Theory of Parallel Hardware May 11, 2004 Massachusetts Institute of Technology Charles Leiserson, Michael Bender, Bradley Kuszmaul

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1 Theory of Parallel Hardware May 11, 2004 Massachusetts Institute of Technology Charles Leiserson, Michael Bender, Bradley Kuszmaul Final Examination Final Examination ffl Do not oen this exam booklet until you are directed to do so. Read all the instructions on this age. ffl When the exam begins, write your name on every age of this booklet. ffl This exam contains 5 roblems, each with multile arts. You have 80 minutes to earn 100 oints. ffl This exam booklet contains 9 ages, including this one. Two extra sheets of scratch aer are attached. Please detach them before turning in your exam at the end of the examination eriod ffl This exam is closed book. You may use two handwritten crib sheets. No calculators 2 or rogrammable devices are ermitted. ffl Write your solutions in the sace rovided. If you need more sace, write on the back of the sheet containing the roblem. Do not ut art of the answer to one roblem on the back of the sheet for another roblem. ffl Do not waste time and aer rederiving facts that we have studied. It is sufficient to cite known results. ffl Do not send too much time on any one roblem. Read them all through first, and attack them in the order that allows you to make the most rogress. ffl Show your work, as artial credit will be given. You will be graded not only on the correctness of your answer, but also on the clarity with which you exress it. Be neat. ffl Good luck! Problem Parts Points Grade Grader Total

2 2 Problem 1. Recurrences (5 arts) [25 oints] For each of the recurrences below, give an examle of a roblem during the term where this recurrence was imortant. Then, give an aysmtotically tight ) bound ( for its solution. (a) f (n) = f (d2n=3e) + (1) Solution: Wallace tree: (lg n) (b) f (n) = 2 f (n=4) + (1) Solution: side of H-tree: ( n)

3 3 (c) f (n) = 2 f (n=4) + ( n) Solution: side of tree of meshes or area-universal fat-tree: ( n lg n) (d) f (n) = f (n=2) + (lg n) Solution: deth of bitonic sorting network: (lg 2 n) (e) f (n) = 4 f (n=4) + n) ( Solution: deth of bitonic sorting (n) network:

4 4 Problem 2. Definitions (4 arts) [16 oints] Give a one-sentence descrition of each of the following terms to show that you know what they mean: (a) carry-lookahead adder Solution: Adding 2 N-bit numbers with O(n) hardware, O(lg n) deth. The idea is to calculate the carries by doing arallel refix in the kg monoid. (b) systolic circuit Solution: a circuit formed of Moore machines. All state and communication with neighboring rocessors is clocked. (c) load factor of a set M of messages on a network R. number of msgs crossing the cut Solution: = max f all cuts of R g caacity of the cut (d) Beneš network Solution: Two butterflies back to back.

5 5 Problem 3. Bounds (5 arts) [25 oints] For each of the following roblems, give the best ossible bound. Since the uer and lower asymtotic bounds are the same for each answer, you can just give the bound without surrounding O s, Ω s, or s. If you wish to justify your answer, lease do so. (a) The deth of a circuit for multilying n-bit two numbers. Solution: n) (lg (b) The deth of a circuit to rereresent the sum 3 n-bit of binary numbers as the sum of 2 ( n + 1) -bit binary numbers. Solution: (1)

6 6 (c) The time to sort n numbers on an n n mesh-of-trees network. Solution: (lg n) (d) The 3-dimensional volume to ermute any n wires. Solution: (n 3=2 ) (e) The time to simulate an n-node 3-dimensional mesh on an (n lg 2 n)-area area-universal fat-tree with n leaves. (That is, simulate on a basic area-universal fat-tree, not the more comlicated one with meshes at the leaves.) 2=3 n Solution: (n 1=6 ) since = n

7 7 Problem 4. Short Answer (3 arts) [18 oints] (a) Prove or give a counterexamle. The number of latches in a synchronous circuit remains invariant under retiming. Solution: This is false and counterexamles are easy to find. The correct statement is that the number of latches around a cycle is reserved. (b) Give a Gray code on 4-bit numbers. Solution:

8 8 (c) Consider an n-inut, n-outut butterfly network. Suose that an adversary destroys a single switch in the network, reventing some inuts from connecting to some oututs in a single ass through the network. We can still find a large S subset of inuts and a large subset T of oututs such that every inut S in can connect to every outut in T. Let m = min fjsj ; jt jg, where S and T are chosen to allow m to be as large as ossible. Give a good asymtotic uer bound n on m. Solution: ( n). Suose a switch at level i is cut. There 2 i are inuts that can reach this switch and 2 lg n i oututs. Routing from any such inut to any such outut will be imossible, but all other aths still exist(the butterfly has unique aths between any inut and outut). So if we either exclude these inuts or these oututs from from the set of m, we will be able to route everything. max i minf2 i ; 2 lg n i g = n

9 9 Problem 5. Butterfly Fat-Tree (3 arts) [16 oints] Consider a butterfly fat-tree in which every 2 out of 3 levels consist of butterfly switches and every 1 out of 3 levels consists of tree switches. (a) What is the asymtotic number of connections to the root of such N -leaf an fat-tree? Justify your answer. Solution: Suose the leaves are level 0. At each level C(i) i, let be the number of connections going out of all nodes at level i. Then, C(i) = C(i 1) if i mod 3 is 0 or 1, and if i mod 3 is 2, and hence C(i) = C(i 1)=2 There are lg n levels ) C(lg n) = ( C(i) = C(i 3)=28i > 0; C (0) = n n ) = ( n 2=3 2 lg n=3 (b) How many switches does such N an -leaf fat-tree contain? Justify your answer. ). P Solution: Note that P the number of switches in one node (number is of out connections ), lg n n so the answer is ( C(i)) = ( ) = ( n) 2 i=3 i=0 (c) Make an educated guess as to what imortant roerty this fat-tree might have. Solution: It is O(lg V ) universal for simulating volume V networks.

10 SCRATCH PAPER Please detach this age before handing in your exam.

11 SCRATCH PAPER Please detach this age before handing in your exam.