Questions tagged [time-complexity]

The amount of time resources (number of atomic operations or machine steps) required to solve a problem expressed in terms of input size. If your question concerns algorithm analysis, use the [runtime-analysis] tag instead. If your question concerns whether or not a computation will *ever* finish, use the [computability] tag instead. Time-complexity is perhaps the most important sub-topic of complexity theory.

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25
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639 views

Largest set of cocircular points

Given $n$ points with integer coordinates in the plane, determine the maximum number of points that lie on the same circle (on its circumference, not its interior). This can be done in $O(n^3)$ ...
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588 views

Complexity of deciding whether there is a winning strategy in the following game

The sum divider game for $n$ starts with the set $M_0 = \{1,\dots,n\}$. Player A chooses a number $m_1$ from $M_0 \setminus \{1\}$ and B has to choose a divider $m_2$ of $m_1$ from $M_1 = M_0 \...
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299 views

P vs NP and the Time Hierarchy

Assuming $P\neq NP$, is it possible that there exists a $k$ such that $P\subseteq\textsf{NTIME}(t^k)$? There reason I ask this is that I assume the following: $$P=NP \implies \forall k\ \exists j.\ \...
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206 views

Complexity of removing edges to eliminate a perfect matching

Suppose $G$ is a bipartite graph which has a perfect matching. I want to find the fewest number of edges to delete from $G$ so that a perfect matching no longer exists. What is the complexity of this ...
8
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968 views

Complexity of Sorting Integers on a Multitape Turing Machine

How expensive is sorting integers on a Multitape Turing Machine? Well known sorting algorithms, like quicksort, tend to rely on jumping / indirect-access being cheap. But MTMs have no indirect access.....
8
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2answers
778 views

Is there a data structure that can find the kth smallest in constant time with logarithmic add and delete operations?

I'm looking for a single or a conjunction of data structures that can find the kth smallest element in constant time, delete the kth smallest element in logarithmic time, and add a new element in ...
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161 views

Overlap Maximization problem

Here's the problem: I have a collection of collections, $C$, where each $c\in C$ is a collection of sets $X\subset U$. Denote $c_i$ as the i-th $X$ in $c$. Informally, I want to map all the sets in ...
6
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286 views

Prove/disprove the existance of a data structure that has O(log N) inserts/deletes and get k-th largest element in O(1)

Consider a sorted array. We can get the $k$-th largest element in $O(1)$, but insertions and deletions cost $O(n)$. Consider an order statistic tree. Insertions and deletions cost $O(\log{N})$, but ...
6
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119 views

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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335 views

Determinant calculation - Bareiss vs. Gauss Algorithm

I've been working on a matrix-library in C++ for a while and amongst other functions, I've implemented two functions for calculating the determinant of a matrix: Gauss-Algorithm: This algorithm is ...
5
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0answers
86 views

Given $n=pq=a^2+b^2$, can we factor $n$?

Just to be clear, $a$ and $b$ are known, while $p$ and $q$ are unknown prime numbers, both congruent to $1$ modulo $4$. Can we design an efficient algorithm to retrieve $p$ and $q$? It is a known ...
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103 views

Specific quadratic 0-1 knapsack problem solvable in linear time?

I am interested in a simple variant of the quadratic knapsack problem. Let $\{w_1, \ldots, w_n\} \in \{0,1\}$ be $n$ weights and $\{v_1, \ldots, v_n\} \in \mathbb{R}$ be $n$ values. Furthermore, ...
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2k views

Time Complexity of a Knapsack-derived problem

Consider the following problem: Let there be a set A of $n$ items $A=\{z_1, ..., z_n\}$, and let $W$ be a strictly positive integer. Each item $z_i$ has a value $v_i$ and a weight $w_i$. Finding a ...
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710 views

Non-deterministic time hierarchy theorem: universal TM overhead

I am currently reading the book of Arora and Barak on computational complexity. In the third chapter (p69-70), two classic theorems regarding time complexity hierarchies are introduced: $\left[f(n)\...
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688 views

Time complexity of finding the largest factor of a number (using a specific oracle)

My question is related to this question posted on math.SE: Given an odd number, what is the quickest (constant-time) algorithm for finding its largest factor and suppose you can call a helper ...
4
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47 views

Simultaneous reachability of NFA states

Suppose I have a $n$-state non-deterministic finite automaton $F$ over alphabet $\Sigma$. Let $S(x)$ be the set of states reachable from the starting state by consuming string $x$. I am interested for ...
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58 views

Algorithm for Unique Selections

Suppose I have $k$ sets with $n$ elements in each. Define a selection as one element taken from each set. A selection is unique if there's one and only one way it can happen—that is, one and only one ...
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49 views

Highest stack of rectangles

Suppose we have a set of $n$ dimensional rectangles $R = \{(x_{i,1}, \ldots, x_{i,n}), i \in 1 \ldots k\}$. We want to create the highest stack in say the first dimension such that each side of the ...
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376 views

Time complexity of obtaining the set of distinct elements in a sequence?

Consider a sequence $s$ of $n$ integers (let's ignore the specifics of their representation and just suppose we can read, write and compare them in O(1) time with arbitrary positions). What's known ...
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103 views

How hard is recovering an invertible binary matrix from its check sums?

This is a follow-up question on my previous one, How hard is recovering a binary matrix from its check sums?. Consider the following problem, which adds an extra restriction that the matrix be ...
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48 views

Complexity class of finding the number of walks of length $k$ that have different vertex sets

Vertex set $A$ is of the form: $A = \{(v_1,r_1),(v_2,r_2),...\}$ where $v_1 \in V$ and $r_1$ refers to the number of times $v_1$ is reached in some walk and $v_j \neq v_i$ whenever $i \neq j$. ...
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69 views

PTAS vs. exact-time sub-exponential algorithms

I have recently summarized several algorithms for the maximum disjoint set problem. This problem is NP-hard, but it has both PTAS and sub-exponential algorithms. These algorithms seem to me closely ...
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728 views

What is the complexity of Hoffman and Pavley's Nth best path algorithm?

I am currently working on a project where I'm using an implementation of Hoffman and Pavley's "Method for the Solution of the Nth Best Path Problem" to find n-th best path through a directed graph. ...
3
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38 views

Make Change in Linear Time

The question is motivated by this post on StackOverflow. Given an integer $n$ and a finite list of distinct positive integers $ds$, let $f(n, ds)$ denote the number of ways $n$ can be expressed as a ...
3
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0answers
31 views

A simple algorithm to solve the MST Sensitivity Analysis problem in linear time when the MST is a path

The problem. Given an undirected, connected, edge-weighted graph $G=(V, E_G; w)$ and a minimum spanning tree (MST) $T=(V, E_T)$ of $G$, the MST sensitivity analysis problem asks to find, for each ...
3
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0answers
18 views

Spanning hypertree which connects the vertices as slowly as possible

I want to find a reference for the following problem or a similar problem for my paper. I found a greedy algorithm for this problem, but writing such an algorithm in a paper is not common in my area, ...
3
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0answers
34 views

What's the best non-amortized disjoint set?

In practice, the amortized O(α(n)) data structure is good for every case. But if I want to be pedantic and require each operation to be under a certain time complexity, what's the currently known best ...
3
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24 views

Bit complexity of computing the sign of an expression evaluated at an algebraic number

I have a univariate polynomial $F(t)\in \mathbb{Z}[t]$ of degree $d$ and maximum bitsize of coefficients equal to $\tau$ and $G(t) \in \mathbb{Z}[t]$ of degree $d^2$ and maximum bitsize of ...
3
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26 views

Complexity of finding an alternating Hamiltonian (x,y)-path in edge bicolored complete graphs

Let $G$ be a simple complete graph with an edge-2-coloring. An alternating Hamilton (x,y)-path is a Hamiltonian path which starts at vertex $x$ and ends at vertex $y$ such that the colors of its ...
3
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63 views

Is the definition of $\textbf{BPP}$ robust for doubly exponential small (or even smaller) error?

$\textbf{BPP}$ is usually defined in terms of probabilistic polynomial-time TMs which have an error probability of at most $\frac{1}{3}$. Furthermore, using the Chernoff bound it can be proven that ...
3
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84 views

Polynomial time algorithms for rank 1 elliptic curves over Q

As an outsider, it sounds like a lot of progress has been made on understanding rank 1 elliptic curves. Much of the BSD conjecture is known for rank 1, and Heegner points provide a way to calculate a ...
3
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1answer
116 views

Survival algorithm for Network deterministic failures

Consider an undirected network $G = (V,E)$ in which edge $e$ $\in$ $E$ fails after (deterministic) time $t(e) > 0$. Network failure occurs at the first instant in which $G$ is no longer connected. ...
3
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0answers
215 views

Is it possible to compute an equality hash for nodes in a *cyclic* directed graph in less than quadratic time?

Calculating hashes for nodes in an acyclic graph is well known using a Merkle tree. With some simplifying assumptions, a simple algorithm will also calculate hashes for nodes in a cyclic graph... but ...
3
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0answers
59 views

Computational complexity of numerically estimating the roots of a polynomial

The Wikipedia article on finding the roots of polynomials mentions all sorts of methods to do so. But it doesn't give, nor can one easily figure out by following the links, known lower and upper ...
3
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0answers
1k views

Analysis of Weighted Quick Union with Path Compression

I have searched the internet for an analysis of why WQUPC is amortized $O( m \alpha (n) ) $ for m operations on n nodes ( $\alpha ( n) $ is the inverse Ackerman function). I understand why it is $O ( ...
3
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0answers
108 views

Are there any algorithms where the recovery of a witness changes the time complexity?

In many algorithms, such as the solution to the longest-subsequence problem using dynamic programming, finding the length of an answer (or signaling the nonexistence of an answer) is easy, but ...
3
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2answers
492 views

Algorithm for scheduling unit time tasks with arrival times and deadlines

Suppose we have $n$ tasks to order over $n$ days. Each tasks takes 1 day to be completed. Each task has a start date when the task becomes available and a deadline when the task must be delivered. ...
3
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1answer
252 views

Theoretical worst case running time of finding a path through a maze?

Given a randomly generated maze of dimensions n x n, with the entrance point always being the top left corner (0,0) and the exit point always being the bottom right corner (n,n) what is the ...
2
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0answers
42 views

Two dimensional recursive function in $O(\log n)$ time complexity

It is well known that a recursive sequence or $1$-d sequence can be calculated in $O( \log n)$ time given that it has the form $$a_n=\sum_{k=1}^{n} C_ka_{n-k},$$ where $C_k$ is a constant. Examples ...
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0answers
60 views

P/NP - Proof that SAT-TM is NP-complete uses certificate

To prove that SAT-TM (Turing machine emulating the satisfiability problem) is NP-Hard $$\text{SAT-TM}:=\{⟨M,p,1^k⟩ \; | \;∃c,\; |c|\leq p(k), \;\text{such that M accepts c in ≤k steps}\}$$ my ...
2
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0answers
24 views

Central Trinomial Coefficients best time complexity

What is the fastest known time complexity for computing central trinomial coefficients? Let $C_n=1,1,3,7,19,51,...$ (OEIS A002426) denote the coefficient of $x^n$ in $(x^2+x+1)^n$ starting at $n=0$. ...
2
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2answers
84 views

Examples of higher order algorithms ($\mathcal{O}(n^4)$ or larger)

In most computer science cirriculums, students only get to see algorithms that run in very lower time complexities. For example these generally are Constant time $\mathcal{O}(1)$: Ex sum of first $n$ ...
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0answers
41 views

Complexity of find histogram bins vs convex hull

For a list of n 2d points, finding the convex hull vertex takes O(n log(n)) time. And O(n) time if it’s sorted lexicon order. Meanwhile What’s the complexity of finding the histogram bin edges of k ...
2
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0answers
38 views

Fastest Algorithm for finding All Pairs Shortest Paths on Sparse Non-Negative Graph

As discussed here Johnson's Algorithm can be used to solve the APSP-Problem in $O(V^2\log V + VE)$ in stead of $O(V^3)$ for Floyd-Warshall. However, Johnsons Algorithm does quite a bit of work for the ...
2
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0answers
32 views

Are there problems that are known to be in ZPP but not in p

Are there any problems that are known to be in ZPP but not in p?
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0answers
57 views

What is the time complexity of the Edmonds-Karp algorithm for finding a maximum cardinality matching in bipartite graphs?

What is the time complexity of the Edmonds-Karp algorithm (not the Hopcroft-Karp algorithm) for finding a maximum cardinality matching in bipartite graphs? Is it still $O(|V||E|^2)$, or it has a ...
2
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0answers
27 views

Can every node of a link/cut tree be accessed in $O(n)$ time?

Per the Sequential Access Theorem we can access every node of a splay tree in $O(n)$ time, when accessing the nodes in a specific order. Given a link/cut tree, is it possible to access all of its ...
2
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0answers
28 views

planar max cut graph with constrains

Given a planar graph $G=(V, E)$ I am looking for a max cut algorithm with the following conditions : some vertices are in one of the partition sets? Is the algo is still polynomial ? I mean a ...
2
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0answers
22 views

Poset data structure to find least element, greater or equal to given

Let $A$ be a finite set, and $S \subset \mathcal{P}(A)$. Is there a data structure for $S$ that would allow to quickly retrieve an element $q \in S $, given a key $p \in \mathcal{P}(A)$, such that $q$ ...
2
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0answers
69 views

Consider vectorization and for loop, do both approaches have the same time complexity?

I am learning this post Fast computation of nearest neighbors is an active area of research in machine learning. The most naive neighbor search implementation involves the brute-force computation ...

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