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<html>
<head>
<link href="final.css" rel="stylesheet" type="text/css"/>
<title>
Design and Analysis of Algorithms Final
</title>
</head>
<body>
<h1>
Design and Analysis of Algorithms Final
</h1>
<p>
Name: _______________________________________
<br>
<br>
NYU NetID: __________________________________
</p>
<hr>
<h3>
Multiple Choice
</h3>
<ol>
<li>Employing the master theorem,
the solution to the recurrence T(n) = 2T(n/4) +
n<sup>0.51</sup> is
<ol type="a">
<li>Θ(n<sup>2</sup>)
<li>Θ(n<sup>2</sup> log n)
<li>Θ(n<sup>0.51</sup>)
<li>the master theorem can't be applied here.
</ol>
<br>
<li>When we have an optimal substructure and only the locally
optimal choice matters, we should probably use:
<ol type="a">
<li>a straightforward recursive solution.
<li>dynamic programming.
<li>a memoized recursive algorithm.
<li>a greedy algorithm.
</ol>
<br>
<li>If we have a binary search tree of height <em>h</em>, then
all searching functions such as min, max, and successor
will run in
<ol type="a">
<li>O(h<sup>2</sup>) time.
<li>O(ln h) time.
<li>O(h) time.
<li>O(lg h) time.
</ol>
<br>
<li>A depth-first search of this tree:
<br>
<img src="graphics/GraphForFinal4.png">
<br>
Will cover the nodes in what order?
<ol type="a">
<li>A-C-B-H-E-F-I-K-G-J-D
<li>A-B-H-C-F-I-K-D-E-G-J
<li>A-B-C-D-E-F-G-H-I-J-K
<li>A-B-C-H-F-I-K-D-E-G-J
</ol>
<br>
<li>According to the binary search tree property
<ol type="a">
<li>for all nodes higher in the tree than x,
x.key ≤ y.key.
<li>for all nodes in the left subtree of x,
y.key ≤ x.key.
<li>for all nodes lower in the tree than x,
x.key ≤ y.key.
<li>all nodes in the right subtree of x will
be the successors of x.
</ol>
<br>
<li>Employing the master theorem,
the solution to the recurrence T(n) = 0.25T(n/2) + 8n is
<ol type="a">
<li>Θ(n<sup>-2</sup>)
<li>Θ(n<sup>-2</sup> log n)
<li>Θ(8n)
<li>the master theorem can't be applied here.
</ol>
<br>
<li>For large n, an algorithm will <em>run the slowest</em>
if it time complexity is:
<ol type="a">
<li>O(n<sup>3</sup>)
<li>O((1.02)<sup>n</sup>)
<li>O(n<sup>5</sup>)
<li>O(500n log n)
</ol>
<br>
<li>Matroids are essentially a theory of
<ol type="a">
<li>independence.
<li>connectivity.
<li>resilience.
<li>linearity.
</ol>
<br>
<li>Consider the following alphabet with frequencies:
<table>
<tr>
<th>
Symbol
</th>
<th>
Frequency
</th>
</tr>
<tr>
<td>
A
</td>
<td>
24
</td>
</tr>
<tr>
<td>
B
</td>
<td>
12
</td>
</tr>
<tr>
<td>
C
</td>
<td>
10
</td>
</tr>
<tr>
<td>
D
</td>
<td>
8
</td>
</tr>
<tr>
<td>
E
</td>
<td>
8
</td>
</tr>
</table>
<p>
Which of the following is a possible Huffman coding for the
string "ABBCDAED"?
</p>
<ol type="a">
<li>11111101011100111110
<li>11001001011100111110
<li>01001001011100111110
<li>11001001011100111110
</ol>
<br>
<li>When we have an optimal substructure and
overlapping sub-problems, we should probably use:
<ol type="a">
<li>a straightforward recursive solution.
<li>dynamic programming.
<li>a merge sort.
<li>a greedy algorithm.
</ol>
<br>
<li>Employing the master theorem,
the solution to the recurrence T(n) = 4T(n/2) + cn is
<ol type="a">
<li>Θ(n<sup>2</sup>)
<li>Θ(n<sup>2</sup> log n)
<li>Θ(cn)
<li>the master theorem can't be applied here.
</ol>
<br>
<li>We have one algorithm for processing customer records
with run time of O(n), and another with run time of O(lg n) + 2500.
<br>In what circumstances <i>might</i> we want
to choose the O(n) algorithm?
<ol type="a">
<li>The algorithm will be distirbuted across a network.
<li>We believe our program will
always be dealing with a
number of records less than 2500.
<li>We are running on a slow CPU.
<li>If <em>n</em> is very large.
</ol>
<br>
<li>We can sometimes be loose with our analysis of
divide-and-conquer algorithms,
in that we might omit details like
<ol type="a">
<li>recursions and function calls.
<li>exponential factors.
<li>floors, ceilings, and boundary conditions.
<li>any n raised to a power less than 3.
</ol>
<br>
<li>Under what circumstances might we
most reasonably expect to
receive worst-case input to our binary search tree?
<ol type="a">
<li>We insert data into the tree as it comes along to us.
<li>The data has been randomly shuffled.
<li>A user types in the data from a sorted list.
<li>All of the above.
</ol>
<br>
<li>When we have overlapping sub-problems and we want to solve
them from the bottom up, we should employ:
<ol type="a">
<li>dynamic programming
<li>a recursive algorithm
<li>a recursive algorithm with memoization
<li>a greedy algorithm
</ol>
<br>
<li>When we randomize an algorithm, we then speak of its
<ol type="a">
<li>necessary running time.
<li>best-case running time.
<li>expected running time.
<li>unbounded running time.
</ol>
<br>
<li>For large n, an algorithm will run the <em>fastest</em>
if it time complexity is:
<ol type="a">
<li>O(n!)
<li>O(n<sup>3</sup>)
<li>O(n<sup>4</sup>)
<li>O((1.02)<sup>n</sup>)
</ol>
<br>
<li>Strongly connected components of a graph are
<ol type="a">
<li>components of a directed graph that can
each be reached from each other.
<li>components in a weighted graph where the
connections have the highest weights.
<li>components of a spanning tree.
<li>an vertices with edges between them.
</ol>
<br>
<li>Which of these functions <em>grows most slowly</em>?
<ol type="a">
<li>lg n
<li>lg* n
<li>ln n
<li>log<sub>10</sub> n
</ol>
<br>
<li>For a greedy algorithm to work, the optimal choice
<ol type="a">
<li>must depend on many over-lapping sub-problems.
<li>must not depend on any future choices.
<li>must be dependent on a global optimal.
<li>must only be available after considering
all sub-problems.
</ol>
<br>
<li>Ω-notation applied to a function implies
<ol type="a">
<li>it is a function we know little about.
<li>it is asymptotically bound from above and below.
<li>only that it is asymptotically bound from below.
<li>only that it is asymptotically bound from above.
</ol>
<br>
<li>In algorithm analysis, we usually analyze algorithms
in terms of
<ol type="a">
<li>actual running time in nanoseconds.
<li>the number of disk operations.
<li>the number of basic operations.
<li>CPU time used.
</ol>
<br>
<li>In proving that a problem has the greedy-choice property,
<br>we often use a proof technique called:
<ol type="a">
<li>cut-and-paste.
<li>the master theorem.
<li>substitution.
<li>recursion-tree analysis.
</ol>
<br>
<li>The chief advantage of a doubly-linked list over a singly-linked list is
<ol type="a">
<li>inserts are faster.
<li>there are fewer pointers to maintain.
<li>we can back up as well as go forward in the list.
<li>less memory is used.
</ol>
<br>
<li>Often, a good alternative to dynamic programming is
<ol type="a">
<li>a simple recursive solution.
<li>a recursive solution with memoization.
<li>a brute-force solution.
<li>all of the above.
</ol>
<br>
<li>A random variable is actually
<ol type="a">
<li>the mean of all possible values.
<li>an algorithm.
<li>a precise real number value.
<li>a function.
</ol>
<br>
<li>In the substitution method for solving recurrences,
we
<ol type="a">
<li>substitute a different recurrence for
the one characterizing our algorithm.
<li>guess a solution and then
use induction to prove it.
<li>substitute a polynomial factor for
an exponential one.
<li>substitute an n<sup>2</sup> wherever
we see a 2<sup>n</sup>
</ol>
<br>
<li>Consider the following recursion tree.
<br>
<img src="graphics/NSquaredTree1.png">
<br>
What closed form best characterizes this tree?
<ol type="a">
<li>O(n/8)
<li>O(n log n)
<li>O(n<sup>2</sup>)
<li>O(log n)
</ol>
<br>
<li>The worst case of hashing with chaining occurs when
<ol type="a">
<li>the input comes in sorted already.
<li>all inputs hash to different values.
<li>the input is purely random.
<li>all inputs hash to the same value.
</ol>
<br>
<li>Consider this recurrence:
<br>
<img src="graphics/RecEq6.gif">
<br>
What is the solution to this recurrence?
<ol type="a">
<li>T(n) = Θ(n<sup>3</sup>)
<li>T(n) = Θ(n<sup>2</sup>)
<li>T(n) = Θ(n<sup>8</sup>)
<li>T(n) = Θ(n<sup>8/3</sup>)
</ol>
<br>
<li>If you have available a simple algorithm with some
acceptable run time, and a more complex algorithm
with slightly faster run time, you should
<ol type="a">
<li>always choose the faster algorithm.
<li>always choose the simpler algorithm.
<li>consider how big your input is likely
to be before choosing either algorithm.
<li>ask your boss what to do.
</ol>
<br>
<li>In graph theory, a tree is
<ol type="a">
<li>a forest with no edges.
<li>a cyclical graph with no forests.
<li>any walk on a graph.
<li>a connected graph with no cycles.
</ol>
<br>
<li>Graph theory is useful for
<ol type="a">
<li>task management.
<li>map coloring.
<li>cellular telephone networks.
<li>all of the above.
</ol>
<br>
<li>A problem with linear probing to resolve hash collisions is
<ol type="a">
<li>the code is extremely complex.
<li>clusters of keys will build up in a linked list.
<li>the hash table clusters around its central value.
<li>once inputs start to cluster in one area of the
hash table, they will become more likely
to do so.
</ol>
<br>
<li>Consider this graph:
<br>
<img src="graphics/GraphForFinal2.png">
<br>
Which of the following would be an adjacency list
representation of this graph?
<ol type="a">
<li>1 (2, 3, 4) 2 (1, 4) 3 (1, 4) 4 (2, 3, 5) 5 (4, 6)
6 (4, 5)
<li>1 (2, 3) 2 (1, 4) 3 (1, 4) 4 (2, 5, 6) 5 (4, 6)
6 (4, 5)
<li>1 (2, 3) 2 (1, 4) 3 (1, 4) 4 (2, 3, 5, 6) 5 (4, 6)
6 (4, 5)
<li>1 (2, 3) 2 (1, 4) 3 (1, 4) 4 (1, 3, 5, 6) 5 (4, 6)
6 (4, 5)
</ol>
<br>
<li>Taking the graph from the previous question,
<br>
which of the following would be an adjacency matrix
representation of this graph?
<ol type="a">
<li>
<table>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>0</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>0</td> <td>1</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>1</td> <td>0</td>
</tr>
</table>
<li>
<table>
<tr>
<td>0</td> <td>1</td> <td>0</td>
<td>0</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>1</td> <td>1</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>1</td> <td>0</td>
</tr>
</table>
<li>
<table>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>0</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>0</td> <td>1</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>1</td>
<td>1</td> <td>0</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>1</td> <td>0</td>
</tr>
</table>
<li>
<table>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>0</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td>
<td>0</td> <td>1</td> <td>1</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>0</td>
<td>1</td> <td>0</td> <td>1</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td>
<td>1</td> <td>1</td> <td>0</td>
</tr>
</table>
</ol>
<br>
<li>Besides running time, we can also measure algorithm performance
by:
<ol type="a">
<li>amount of disk-memory communication
<li>number of processors used
<li>amount of network bandwith used
<li>all of the above
</ol>
<br>
<li>An advantage of the adjacency matrix representation
over the adjacency list representation is
<ol type="a">
<li>it is smaller.
<li>it allows quicker searches for determining whether some
edge exists.
<li>it can represent weighted graphs.
<li>it can represent directed graphs.
</ol>
<br>
<li>Which of the following matrices represents this directed
graph:
<br>
<img src="graphics/GraphForFinal3.png">
<br>
<ol type="a">
<li>
<table>
<tr>
<td>0</td> <td>0</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>0</td> <td>1</td> <td>0</td>
</tr>
</table>
<li>
<table>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
</table>
<li>
<table>
<tr>
<td>0</td> <td>0</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td> <td>1</td>
</tr>
</table>
<li>
<table>
<tr>
<td>0</td> <td>0</td> <td>1</td> <td>0</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td> <td>0</td>
</tr>
<tr>
<td>0</td> <td>1</td> <td>0</td> <td>1</td>
</tr>
<tr>
<td>1</td> <td>0</td> <td>0</td> <td>0</td>
</tr>
</table>
</ol>
<br>
<li>A breadth-first search of this graph:
<br>
<img src="graphics/GraphForFinal4.png">
<br>
Will cover the nodes in what order?
<ol type="a">
<li>A-B-C-D-E-F-G-H-I-J-K
<li>A-C-B-H-E-F-I-K-G-J-D
<li>A-B-H-C-F-I-K-D-E-G-J
<li>A-B-C-H-F-I-K-D-E-G-J
</ol>
<br>
<li>Θ-notation applied to a function implies
it is
<ol type="a">
<li>a function we know little about.
<li>asymptotically bound from above and below.
<li>asymptotically bound from below only.
<li>asymptotically bound from above only.
</ol>
<br>
<li>Using the master theorem, the solution to the recurrence
<br>T(n) = 3T(n/3) + n/2 is:
<ol type="a">
<li>Θ(n/3)
<li>Θ(n log n)
<li>Θ(n)
<li>the master theorem can't be applied here.
</ol>
<br>
<li>Dynamic programming is preferred to simple recursive
solutions when
<ol type="a">
<li>the recursive solution is hard to grasp.
<li>all of the sub-problems are completely
independent of each other.
<li>the recursion goes beyond 4 levels.
<li>the same sub-problems must be solved multiple
times.
</ol>
<br>
<li>Direct dictionaries are only practical when
<ol type="a">
<li>the number of keys is small.
<li>we don't care how slow lookups will be.
<li>we aren't worried about the large number of collisions
hashing will produce.
<li>linear probing is appropriate.
</ol>
<br>
<li>A greedy algorithm is appropriate when
<ol type="a">
<li>we need to consider the global situation
when making any choice.
<li>there is extensive over-lap among sub-problems.
<li>only the locally best choice matters.
<li>greed is the goal of the algorithm choice.
</ol>
<br>
<li>We can use a greedy algorithm to solve
<ol type="a">
<li>a rod-cutting problem.
<li>a minimum-spanning-tree problem.
<li>a matrix-parenthisization problem.
<li>a shortest-path problem.
</ol>
<br>
<li>If f(n) = ω(g(n)), that means
<ol type="a">
<li>g dominates f asymptotically
<li>f is bounded below by g asymptotically
<li>f is bounded above by g asymptotically
<li>f dominates g asymptotically
</ol>
<br>
<li>Using the master theorem, the solution to the recurrence
<br>T(n) = 64T(n/8) - n<sup>2</sup> log n is:
<ol type="a">
<li>Θ(n<sup>2</sup>)
<li>Θ(n<sup>4</sup> log n)
<li>Θ(n<sup>2</sup> log n)
<li>the master theorem can't be applied here.
</ol>
<br>
<li>The worst-case running time of a search on a binary search
tree is
<ol type="a">
<li>θ(n<sup>2</sup>)
<li>θ(n)
<li>θ(lg n)
<li>θ(1)
</ol>
<br>
<li>A toppological sort on a tree makes use of
<ol type="a">
<li>breadth-first search.
<li>finding the minimum spanning tree.
<li>depth-first search.
<li>finding the shortest path.
</ol>
<br>
</ol>
<h2>
Problems
</h2>
<ol>
<li>Let us say we have an algorithm that runs
in 10n<sup>2</sup> + 10 time.
We want to show its order of complexity is
θ(n<sup>2</sup>). Give
an example of a set of k<sub>1</sub>, k<sub>2</sub>,
and n<sub>0</sub> that would do this.
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<li>How will a depth-first search process the following tree?
<br>
Write a number above each vertex indicating the order in which
it will be visited:
<br>
<img src="graphics/GraphForFinal.png">
<br>
<li>Use an indicator random variable to show how many numbers we
have to pick from a set of <em>n</em> numbers before we are
more likely than not to have drawn the same number twice.
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<li>Describe why the fractional knapsack problem can be solved with
a greedy algorithm, but the whole unit problem cannot.
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<li>Prove that Kruskal's algorithm can correctly find a minimum
spanning tree.
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
<br> <br> <br> <br> <br>
</ol>
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