課程名稱︰資料結構與演算法下
課程性質︰必修
課程教師︰蔡欣穆
開課學院：電資學院
開課系所︰資工系
考試日期（年月日）︰2011.4.22
考試時限（分鐘）：180
試題 :
Data Structure and Algorithm II, Spring 2011
Midterm Examination
140 points
Time: 9:10am-12:10pm (180 minutes), Friday, April 22, 2011
Problem 1.
In each of the following question, please specify if the statement is true or
false. If the statement is true, explain why it is true. If it is false,
explain what the correct answer is and why. (20 points. For each question,
1 point for the true/false answer and 3 points for the explanations.)
1. n·log(n) is polynomially larger than n.
1 + 10^(-100)
2. n is polynomially larger than n·log(n).
3. As long as a problem exhitbits optimal substructure, it can be solved by
using a greedy algorithm.
4. The solution of T(n) = 4T(n/2) + n^2 is T(n) = Θ(n·log(n)).
5. As long as a recurrence is of the form T(n) = a·T(n/b) + f(n), it can be
solved by using the master method.
Sol: 1. False. n·log n is not larger than n by a factor n^ε, for any ε > 0.
10^(-100)
2. True. Let f(n) = n and g(n) = log(n). We can see from
log(n) = O(n^ε), for any ε > 0, that f(n) is polynomially larger
than g(n).
3. False. Take the 0-1 knapsack problem as a counterexample.
4. False. By master theorem, the solution is Θ((n^2)·log(n)).
5. False. The recurrence of the form T(n) = 2·T(n/2) + n·log(n) can't
be solved the master theorem, since n·log(n) is not polynomially
larger than n. (However, this can be solved by using the extended
version ofthe master theorem.)
Problem 2.
"Short answer" questions: (25 points)
1. What is the main job of a Quality Assurance (QA) engineer in a software
company? (4 points)
2. Give a formal definition of "uniform random permutation". (4 points)
3. Derive an optimal Huffman code for the first n Fibonacci numbers, i.e.,
{c1: 1, c2: 1, c3: 2, c4: 3, c5: 5, c6: 8, c7: 13, c8: 21, ... cn: F(n)}.
In other words, show the general form of the codeword for the i-th
character ci (with F(i) as its frequency). (5 points)
4. Give two reasons for implementing the paper prototype instead of the
"real" prototype in a software project. (4 points)
5. How do we roughly estimate the cost of a software product? (4 points)
6. Give two reasons for writing down the specifications of a software
project before we start writing codes. (4 points)
Sol: 1. Quality Assurance (QA) engineers perform tests on the software
function to guarantee the stability of the product. They supervise and
report the software utility to see if they reflects the quality
requirements from the specification.
2. A uniform random permutation is the one in which each of the n!
possible permutations is equally likely to occur.
3. The Huffman tree for the first n Fibonacci numbers is a full binary
tree of height n with exactly one external node in all levels, except
for the first level which contains only the root and the last level
which contains two external nodes, since nodes are being merged in the
increasing order of their frequency. We claim this by observing that
Σ(i = 1 to n) F_i = F_(n+2) - 1, for all n ≧ 1, which can be
verified by induction. Based on this formulation, the internal node
with frequency Σ(i = 1 to k) Fi (which is smaller than F_(k+2)) and
the external node representing c_k with frequency F_(k+1) are the two
nodes with the least frequency at the k-th merging step. Therefore,
the Huffman code for c_k is 1 ... 10, in which there are (n - k) 1s
precedes 0, for k = 2 to n, and the code for c_1 is a bit string of
length (n - 1) containing no 0 in it.
4. The main idea of using paper prototype is to keep down the cost of
time spent on implementing a real prototype, so that the total
producing time can be well managed. The productivity gains can be
significant once the time is less consumed.
5. The cost of producing a software product consists mostly of human
resources, we can roughly estimate on the number of members involved
in and the time required to finish the project. The cost of marketing
can be estimated based on the scale of market and channels of
distribution.
6. Writing down the specifications would save lots of time to design.
Communications between divisions would be more efficient, while the
product content is not only recognized by software engineers, but also
people from other departments.
Problem 3.
Given a list of n distinct numbers (not sorted), please derive a
divide-and-conquer algorithm to return the first k smallest numbers in the
list. Your algorithm should have a running time of O(n). Note that you cannot
use the number selection algorithm taught in the class for k times, as the
running time will be Θ(kn) = O(n^2). Sorting does not work either as it will
take O(n·log(n)). In addition to the algorithm, please write down the
recurrence representing the running time, solve the recurrence, and prove your
solution by induction. (15 points, 7 points for the algorithm, 3 points for
the recurrence, and 5 points for the proof)
Sol: ────────────────────────────────────
Algorithm 1 QuicksortFindFirstK(list, left, right, k)
────────────────────────────────────
1: if right > left then
2: select median pivot between left and right using median of median
3: newPivot ← Partition(list, left, right, pivot)
4: if newPivot > k then
5: QuicksortFindFirstK(list, left, newPivot - 1, k)
6: else if newPivot < k then
7: QuicksortFindFirstK(list, newPivot + 1, right, k)
────────────────────────────────────
The algorithm 1 proposes a method to find the first k smallest integers
using divide and conquer method. At first, we divide the list into
5 groups and find median of median. This part costs T(n/5). The second
part is the partition function of quicksort. It runs in O(n). Lastly, we
use newPivot to call the function recursively. The running time of this
part is the max length between left to newPivot - 1 and newPivot + 1 to
right, max(T(3n/10), T(7n/10)). Thus, the running time of the last part
is T(7n/10). The equation 1 represents for the total running time of the
algorithm 1. The similar proof could be found from the course materials.
Therefore, the total running time is T(n) = θ(n).
T(n) = T(n/5) + θ(n) + T(7n/10)
Problem 4.
Use the recursion-tree method to derive an asymptotic tight upper bound for
T(n) = T(n/2) + T(n/4) + T(n/8) + n. You can assume that T(n) is a constant
when n is sufficiently small. Prove that your bound is correct by induction.
(10 points, 5 points for the recursion-tree and 5 points for the proof)
Sol: By observing that expanding the i-th level in the recursion tree costs
((7/8)^i)n, we claim that the cost of expanding all nodes in the
recursion tree is no more than n + (7/8)n + ((7/8)^2)n + ... = Θ(n). We
assume in our induction hypothesis that T(n) ≦ cn, for some positive
constant c. Then, T(n) = T(n/2) + T(n/4) + T(n/8) + n
≦ c(n/2 + n/4 + n/8) + n = ((7/8)c + 1)n ≦ cn, as long as we pick
c ≧ 8. Since it is obvious that n is the minimum requirement for T(n),
the tight upper bound for T(n) is O(n).
Problem 5.
For bit strings X = x_1 ... x_m, Y = y_1 ... y_n and Z = z_1 ... z_(m+n), we
say that Z is an interleaving of X and Y if it can be obtained by interleaving
the bits in X and Y in a way that maintains the left-to-right order of the
bits in X and Y . For example if X = 101 and Y = 01 then
x_1 x_2 y_1 x_3 y_2 = 10011 is an interleaving of X and Y , whereas 11010 is
not. (15 points)
1. Please come up with the definition of the subproblem. Use your definition
and prove that this problem has optimal substructure. (5 points)
2. Give the most efficient algorithm you can to determine if Z is an
interleaving of X and Y. (7 points)
3. Analyze the time complexity of your algorithm as a function m = |X| and
n = |Y|. (3 points)
Sol: Consider x = x_1 ... x_m, y = y_1 ... y_n, and z = z_1 ... z_(m+n). We
claim that if z is the interleaving of x and y, it contains a subproblem
z' = z_1 ... z_(m+n-1) and the last bit z_(m+n) must be x_m or y_n. We
will have the following substructure and build an two-dimension array
D[m,n] to check whether z is the interleaving of x and y. Since all cases
D[i,j] comes from one of the answers of the subproblems, this problem
exists the optimal substructure property. At the end of this problem, we
present a T(n) = O(mn) dynamic programming algorithm 2 to solve this
problem.
╭ true, if i ＝ j ＝ 0
│ false, if x_i ≠ z_(i+j) and y_j ≠ z_(i+j)
D[i, j] = ┤ D[i-1, j], if x_i ＝ z_(i+j) and y_j ≠ z_(i+j)
│ D[i, j-1], if x_i ≠ z_(i+j) and y_j ＝ z_(i+j)
╰ D[i-1, j] or D[i, j-1], if x_i ＝ z_(i+j) and y_j ＝ z_(i+j)
Problem 6.
Consider the problem of making change for n dollars using the fewest number of
coins. Assume that each coin's value is an integer. The same coin can be used
for any number of times in the change. (25 points)
1. Prove that the problem exhibits optimal substructure. (5 points)
2. Assume the following set of coins is available to you: 1, 5, 10, 50.
Prove that under this condition, the problem has the greedy property.
(5 points)
3. Describe a greedy algorithm to solve the problem with the coin set in 2.
(5 points)
4. Give a set of coins for which the greedy algorithm in 3 does not yield an
optimal solution. Your set should include a one-dollar coin so that there
is a solution to every value of n. Explain the intuition behind your
choice of coins in the set. (3 points)
5. Describe a dynamic programming algorithm which solve the problem with any
coin set with k different coins, assuming that one of them is a
one-dollar coin. (7 points)
Sol: 1. The coin changing problem exhibits optimal substructure in the
following explanation. Consider we have a coin set C = {c_1, ..., c_k}.
Suppose we want to make change for n dollars, we divide n dollars into
left-part (n - m) dollars and right-part m dollars. Therefore, the
solution to (n - m) dollars is s_L = {c_1, c_1, c_2} and the solution
to m dollars is s_R = {c_1, c_2, c_4}. We claim that the left-part of
solution must be the optimal way using coin set C and the right-part
of the solution must be the optimal way using coin set C so that the
optimal solution to n dollars is s = s_L + s_R.
Proof. By contradiction, suppose there exists a solution s'_R better
than right-part solution s_R. The right-part solution could be
replaced by the better solution s'_R, yielding a solution
s' = s_L + s'_R which is better than s = s_L + s_R. This
contracts to our previous assumption.
Therefore, the optimal solution to coin changing problem is composed
of smaller subproblems of optimal solutions.
2. It it clear that there are at most 4, 1, 4 coins with value 1, 5, 10,
respectively, in an optimal solution. Observing that 4*1 + 1*5 + 4*10
= 49 < 50, there's no way to not choose 50 to have an optimal solution
while n > 50. With the same reason, there's no way to not choose 10 to
have an optimal solution while 50 > n > 10, and so on. Therefore, the
optimality is guaranteed by the greedy approach, which selects the
largest available coin at each step.
3. ─────────────────────────
Algorithm 3 minNum = GreedyCoinChange(n)
─────────────────────────
1: numOfCoins ← 0
2: for c from c_k to c_1 do
3: if c > n then
4: numOfCoins ← numOfCoins + floor(n/c)
5: n ← n mod c
6: return numOfCoins
─────────────────────────
Suppose the coin set has the greedy property, the algorithm 3 is
developed to solve this problem. We assume that for all
c ∈ C = {c_1, ... , c_k} if i < j then c_i < c_j. The main concept of
this algorithm is always select the biggest possible coin in order to
make the least number of the coins.
4. Consider the list {1, 6, 7}. If n = 12, greedy algorithm give the
solution {7, 1, 1, 1, 1, 1}. However, the optimal solution for n = 12
is {6, 6}, which is much less than the previous solution. This is
because this coin list does not contains greedy property in the coin
changing problem.
5. ──────────────────────────
Algorithm 4 DPCoinChange(n)
──────────────────────────
1: D[0] ← 0
2: S[0] ← 0
3: for p from 1 to n do
4: for c from c_1 to c_k do
5: if c ≧ p and 1 + D[p - c] ＜ D[p] then
6: D[p] ← 1 + D[p - c]
7: S[p] ← c
──────────────────────────
Here we present a DP algorithm 4 to solve the coin changing problem.
D[1..n] array stores the optimal solution using coin set C. The S[1..n]
array is the solution to the combination of the coins. For certain
D[p], it stores the optimal solution to number of coins for p dollars
using coin set C = {c_1, ... , c_k}. The following equation give the
concept of the our algorithm.
╭ 0, if p ＝ 0
D[p] = ┤
╰ min {1 + D[p - c_i]}, if p ＞ 0
i:c_i≦p
Problem 7.
This problem examines three algorithms for searching for a value x in an
unsorted array A consisting of n elements (x appears in A for k times, k ≧ 0):
(20 points)
˙ Algorithm α:
We pick a random index i into A. If A[i] = x, then we terminate;
otherwise, we continue the search by picking a new random index into A.
We continue picking random indices into A until we find an index j such
that A[j] = x or until we have checked every element of A. Note that we
pick from the whole set of indices each time, so that we may examine a
given element more than once.
˙ Algorithm β:
The algorithm searches A for x in order, considering A[1], A[2], ..., A[n]
until either it finds A[i] = x or it reaches the end of the array. Assume
that all possible permutations of the input array are equally likely.
˙ Algorithm γ:
We uniformly and randomly permute the input array A and then run
Algorithm β.
1. For each algorithm, derive the expected running time when
(a) k ＝ 0,
(b) k ≧ 1.
(18 points, 3 points for each answer)
2. Which algorithm would you use? Please explain your answer. (2 points)
Sol: 1. ˙ Algorithm α:
(a) Let T be the time to collect all n elements in A, and t_i be
the time to collect the i-th element after (i - 1) elements
have been collected. Since the probability of picking a new
element given (i - 1) elements already being selected is
p_i = (n-i+1)/n, the expected value of the geometrically
distributed random variable t_i is E(t_i) = 1/p_i = n/(n-i+1).
Therefore, E(T) = Σ(i = 1 to n) E(t_i) =
n·Σ(i = 1 to n) 1/(n-i+1) = n·H_n, where H_n is the n-th
harmonic number. Thus the expected running time is approximately
n·ln(n).
(b) Let x = (n-k)/n. The expected running time for k ≧ 1 is
╭ k ╮ ╭ n-k ╮╭ k ╮ ╭ n-k ╮2╭ k ╮
1│──│ + 2│───││──│ + 3│───│ │──│ + ...
╰ n ╯ ╰ n ╯╰ n ╯ ╰ n ╯ ╰ n ╯
k ∞ ╭ n-k ╮i k ╭ ∞ i ∞ i╮
=── Σ (i + 1)│───│ =──│ Σ (i)(x) + Σ (x) │
n i=0 ╰ n ╯ n ╰i=0 i=0 ╯
k ╭ x 1 ╮ n
=──│─────+ ──│= ──
n ╰ (1-x)^2 1-x ╯ k
˙ Algorithm β:
(a) The expected running time is n when k = 0.
(b) Since the search will end until the first x is read, the
expected time to read x is 1. For every y in A, y ≠ x, the
probability of y has been read before the first x appears is
1/(k+1), and there are (n - k) elements in A which is not equal
to k, thus the expected time of reading these non-x elements is
(n-k)/(k+1). Therefore, the expected running time is
1 + (n-k)/(k+1) = (n+1)/(k+1).
˙ Algorithm γ:
(a) The expected running time is n + n = 2n.
(b) The expected running time is n + (n+1)/(k+1).
2. If we do not have any information about the input data, Algorithm β
seems better in overall evaluation, while the expected cost of both
the cases are asymptotically linear to n and will never exceed n.
Problem 8.
This semester we made lots of changes in the course compared to last semester
(homeworks, programming assignments, course content), and we are curious about
how you feel about the current form of the course. If you have the power of
changing 3 things in the course, what would these 3 things be and how would
you change them? Your feedbacks are very important to the teaching team; we
thank you for your valuable opinion. :) (10 points)
Sol: Skipped.
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