CS570
Analysis of Algorithms
Summer 2017
Exam II
Name: _____________________
Student ID: _________________
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Maximum Received
Problem 1 20
Problem 2 20
Problem 3 20
Problem 4 20
Problem 5 20
Total 100
Instructions:
1. This is a 2-hr exam. Closed book and notes
2. If a description to an algorithm or a proof is required please limit your description or
proof to within 150 words, preferably not exceeding the space allotted for that
question.
3. No space other than the pages in the exam booklet will be scanned for grading.
4. If you require an additional page for a question, you can use the extra page provided
within this booklet. However please indicate clearly that you are continuing the
solution on the additional page.
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1) 20 pts
Mark the following statements as TRUE or FALSE. No need to provide any
justification.
[ TRUE/FALSE ]
In the Ford-Fulkerson algorithm, choice of augmenting paths can affect the min cut.
[ TRUE/FALSE]
A dynamic programming algorithm with n2 unique subproblems, could have a time
complexity that is better than O(n2).
[ TRUE/FALSE ]
A dynamic programming algorithm with n2 unique subproblems, could have a
memory requirement that is better than O(n2).
[ TRUE/FALSE ]
A dynamic programming solution when implemented using iteration will always run
faster compared with the same dynamic programming solution implemented using
recursion and memoization.
[ TRUE/FALSE ]
It is possible for a dynamic programming algorithm to have exponential running time.
[ TRUE/FALSE ]
In the final residual graph constructed during the execution of the Ford-Fulkerson
Algorithm, there’s no path from source to sink.
[ TRUE/FALSE ]
Bellman-Ford algorithm is guaranteed to work on directed graphs that don’t contain
any negative weight cycles.
[ TRUE/FALSE ]
Bellman-Ford algorithm is not guaranteed to work on all undirected graphs.
[ TRUE/FALSE ]
If we replace each directed edge in a flow network (except for edges incident on s or
t) with two directed edges in opposite directions with the same capacity and
connecting the same vertices, then the value of the maximum flow remains
unchanged.
[ TRUE/FALSE ]
Let (S,V −S) be a minimum (s,t)-cut in the network flow graph G. Let (u, v) be an
edge that crosses the cut in the forward direction, i.e., u � S and v � V − S. Then
increasing the capacity of the edge (u, v) necessarily increases the maximum flow of
G.
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2) 20 pts
A message containing only letters is encoded in the following ways:
‘A’ à 1
‘B’ à 2
… …
‘Z’ à 26
Design an algorithm to calculate the number of ways to decode a given message M.
For example, for the message M=‘12’, we can decode it as ‘AB’ or ‘L’, so there are 2
ways to decode it.
Maintain an array dp of size n+1, where dp[i] represents the number of ways
decoding the first i numbers in the given message.
The solution to our problem will be dp[n] where n=number of input numbers
We define a substring is valid as following:
If the string starts with a “0”, it is not valid digit.
If the string is larger than 0 and smaller than 27, it is a valid digit.
If the sting is larger than 26, it is not a valid digit.
We use M[i-1,i] to denote the substring only containing the ith letter and M[i-2,i]
to denote the substring containing (i-1)th letter and ith letter.
The base case is dp[0] = 1 as there is only one way to decode empty message. Set
dp[1] = 2 if M[-1,1] is valid, dp[1] = 0 if neither M[0,1] nor M[-1,1] is valid,
dp[1] = 1 if only M[0,1] is valid
We can calculate dp[i] by considering following cases:
Case 1: both substring M[i-1,i] and M[i-2,i] are valid
Case 2: only M[i-1,i] is valid
Case 3: only M[i-2,i] is valid
In case 1, dp[i] = dp[i-1] + dp[i-2]
In case 2, dp[i] = dp[i-1]
In case 3, dp[i] = dp[i-2]
Below is the pseudocode:
Input: message M
Output: number of ways to decode M
n = length of M
Initialize dp as an array of length (n+1) and all elements are 0
dp[0] = 1
if M[-1,1] is valid::
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dp[1] = 2
elseif M[0,1] is not valid or M[-1,1] is not valid:
dp[1] = 0
else:
dp[1] = 1
endif
for( i = 2, I <=n, i++)
if M[i-1,i] is a valid substring:
dp[i] += dp[i-1]
endif
if M[i-2,i] is a valid substring:
dp[i] += dp[i-2]
endif
endfor
return dp[n]
Grading:
5 points for the meaning of dp[i] and pointing out the solution is dp[n]
6 points for the two initial states, 3 points for each
9 points for the analysis of three different cases when calculating dp[i], 3
points for each
3) 20 pts
Consider the flow network in the following figure with the indicated s-t flow f: every
edge is labeled a/b where a is the flow and b is the capacity of the edge.
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a) Draw the residual network with respect to the flow f. (5 pts)
Grading:
Each wrong edge (-0.5 points)
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b) Is f a valid flow? Explain why. (5 pts)
Yes,
1) for any node in this graph, the out degree equals the in degree and (2.5 points)
2) the flow doesn’t exceed any edge capacities. (2.5 points)
c) Is f a maximum flow in G? Explain why. (5 pts)
Yes. By following Ford-Fullkerson algorithm, this is the max-flow in G since
there is no more s-t paths in the residual graph.
d) Give a minimum s-t cut of the (original) network by specifying below the two sets
S and T of the corresponding partition of the nodes, where s is in S, and t is in T.
(5 pts)
The residual network does not contain any path from s to t, hence f is a maximum
flow. A minimum cut can be derived by letting S be the set of nodes reachable
from s in the residual network and T the set of remaining nodes:
S = { s, A, E, F } T = { t, B, C, D }
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4) 20 pts
Assume we have N workers. Each worker is assigned to work at one of M factories.
For each of the M factories, they will produce a different profit depending on how
many workers are assigned to that factory. We will denote the profits of factory i with
j workers by P(i,j). Develop a dynamic programming solution to find the assignment
of workers to factories that produces the maximum profit, by completing the
following steps
a) Describe a unique subproblem (in English) (5 pts)
Assignment of j workers to the first m factories that produces the maximum profit
Grading:
- Didn’t mention maximum profit (-2 point)
- Didn’t mention how many workers (-1 point)
- Didn’t mention how many factories (-1 point)
b) Give a recurrence formula to compute the value of an optimal solution. (5 pts)
Let B(m, j) denotes the maximum profit of the assignment of j workers to the first
m factories.
An optimal solution of j workers to the first m factories assigns some number of
workers, k, to factory m, and must assign j-k workers with maximal profit to the
first m - 1 factories, which, by induction, has cost B(m-1, j-k). Thus, we have
B(m, j) = max0≤k≤j P(i, k) + B(m-1, j-k).
Grading:
- Wrong recurrence formula in dynamic programming recurrence formula form (-2
points)
- Formula not in dynamic programming recurrence formula form (-5 points)
c) Write pseudo-code to compute the value of an optimal solution using iteration.
(5 pts)
B(0,0) = 0
For m = 1, 2, …, M
For j = 1, 2, …, N
B(m, j) = max0≤k≤j P(i, k) + B(m-1, j-k)
Endfor
Endfor
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https://www.coursehero.com/file/26269115/CSCI570-Midterm-Exam-2-Summer-2017-Grading1pdf/
Grading:
- No initialization (-1 point)
- Used only one for-loop (-2 points)
- Didn’t include recurrence formula (-3 points)
d) What is the complexity of the code in section c? (5 pts)
The time to compute each value B(m, j) is O(N) and since there are MN
subproblems, the total runtime is O(N2M).
Grading:
- Wrong time to compute each value B(m, j) (-2 points)
- Wrong number of subproblems (-2 points)
- Missing O( ) notation (- 1 point)
5) 20 pts
Formulate the following optimization problem as a network flow problem:
There are m houses H1, …, Hm for sale, n buyers B1, …, Bn looking to buy a house,
and k real-estate agents A1, …, Ak. Each agent Ai knows a subset hi ⊆ {H1, …, Hm}
of the properties and a subset bi ⊆ {B1, …, Bn} of the potential buyers. Due to
workload constraints, each agent Ai can close at most ai real-estate transactions. What
is the maximum total number of transactions that can be arranged?
Construct a network with the nodes s (source), H1, …, Hm, A1, …, Ak, A’1, …, A’k, B1,
…, Bn, and t (sink). Note that there are two nodes for each agent. There is an edge of
capacity 1 from s to each Hj node. There is an edge of capacity 1 from Hj to Ai if and
only if Hj ∈ hi. There is an edge from Ai to A’i of capacity ai. There is an edge of
capacity 1 from A’i to Bj if and only if Bj ∈ bi. There is an edge of capacity 1 from
each Bj node to t.
Grading:
- Wrong capacities (-1 point for each group of edges)
- Not mentioning which houses (/buyers) get connected to which agents (-1 point)
- If your three layers are H—A—B: This does not limit agent A_i being in at most a_i
transactions (even if you add a demand of a_i to node A_i) (-10 points)
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- If your three layers are A—H—B: This does not limit a house H_j being assigned to only
one buyer (-10 points)
- If your three layers are A—B—H: This does not limit a buyer B_j being assigned to only
one house (-10 points)
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