Capacity of a discrete memoryless channel

For an integer $ I$ , the input-output relationship of a discrete memoryless channel is given by:

$ Y = X + Z$ (mod I, i.e. sum indicates a modular addition)

where $ I ≥ 2$ , and

• X is an integer chosen from the alphabet Ax = {1,…,2I},

• Z is noise which is a uniform Bernoulli random variable. This means that Az = {0,1}, and

Pr{Z = 0} = Pr{Z = 1} = 0.5.

How can we calculate the capacity of this channel?

0-1 knapsack problem with minimum and maximum weight capacity

In classical 0-1 knapsack problem we have maximum allowed value for the weight – weight capacity.

Let’s restrict total knapsack weight by min and max values

$ $ M \leq \sum_{i=1}^{n}{w_i x_i} \leq W $ $

Is there any known algorithm for this problem? Is there anything known which is better than brute force?

I failed to find anything about the given knapsack problem variation.

Carrying Capacity and Spells

So, as I go into detail on the spells I noticed something that I hadn’t noticed before. For example, with Fly, it states:

The subject of a fly spell can charge but not run, and it cannot carry aloft more weight than its maximum load, plus any armor it wears.

I had always assumed that armor weight counted toward the carrying capacity and so it often pushed characters into carrying a medium load on its own. However, this implies it might not. In the starting of the Adventuring chapter, however, it explicitly states:

If you want to determine whether your character’s gear is heavy enough to slow him or her down more than the armor already does, total the weight of all the character’s items, including armor, weapons, and gear.

Are one or other of these a misprint or am I missing something in understanding why it would say the gear plus the weight of armor? Can encumbered characters become unencumbered by a flight spell, if subtracting the weight of their armor makes the difference, for example?

Should weight of PC itself counts at weight capacity calculation?

Imagine dwarf Alice (200 pounds) and gnome Bob (40 pounds). Both characters have 10 Strength.

So, as PHB says, both of them are able to carry 10 × 15 = 150 pounds.

Does it mean that Bob’s effective capacity is 150 – 40 = 110 pounds and Alice’s effective capacity is 150 – 200 = -50, which means she has problems with carying her own body?

Or maybe character weight does not count for capacity calculation purposes?

what happens to max flow if we decrease the capacity of every edge by some constant?

Given a graph $ G = (V,A)$ , with source $ s$ , sink $ t$ , edge capacity larger than 1 (but not all equal), I know that if we decrease the capacity of one edge by 1, the $ s,t$ -maximum flow decreases by at most 1. But I would like to know what happens to max flow if we decrease (or increase) the capacity of all edges by 1. I’d appreciate any comments/insights on this. Thanks!

Proof that quantum entanglement does not increase the asymptotic capacity of classical channel

Consider a classical channel $ N_{X\rightarrow Y}$ which takes every input alphabet $ x\in X$ to output alphabet $ y\in Y$ with probability $ P(y|x)_{Y|X}$ . It is stated in many papers that even if the sender and receiver share entangled quantum states (or even no-signalling resources like a Popescu-Rohrlich box), they cannot increase the asymptotic capacity.

What is the proof of this statement?

Notes

A half-answer to this question is in this work, where the authors comment

By itself, prior entanglement between sender and receiver confers no ability to transmit classical information, nor can it increase the capacity of a classical channel above what it would have been without the entanglement. This follows from the fact that local manipulation of one of two entangled subsystems cannot influence the expectation of any local observable of the other subsystem

However, this is a little mysterious to me since it is known (see here) that in other regimes (one shot, zero error, etc.), entanglement between the sender and receiver can be used to boost the capacity of a classical channel. However, all these advantages die away in the case where one uses the channel $ n$ times and $ n\rightarrow\infty$ . Why is this so?

Creating capacity graph for a list of flights?

I have a list of flights and for each flight, I have information like source, destination, flight capacity, arrival time, departure time. There are only 8 distinct values that are populated in the source and destination column. There are multiple flights between any two nodes, i.e. multiple flights from LAX to PHX and similarly multiple flights from PHX to other destinations. Now, I have to find the capacity of the whole network given our source is LAX and our destination is JFK. I am completely stuck on how to create a capacity graph or find the capacity of the whole network. Any idea or hint is greatly appreciated. Sample data

Relative vertex capacity in max flow algorithm

I am designing a network for a max flow and would appreciate the following feature:

Say there is a flow incoming to a vertex. I would like to consume some specified amount of that flow and let through the rest. Unlike vertex capacity, which says “let though at most N units of flow”, I need to “let through everything but N units of flow”.

An example:

  • Vertex of relative capacity 10
  • Incoming flow = 15. Outgoing flow will be 15 – 10 = 5.
  • Incoming flow = 10. Outgoing flow will be 10 – 10 = 0.
  • Incoming flow = 5. Outgoing flow will be 5 – 10, which is less than 0, so 0.

Does such a feature exist? And if yes, which edges/vertices/edge capacities do I need to add/remove/modify and how?

Thanks a lot!