Lectures 6-8 homework:

Lecture 6 homework:

1. [1 point each] Exercises.  Do any of the exercises &/or problems from chapter 3, "Introduction to Computer Science," in Nielsen & Chuang.

Lecture 7 homework:

1. [1 point each, max 5 points] Analysis.  In this exercise, you will pick one or more specific popular multiprocessor interconnection topologies (1 point each), and prove that they are not scalable (not consistent with the assumption of constant-time links).  It may be helpful for you to read the Vitanyi and/or Bilardi-Preparata papers before doing this assignment, so you can understand how the argument works more generally.  Below are some suggestions for topologies to tackle, although you are free to find others.  If you don't know what these are, the Rosen discrete math textbook and the Hennessy & Patterson graduate architecture textbook describe some of these, or you can look them up on the web.
• Fully-connected network.
• Hypercube or n-cube.
• Binary tree.
• Fat tree.
• Butterfly network.
Here is a guideline of some suggested steps that your analysis could follow.  Other ways to prove the point are also possible.
• (a) Characterize the size of your network by a parameter n, which could be, e.g., the number of dimensions in a hypercube, or the number of levels in a tree or butterfly network.
• (b) How does the number of processing nodes N scale as a function of n?
• (c) Assuming that each node has a constant minimum physical volume, e.g., 1 cubic millimeter, what is the minimum physical volume of the entire network, in cubic millimeters, as a function of n?
• (d) Therefore, what is the minimum physical diameter of the network, in millimeters, as a function of n?  (Hint: What shape has minimum diameter for a given volume?)
• (e) Therefore, what is the minimum speed-of-light travel time, in picoseconds, as a function of n?
• (f) As a function of n, what is the maximum number of hops (from a node to its neighbor) required for a message to get from any node on the network to any (single) other node?
• (g) Therefore (from e and f), what is the minimum average time taken per hop, as a function of n?  Note that some of the hops must take at least as long as the average time per hop.
• (h) Therefore, with the topology you've chosen, is it consistent to assume that the time per hop is upper-bounded by a constant that is independent of network size?  I.e., is the topology scalable?
• (g) At what number of nodes N does the minimum average worst-case time-per-hop exceed, say, 100 ps?  (1 cycle of a near-future 10 GHz CPU)

Lecture 8 homework:

1. [1 point each, max 3 points] Calculation.  Choose 3 example systems that are interesting to you (which may be anything from an proton to a transistor to a galaxy), estimate their size and energy content from any sources you may have available, and then calculate their maximum entropy, according to the Smith bound (assuming 2 massless particle states), and the Bekenstein bound?  How many distinguishable states does each system have, according to each bound (assuming that the bound is tight)?  What is the system's average entropy density throughout its volume, in bits per cubic Angstrom?  Also, what temperature would the object (if converted to a photon gas) need to have to achieve Smith's bound, based on eq. 2.10 in sec. 2.2.3 (p.36) of the green book?
2. [1 point each, max 3 points] Calculation.  Locate a source of physical thermochemical data for materials, such as the CRC Handbook of Chemistry and Physics (which can probably be found in the reference section at Marston), or a web site such as webelements.com.  Based on the information available (e.g., entropy per mole, atomic weight, density), calculate the number of bits per atom and per cubic Angstrom for several materials.  (1 point each)
3. [2 points] Analysis.  From the previous problem, you should have obtained entropy data for some material at a given temperature and pressure (probably room temperature and atmospheric pressure).  In this problem you are going to estimate entropy data at a different temperature (but the same pressure).  For example, you can choose the melting point of the material.
• (a) Express the change in entropy delta-S from T₁ (the lower temperature) to T₂ (the higher temperature) as a definite integral of dS over that range of temperatures.
• (b) You know that T=dE/dS.  Solve this for dS and plug that into your integral to get an integral involving dE.
• (c) Heat capacity C(T) (usually a function of temperature) is defined as dE/dT, the heat input required for a given increase in temperature.  Using this definition, change your integral to an integral involving dT.
• (d) In most elemental materials, the heat capacity per atom (which, note, has the same units as entropy) empirically approaches 3 k_B per atom at high temperatures, one nat of heat capacity for each of the 3 degrees of freedom of atomic motion in 3 dimensions.  Using the simplifying assumption that C(T) is exactly 3k_B over the entire range of temperatures for your material, solve your integral (reduce it to a closed-form expression which should involve T₁ and T₂).
• (e) Using your solution, estimate the entropy per atom of each of your materials from the previous problem at the new temperature you selected.  Estimate the entropy density also at this temperature. (For this you will need a coefficient of expansion, or the mass density at the other temperature.  If you can't find any data, assume the mass density is the same.)
• (f) If your temperature range included a phase transition (solid-liquid or liquid-gas), then you will need to revise your answer from part (e) to take into account the change in entropy at the phase transition.  Figure out how to do this yourself and include it in your calculation for 1 point extra credit.