Proving that the Bellman-Ford algorithm contains negative circuit

Let $ D=(V,B), n=|V|$ be a directed graph. Then the graph contains a circuit of negative length from $ s$ if and only if $ f_n(v) \neq f_{n-1}(v),$ where $ v \in V,$ and $ f_k(v)=$ min$ \{l(P)|P$ is an $ s-v$ walk traversing at most $ k$ arcs }.

I do not understand what a $ s-v$ walk means and what is the meaning of the function $ f_n$ .

Can somebody help me prove the two directions of the above statement ? Many thanks.

Strongly connected subgraph that contains no negative cycles

Is there an algorithm that solves the following decision problem:

Given a strongly connected weighted directed graph G, defined by its transition matrix, is there a strongly connected spanning subgraph of G that has no negative cycles?

A strongly connected spanning subgraph of G is a strongly connected subgraph of G that shares the same vertexes with G. You can look to this paper for the definition of strongly connected spanning subgraph. In this paper they present an approximation for the minimum strongly connected subgraph problem, wich is different from the problem stated in this question.

A naive approach to this problem is to find a negative cycle of the graph using the Ford-Bellman or Floyd-Warshall algorithm, deleting an edge from this cycle, and repeating while the graph is still strongly connected. But this naive approach has poor time complexity because we will potentially run the Ford-bellman algorithm and check for strong connectivity many times — moreover I am unable to prove if this algorithm is correct in all instances.

I’m hoping to find experts here who can tell me if this decision problem can be solved in a polynomial time and what algorithm does so. Many thanks in advance.

What is dysfunctional, controversial, or negative about Epic skill checks?

I often see it said that the entire Epic Level Handbook should be dismissed as broken and frequently dysfunctional. However, outside of claims that the entire book is bad, I never see Epic skill checks listed as an example of something negative, controversial, or objectionable. The only near example that I can think of is Diplomancy, but that’s an issue that was largely inherited from Core. There are of course some absurd examples like the Arseplomancer, but Epic levels are all about performing absurd feats of skill.

This leads me to my question – it is frequently said that the entire Epic Level Handbook is bad, but what specifically is dysfunctional, controversial, or often seen as negative about Epic skill checks?

Minimum number of bits to represent negative number

Minimum number of bits required to represent $ (+32)_{base10}$ and $ (-32)_{base10}$ in signed two’s compliment form?

My attempt:

32 = 0100000 ( 1st zero – sign bit as positive)

So to represent +32 we need 7 bits

-32 = 1100000 (1st bit 1 – sign bit as negative)

So to represent -32 we need 7 bits

But answer is given as 6 bits. His reason – one 1 bit enough to represent negative number. I am confused. Please clarigy here

Also i have following Questions:-

Can we say number of bits required to represent a negative number is strictly less than( or less than equal to) number of bits required to represent that corresponding positive number?

how can we generalise minimum number of bits required to represent a given positive and negative number in signed magnitude representation, signed 1’s complement notation and signed two’s compliment notation.

I know that minimum number bits will be of order of logn to base 2. But exactly how much i am not able to think.

I know range of numbers in signed magnitude and signed one’s complement is $ -(2^{n-1} – 1) $ to $ +(2^{n-1} – 1) $ while range of numbers in signed two complement representation is $ -(2^{n-1}) $ to $ +(2^{n-1} – 1) $

Negative words in url/domain

I have added a negative word in ser and ran the project but still many links got built having the neg word in url and domain. I have tried running multiple projects but still happening the same.

Also in using domain extension skip filters like !edu!gov, still lots of redirects got build which dont have the edu or gov extensions.

Actually facing this from a long time but was trying to put some time in manual links removal but still I am reporting to get it checked. I hope it will be checked and worked on. Thanks

What are the smallest and biggest negative floating point numbers in IEEE 754 32 bit?

I am stuck with a question that asks for smallest and biggest negative floating point numbers in IEEE 754 32-bit (their representation and decimal numerical value from which one can approximate the precision of the number)? So -0, NaN and Infinity do not belong to negative rational numbers.

I have stumbled upon -3.403 x 10^38 and 2^-126. I came close to the first one actually. I tried to do some calculations but got kind of lost in the process as floating point representation is counter-intutive for me, especially when calculating negative numbers. Can someone help me to clarify my thought process for the calculations so that I can find the numbers?

How do I present ‘negative’ sounding information without discouraging the user?


I’m currently working on a project that essentially will become a website that guides the user to the optimal recruitment strategy for their specific hiring needs.

Before displaying the recommended path for the generated recruitment plan, I want to display an infographic that shows how difficult / easy it is to recruit for the position that the user has specified (based on data gathered from thousands of employers twice a year).

User testing shows that displaying information such as “Employers in XX county think it’s difficult to recruit for the YY position” can feel discouraging for the user to see, while sometimes it’s a good wake up call.

While it’s possible in some cases that the infographic tells the user that it’s easy to hire, it’s almost always at least challenging.

Our goal is to inform the user that they can’t expect to hire over night, which other user testing seem to imply that our users think. While simultaneously inspiring the user to use the recommended plan to make it easier.


How do I inform the user that it’s going to be difficult to hire for the chosen position, without discouraging the user from trying?


How do I inspire the user do perform an action that isn’t going to be easy?