szorenyi's paper
I am asking about the proof in Theorem 5 (page 6) of this paper,
http://www.inf.u-szeged.hu/~szorenyi/Cikkek/sq_d0_ext.pdf
Quite a few things about this short argument seem unclear to me,
Towards the ...
- 1,453
7
votes
1
answer
184
views
Lower bound for enumerating k closest pair of points
Consider 1 dimensional points $x_1, \dots x_n, x_i \in \mathbb{R}$ where the distance between any two points is defined as $d(x_i, x_j) = \vert x_i - x_j \vert$. The goal is to enumerate all $n^2$ ...
- 1,187
3
votes
0
answers
100
views
Lower bound for reversing a list using queues
How do you prove (or disprove) that a list of length $n$ cannot be reversed in time $o(n \log n)$ using $O(1)$ queues?
Each queue is FIFO. Time refers to the number of operations on the queues.
...
- 2,742
1
vote
0
answers
68
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\alpha-path on Euclidean graphs
Consider the following problem:
Suppose we are given a G=(V, E) Euclidean Graph in the plane and a real $\alpha > 0$. For simplicity assume, there exists only one path whose summation of weights ...
- 11
5
votes
1
answer
157
views
Reference request: complexity of $k$-partite $k$-SAT
Let's consider following variation of $k$-SAT that I will call $k$-partite $k$-SAT:
given $n$ variables that are divided into $k$ groups and a $k$-SAT formula $\phi$ such that each clause has literal ...
- 901
5
votes
1
answer
260
views
Hardness of Subgraph isomorphism problem for sparse pattern graph
Subgraph isomorphism problem is a well studied problem: given graphs $G$ and $H$, one needs to answer if $H$ contains $G$ as a subgraph. It was proven that this problem requires $|H|^{\theta(|G|)}$ ...
- 901
6
votes
1
answer
405
views
How much memory is needed for counting distinct elements in a stream exactly with high probability
Assume we know a parameter $n\in\mathbb N$, and then get to observe a sequence of elements $x_1,\ldots, x_n$, one at a time.
Our goal is to count the number of distinct elements in $x_1,\ldots, x_n$, ...
- 9,508
8
votes
3
answers
279
views
Showing that interval-sum queries on a binary array can not be done using linear space and constant time
You are given a $n$-sized binary array.
I want to show that no algorithm can do the following (or to be surprised and find out that such algorithms exist after all):
1) Pre-process the input array ...
- 9,508
3
votes
0
answers
187
views
Finding median in a changing array
Consider the problem of needing to support an $n$ integers array structure with two operations:
Set(k,v) - set the $k$'th integer to value $v$ (i.e. $A[k]=v$).
Median() - return the median value of ...
- 9,508
0
votes
2
answers
159
views
Mergeable Exact Order Statistics Data Structure
Given $n$ sets of integers $S_1, S_2, \cdots, S_n$, it is guaranteed that
$$
x < y, \text{ for } \forall x \in S_i \text{ and } \forall y \in S_{i+1}
$$
and let's denote this relationship as $S_i &...
user22369
6
votes
0
answers
119
views
Lower bounds for randomized frequency estimation algorithms
Consider a stream of elements $s_1s_2\ldots s_N$.
A counter-based frequency estimation algorithm uses $m$ counters and is required to answer queries of the form "How many times did $x$ appear"?
It ...
- 9,508
2
votes
1
answer
463
views
Lower bound for finding repeated elements in sorted array
This is inspired by [1] (which still needs answers).
What is the tight lower bound (or optimal algorithms) for the "finding repeated elements" problem: Given a sorted integer array of size $n$, how ...
- 2,329
6
votes
1
answer
901
views
Lower bounds for inversion counting in comparison model?
For counting the number of inversions in an array, there are many $O(n \log n)$ algorithms, e.g. the one that modifies Merge Sort. There is an easy $\Omega(n)$ lower bound simply because you have to ...
- 63
1
vote
0
answers
141
views
Consequences of the existence of the following algorithm: does it imply any complexity class separation / collapse?
Let $G$ be a $3$-regular graph. Let $O$ be the number of vertex covers of $G$ having odd cardinality, and let $E$ be the number of vertex covers of $G$ having even cardinality. Let $\Delta = O - E$. ...
- 6,922
25
votes
3
answers
4k
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Nontrivial algorithm for computing a sliding window median
I need to calculate the running median:
Input: $n$, $k$, vector $(x_1, x_2, \dotsc, x_n)$.
Output: vector $(y_1, y_2, \dotsc, y_{n-k+1})$, where $y_i$ is the median of $(x_i, x_{i+1}, \dotsc, x_{i+k-...
- 11.6k
12
votes
0
answers
290
views
Hardness of optimal sorting
For comparison-based sorting algorithms, asymptotically optimal algorithms in worst-case $\Theta(n\log n)$ comparisons are well known.
From a purely theoretical perspective, however,
exactly optimal ...
- 1,095
11
votes
4
answers
656
views
Lower bound for testing closeness in $L_2$ norm?
I was wondering if there was any lower bound (in terms of sample complexity) known for the following problem:
Given sample oracle access to two unknown distributions $D_1$, $D_2$ on $\{1,\dots,n\}$, ...
- 4,491
3
votes
0
answers
222
views
Proving greedy algorithm is optimal for a scheduling problem
First, the problem discription:
For a sequence of $4n$ tasks, $a_1a_2\dots a_{4n}$, where $a_i\in\{0,1\}\forall i$, put them sequentially to the tail of one of the two initially empty queues of ...
- 131
9
votes
1
answer
704
views
On optimality of Grover algorithm with high success probability
It is well-known that bounded error quantum query complexity of the function $OR(x_1,x_2,\ldots, x_n)$ is $\Theta(\sqrt{n})$. Now the question is what if we want our quantum algorithm to succeed for ...
9
votes
1
answer
1k
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How to generate a permutation uniformly by repeating using an one-bit uniform random generator?
If I have an one-bit uniform random generator, how can I use it to generate a permutation uniformly for the sequence {1, 2, ..., n}.
I have a solution: run the one-bit random generator n*n times to ...
- 207
0
votes
1
answer
164
views
Functions and Counting Problems in Streaming Computation
I have read a stream computation paper in STOC07(Paul Beame, T. S. Jayram, and Atri Rudra. Lower bounds for randomized read/write stream algorithms.) and FOCS08（Paul Beame and Trinh Huynh. On the ...
- 327
6
votes
0
answers
222
views
Lowerbounds for in-situ permutation
What is the best known lowerbound for the worst case complexity of in-situ permutation (also called in-place rearrangement)? Has there been any reported progress after the 1970's article by Knuth (...
- 161
13
votes
2
answers
2k
views
Reversing a list using two queues
This question is inspired by an existing question about whether a stack can be simulated using two queues in amortized $O(1)$ time per stack operation. The answer seems to be unknown. Here is a more ...
- 1,478
7
votes
4
answers
1k
views
What are the best known upper bounds and lower bounds for computing O(log n)-Clique?
Input: a graph with n nodes,
Output: A clique of size $O(\log n)$,
Providing links to references would be great
- 21.1k
124
votes
18
answers
10k
views
Examples of the price of abstraction?
Theoretical computer science has provided some examples of "the price of abstraction." The two most prominent are for Gaussian elimination and sorting. Namely:
It is known that Gaussian ...