Sample exam questions

Note that final exam also covers the material covered by midterm 1 and midterm 2

Divide and Conquer

Binary Multiplication

In the following, suppose x and y are two n bit numbers, where n = 2ᵏ for some integer k. The function shiftl(a, b) returns 2b × a, and takes Θ(b) time. The function plus(a, b) returns a + b and takes Θ(log a + log b) time (i.e. plus(x, y) would take Θ(n) time). Analyze the complexity of the following algorithm.

function Mult(x, y)
    Let xL (yL ) be the left n/2 bits of x (y).
    Let xR (yR) be the right n/2 bits of x (y).
    z ← shiftl(Mult(xL , yL ), n)
    z ← plus(z, shiftl(plus(Mult(xL , yR ), Mult(xR , yL )), n2 ))
    z ← plus(z, Mult(xR , yR ))
    return z
end function

Recurrences

Let T (n) = T (n/4) + T (n/2) + Θ(n). Prove by substitution that

T (n) ∈ O(n)

Randomized Algorithms

Recursive example 1

function foo(n)
   if n ≤ 1
        return
   else
        i = random(n)
        foo(i)
   end
end

Suppose that random(n) takes Θ(1) time to return a random integer between 1 and n inclusive.

• Give a recurrence for the running time of foo(n). Use random variables to incorporate the behaviour of random() into your recurrence.

• Derive a recurrence for the expected running time of foo(n).

Recursive example II: Loopy

Suppose that random(n) returns a random integer between 1 and n inclusive. What is the expected running time of Loopy as a function of n and a. You may assume random(n) takes Θ(1) time, and that 1 ≤ a < n/2.

function Loopy(n, a, j)
    r ← random(n)
    if (r ≤ a or r > n − a) then
        return Loopy(n, a, j + 1)
    else
        return j
    end if
end function

Minimum Spanning Tree

Local improvement

Let G = (V, E) be a connected, undirected graph with distinct edge weights. Let T be a spanning tree of G. Let e be an edge of G not in T. - Explain how to create a new spanning tree T' that contains e, and as many edges of T as possible. - Give a condition on e and T that guarantees T' has smaller total weight than T.

Heaviest Edge

• Give sufficient conditions for the unique heaviest edge in a graph not to be in any minimum spanning tree.
• Prove your conditions.

Topological Sort

• Explain what a topological sort of a DAG is.
• Give an algorithm to produce a topological sort.
• Prove your algorithm is correct.

Dynamic Programming

Longest Alternating subsequence

Given a sequence of nonzero integers, the longest alternating subsequence is the longest subsequence such that succesive elements of the subsequence change sign. For example given the sequence

  1, 7, −3, −4, 5, 7, −9,

a (non-unique) longest alternating subsequence is 7, −3, 5, −9. We interested here in computing the length of the longest alternating subsequence.

Give a dynamic programming algorithm for this problem

• based on memoization
• without using memoization

Longest Divisor Sequence

This is a midterm question based on the previous question (also very similar to Longest Increasing Subsequence from class).

A divisor sequence from a list (array) S, is a subsequence of not-necessarily consecutive elements from S such that each element of the subsequence after the first divides the previous on. Given the input S

  28, -3, 14, 7, −3, −4, 5, -7, −9, 1

The longest divisor sequence is 28, 14, 7, -7, 1

• Give a recursive function that calculates the length of the longest divisor sequence. Your answer should use the mathematical function form from class, with two or cases. Do not write procedural code. Explain what your parameters mean.

• Give a memoization based algorithm based on the first part.

• Give a non-memoization (non-recursive) based algorithm based on the first part.

Edit Distance

Edit distance measures the number of single character inserts, deletes, and substitutions needed to transform one string into another. Each set of editing operations can be represented as an alignment, i.e. a table where all non-matching positions cost one operation. For example the following alignment costs 5.

exam__ination
_caffeination

• Prove that if we remove any column from an optimal alignment, we have an optimal alignment for the remaining substrings.
• Give a recurrence to find the edit distance.
• Describe the DAG of subproblems for dynamic programming.
• Give an iterative (non-recursive) dynamic programming algorithm to compute edit distance.

Union Find

Prove that if we only union root elements x and y with rank(x) == rank(y), the invariant ∀ z rank(z) ≤ height(z) is maintained

Multithreaded algorithms

function mergesort(A, lo, hi) // Sort A[lo...hi-1]
    if lo+1 < hi then  // at least two elements.
        mid =⌊(lo + hi) / 2⌋
        spawn mergesort(A, lo, mid)
        mergesort(A, mid, hi)
        sync
        parallel_merge(A, lo, mid, hi)
    end
end

Assume that parallel_merge takes Θ(k) work and has span Θ(\log^2(k)) when called on k elements (i.e. hi-lo = k).

1. Give a recurrence for the work T₁ of mergesort.
2. Give a Θ bound for recurrence of part (1).
3. Give a recurrence for the span T_∞ of mergesort.
4. Give a big-O bound for the recurrence of part (3).

Backtracking, SAT

Give a fast algorithm for solving monotone SAT problems, i.e. the input is CNF either without any negated variables, or with all negated variables.