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Questions tagged [upper-bound]

68 questions
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Upper bound of of fib(n+2)

I have a homework problem that is perplexing me because the math is beyond what I have done, although we were told that it was unnecessary to solve this mathematically. Just provide a close upper bound and justify it. Let $$f(n) = |\{w ∈ \{a,…
Wawa
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Can we compute in polynomial time, an upper bound on an optimal solution of an integer linear program?

Consider the following integer linear program (where $A$ is an integer matrix, $b$ an integer vector, and $c$ a positive integer vector): $$ \text{minimize}~~~ c\cdot x \\ \text{subject to}~~~ A\cdot x \geq b, ~~~ x\geq 0, ~~~ x~\text{is an integer…
Samuel Bismuth
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What is the runtime of the 'Risch Algorithm'?

I have been trying to find the upper bound on the runtime of the 'Risch algorithm' used for finding the integral of mathematical functions, but have been unable to do so. https://en.wikipedia.org/wiki/Risch_algorithm Does anyone know where I would…
Gurpreet Handa
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Finding the hidden treasure

Let's assume I am trying to find a hidden treasure. The treasure is hidden at an uknown position x. We know that the position x of the treasure is somewhere on the integer axis (in other words x is an integer). In order to find the treasure I have…
MJ13
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The upper bound on a Nondeterministic Finite Automata's configurations number

In "Engineering: A Compiler" 2nd ed. by Cooper and Torczon, in 2.4.1 "Nondeterministic Finite Automata" of Chapter 2, section "Equivalence of NFAs and DFAs" discusses the upper bound of a NFA's configurations number. Here NFA stands for…
Myk
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Maximum number of cliques of size $\ge 2$ of a graph with exactly $m$ edges

Let $G=(V,E)$ be an undirected graph with $|E|=m$ edges. Given that any clique of size $\ge 2$ can be identified with its corresponding edges, and at most every subset $S\subseteq E$ creates a clique, $G$ can have at most $|P(E)|= 2^m$ cliques of…
Sudix
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Upper bounding randomized k-SAT solver

Problem: Consider a k-SAT solver that assigns values independently uniformly at random to all the variables. Given the formula has $m$ clauses provide an upper bound for the probability of a random assignment to be unsatisfiable. The issue is that…
ddnomad
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Sum of size of distinct set of descendants $d$ distance from a node $u$, over all $u$ and $d$ is $\mathcal{O}(n\sqrt{n})$

Let's consider a rooted tree $T$ of $n$ nodes. For any node $u$ of the tree, define $L(u,d)$ to be the list of descendants of $u$ that are distance $d$ away from $u$. Let $|L(u,d)|$ denote the number of nodes that are present in the list…
Banach Tarski
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Number of planar graphs, given an embedding

I want to find an upper bound on the number of planar graphs with $n$ vertices, assuming that we are given some embedding for those vertices beforehand. In particular, Im interested in either showing for every embedding there is $2^{o(n\log(n))}$…
nir shahar
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Upper bound for size of minimal covers of a set

Would appreciate any insight about the following regarding set covers: Begin with a universe set $X = \{x_1,x_2,...,x_n\}$ and a set $S=\{s_1, s_2,...,s_p\}$ such that each $s_i \subseteq X$ and $\bigcup S = X$. Consider the task of finding all…
Robert Rovetti
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Anagram sorting with inversion count oracle

Given a permutation $P$ of an unknown array $U$ of length $N$ and a function $f(Q)$ that calculates the minumum number of swaps between consecutive elements of array $Q$ to reach $U$, what is the minimum number of invocations of $f$ to discover…
chubakueno
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Maximum value reached in extended binary GCD

Given positive integer inputs $x$ and $y$ , with $0
fgrieu
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Tight bound on the number of intersections between a line and a triangulation

I'm interested in the maximum number of intersections that a line and a triangulation on $n$ points could have. More specifically, given $n$, we are interested in the worst-case (maximum) number of intersections that any line could have with any…
PlsWork
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What's the reason for sqrt(n) bounds in online learning?

I have a question regarding no-regret algorithms (of online learning). As far as I can see, such algorithms allow the absolute regret up to round $n$, which is $R_n$, to grow by $\sqrt{n}$. So, in the limit, the per round regret ($R_n/n$)…
Christian
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Finding the (probable) maximum of a large set of integers *without* iterating over all of the values

As in the title, I am trying to find the largest (aka least upper bound) of a (very large) set of integers. Importantly, I do not have direct access to the full list of integers, but I do have a function $f(n)$ which returns true/false if $n$ is in…
R. Granton
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