# Questions tagged [upper-bound]

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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 ...
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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/...
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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 ...
48 views

### 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 ...
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### Quickly obtaining sums of sets of numbers

We are given a set of $n$ bits, call them $a_1$, $a_2$,...,$a_n$. We are also given a set of $m$ sums, where the sums $s_1$, $s_2$,...,$s_k$,...,$s_m$ are given as sums of some of the bits. For ...
24 views

### 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 ...
112 views

### Maximum value reached in extended binary GCD

Given positive integer inputs $x$ and $y$ , with $0<x<y$ and $y$ an odd prime (or $\gcd(x,y)=1$ and $y$ odd), the following algorithm computes $x^{-1}\bmod y$ per the (half-)extended binary GCD. ...
44 views

### 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 ...
67 views

### What is the largest number of Byzantine failures that can be tolerated in an $m$-dimensional hypercube for the consensus problem?

We are given a system with $n$ nodes which have been arranged into the topology of a hypercube of $m$ dimensions. I would like to derive a tight bound on the maximum number of Byzantine failures that ...
29 views

### Algebra for min/max bounds

I am trying to model some set operations which are only well-defined if one is a subset of the other. The way the sets are constructed, I'll have a series of constraints of the form $x \subseteq y$, ...
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### How to find sets of polynomially bounded numbers whose subset sums are different?

Let $n$ be any positibe integer and set $N=\{1,\dots,n\}$. Now select two arbitrary but different subsets of $N$, say $S$ and $S'$. We are interested in finding a function $\pi(A)=\sum_{i\in A}a_i$ ...
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### Communication Complexity for Product Distributions

In general for the (two-party) set disjointness problem for inputs of length n, we know that the parties need to communicate $\Omega(n)$. Surprisingly, today I discovered (if I understood correctly) ...
38 views

### Asymptotics of a sinusoid

Consider the function $$f(n) = 2n^2 |\sin(\pi \cdot n/2)|.$$ Which of the following classes does $f(n)$ belong to? $$O(n^2), \Omega(n^2), \Theta(n^2), \omega(n^2), o(n^2).$$ I'm working in this ...
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### Upper bound for runtime complexity of LOOP programs

Recently I learned about LOOP programs, which always terminate and have the same computational power as primitive recursive functions. Furthermore primitve recursive functions can (as far as I ...