Fall 2008 CSE Qualifying Exam. September 13, 2008

Transcription

1 Fall 2008 CSE Qualifying Exam September 13,

2 Architecture 1. (Quan, Fall 2008) Your company has just bought a new dual Pentium processor, and you have been tasked with optimizing your software for this processor. You will run two applications on this dual Pentium, but the resource requirements are not equal. The first application needs 75% of the resources, and the other only 25% of the resources. (a) Given that 60% of the first application is parallelizable, how much speedup would you achieve with that application if run in isolation? (b) Given that 95% of the second application is parallelizable, how much speedup would this application observe if run in isolation? (c) Given that 60% of the first application is parallelizable, how much overall system speedup would you observe if you parallelized it, but not the second application? (d) How much overall system speedup would you achieve if you parallelized both applications, given the information in parts (a) and (b)? 2. (Quan, Fall 2008) Assume a five-stage single-pipeline microarchitecture (fetch, decode, execute, memory, write back) and the code below. There is no forwarding. Loop: LW R3,0(R0) LW R1,0(R3) ADDI R1,R1,#1 SUB R4,R3,R2 SW R1,0(R3) BNZ R4, Loop (a) Show the phases of each instruction per clock cycle for one iteration of the above loop. (b) How many clock cycles per loop iteration are lost to branch overhead? (c) Assume a static branch predictor predicting always taken in the Decode stage. Now how many clock cycles are wasted on branch overhead for this segment of code? 3. (Quan, Fall 2008) Suppose you have a computer with the following characteristics: 1) the processor pipeline can run an instruction each cycle 2) the cache can provide data every cycle (i.e. no penalty for cache hits) 3) the instruction cache miss rate is 1% 4) the data cache miss rate is 5% 5) 20% of instructions are memory instructions 6) the cache miss penalty is 80 cycles. 2

3 Assume that you have decided to purchase a new computer. For the budget allocated, you can either 1) purchase a machine with a processor and cache that is twice as fast as your current one (memory speed is the same as the old machine, i.e., the cache miss penalty is 160 cycles), or 2) purchase a machine with a processor and cache that is the same speed as your old machine but in which the cache is twice as large and the cache miss rate for the programs you run will drop by 40% with this larger cache. Which computer are you best off purchasing? Explain in detail, showing the relative performance of each choice. 3

4 Algorithms 1. (Fenner, Fall 2008) Let α be a real constant such that 0 < α < 1. Let T (n) be given by the recurrence n 1 T (n) = 1 + T (i) α for all n 0. (So in particular, T (0) = 1.) Using the substitution method, show that there exists a constant k 0 such that T (n) = O(n k ). 2. (Fenner, Fall 2008) Let SubsetSum be a Boolean-valued function that takes an array S[1... k] of positive integers and an integer n 0. SubsetSum(S, n) returns true if and only if n can be expressed as the sum of elements of S, where each element can be used at most once. (Equivalently, SubsetSum(S, n) returns true iff there exist b 1,..., b k {0, 1} such that n = k i=1 b is[i].) Give an algorithm to compute SubsetSum(S, n) that runs in time O(kn), where k is the number of elements of the array S. [Hint: use dynamic programming.] 3. (Fenner, Fall 2008) Let G = (V, E) be an undirected, connected graph with edge weight function w : E R such that all edge weights are distinct. Also assume that E V. (a) (100% credit) Show that G s minimum spanning tree is unique. [Hint: If there are two distinct MSTs for G, then their union has a cycle.] (b) (extra credit because it s hard) Let T be the (unique) minimum spanning tree of G. Show that there exist edges (u, v) T and (x, y) / T such that (T {(u, v)}) {(x, y)} is a second-best minimum spanning tree of G. (A second-best MST is a spanning tree whose weight is minimum among all spanning trees that are not MSTs.) i=0 4

5 Theory 1. (Fenner, Fall 2008) Fix a finite alphabet Σ. For two strings w, x Σ, we say that x is a subsequence of w (denoted x w) if x can be obtained by removing zero or more characters from w. (In other words, x = x 1 x n for some x 1,..., x n Σ, and w Σ x 1 Σ Σ x n Σ.) Show that if L Σ is regular, then the language L := {w Σ ( x L) x w} is also regular. (L is the language of all strings that have at least one subsequence in L.) [Hint: Given a DFA or NFA for L, find an NFA for L.] 2. (Fenner, Fall 2008) Let f be a function mapping strings to natural numbers that bounds the run time of every accepting computation. That is, for every TM M and every string w, if M accepts w in some number of steps s, then f( M, w ) s. (If M does not accept w, then f( M, w ) is completely arbitrary.) Show that no such f can be computable. 3. (Fenner, Fall 2008) Assume that P = NP. Show that there is a polynomial-time computable function f such that for any Boolean formula ϕ, f(ϕ) outputs a satisfying truth assignment for ϕ if there is one (otherwise, f(ϕ) outputs unsatisfiable. ) 5

6 Computational Biology 1. (Valafar, Fall 2008) Describe and compare the Needleman-Wunsch and Smith-Waterman algorithms. Prove the optimality guarantee of the Needleman-Wunsch algorithm. 2. (Valafar, Fall 2008) Derive the rotation operators about the Z and Y axes (denoted by R z (α) and R y (α) respectively) in the Cartesian space. Using these operators, derive a rotation operator R V (α) that denotes a rotation of α radians about an arbitrary rotor V in space. Note: you are free to use a representation of the vector V in any coordinate system such as Cartesian or spherical. 3. (Valafar, Fall 2008) Describe a complete force-field used in folding of proteins from first principles. Fully describe and define each of these terms. 6

7 Networks 1. (Nelakuditi, Fall 2008) What is count-to-infinity problem associated with distance vector routing? Why does distance vector routing suffer from count-to-infinity problem? How come link state routing does not have this problem? 2. (Nelakuditi, Fall 2008) In some networking environments such as a wireless communication environment, packets may be lost due to link-layer transmission errors instead of network congestion. What impact does such packet loss have on the performance of TCP? What remedies would you suggest for addressing this? 3. (Nelakuditi, Fall 2008) What is hidden terminal problem in wireless networks and how is it addressed by ? Why is it not a problem for wired networks such as Ethernet? Under what conditions would it be possible to use CSMA/CD in a wireless LAN? 7