CPSC 626: Parallel Algorithm Design and Analysis
Homework Assignment #2
Fall 2008

Due: Thursday September 18, 2008 in class.

General Guidelines for Homework

• Be clear and precise. Write neatly and legibly. Justify all answers, even if not specified in the question. Use good judgement concerning how detailed to make your answer.
• If the question is not fully specified, you may need to make some assumptions. In this case, you must state any assumptions you make, and justify why they are reasonable.
• Everyone must turn in their own copy of the assignment. You may consult outside material or work with others (this is encouraged), but you must reference your sources (people, texts, solutions, etc.). These must be listed on the cover sheet mentioned below.
• All assignments in this course must include a completed and signed cover sheet available on the course homepage. Any assignment turned in without a fully completed and signed coverpage will receive ZERO POINTS.

1. Given a sequence of numbers x₁, x₂, ..., x_n, the prefix sums are the partial sums
   s₁ = x₁
   s₂ = x₁ + x₂
   ...
   s_n = x₁ + x₂ + ... + x_n

   (a) Describe an algorithm to compute the prefix sums on a PRAM with n processors in O(log n) time. Analyze the running time of your algorithm and argue, at least informally, its correctness. Which PRAM model does your algorithm use (e.g., EREW, CREW, CRCW)? Does your algorithm require a synchronous PRAM?

   (b) Describe an algorithm to compute the prefix sums on an n processor mesh in O(sqrt(n)) time. Assume sqrt(n) is an integer. Analyze the running time of your algorithm and argue, at least informally, its correctness.

   (c) Describe an algorithm to compute the prefix sums on an n processor hypercube in O(log² n) time (or faster). Assume n is a power of 2. Analyze the running time of your algorithm and argue, at least informally, its correctness.

2. Repeat the previous question, but this time consider the case when p < n, i.e., the number of processors is less than the number of input elements. For the hypercube and the mesh, assume that each processor initially holds n/p input elements. Analyze the running time of your algorithms and argue, at least informally, their correctness. (If an algorithm is similar to the corresponding one from 2, you need only discuss the modifications.)