1 Write a multithreaded algorithm to compute the product of an n × n matrix and an n-vector.

• Write a O(log n) span algorithm to multiply one row of a matrix by a vector.

• Use a parallel for loop, along with your algorithm from part (a) to multiply a matrix by a vector.

• Analyze the work and span of the resulting algorithm.

2 Consider the following parallel search algorithm.

function ParSearch (A, p, q, k)
  If p > q return FALSE
  If A[p] = k  return TRUE
  if p = q return FALSE
  right ← spawn ParSearch(A, p + 2, q, k)
  left ← ParSearch(A, p+1,p+1, k)
  sync
  return left OR right
end function

• Draw a computation DAG for ParSearch(A,1,4,k) for some k ∉ A
• Give recurrences for span and work of this algorithm
• Solve the recurrences

3 (ex 27.1-7 from Cormen et. al) Consider the following multithreaded pseudocode for transposing an n×n matrix A in place:

        P-TRANSPOSE (A)
            n ← A.rows
            parallel for j ← 2 to n
                parallel for i ← 1 to j - 1
                    swap A[i,j],A[j,i]

Analyze the work, span, and parallelism of this algorithm.