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[SOLVED] Cs-310 assignment 1 problem 1. (10 points) gale and shapely published their paper on the stable matching

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Problem 1. (10 points) Gale and Shapely published their paper on the Stable Matching
Problem in 1962; but a version of their algorithm had already been in use for ten years by
the National Resident Matching Program, for the problem of assigning medical residents
to hospitals.Basically, the situation wa the following. There were m hospitals, each with a certain number of available positions for hiring residents. There were n medical students
graduating in a given year, each interested in joining one of the hospitals. Each hospital
had a ranking of the students in order of preference, and each student had a ranking
of the hospitals in order of preference. We will assume that there were more students
graduating than were slots available in the m hospitals.The interest, naturally, was in finding a way of assigning each student to at most one
hospital, in such a way that all available positions in all hospitals were filled. (Since we
are assuming a surplus of students, there would be some students who do not get assigned
to any hospital.)We say that an assignment of students is stable if neither of the following situations
arise.
1. First type of instability: There are students s and s
0 and a hospital h, so that
• s assigned to h, and
• s
0
is assigned to no hospital, and
• h prefers s
0
to s.2. Second type of instability: There are students s and s
0 and hospitals h and h
0
, so
that
• s assigned to h, and
• s
0
is assigned h
0
, and
• h prefers s
0
to s, and
• s
0 prefers h to h
0
,So, we basically have the Stable Matching Problem, except that (i) hospitals generally
want more than one resident, and (ii) there is a surplus of medical students.
Show that there is always a stable assignment of students to hospitals, and give an
algorithm to find one.Problem 2. (18 points)Suppose you have algorithms with the six running times listed
below. (Assume these ar the exact number of operations performed as a function of the
input size n.) Suppose you have a computed that can perform 1010 operations per second,
and you need to compute a result in at most an hour of computation.For each of the
algorithms, what is the largest input size n for which you would be able to get the result
within an hour?
a) n
2
b) n
3
c) 100n
2
d) nlogn
e) 2
n
f) 2
2
n
Problem 3. (5 points) Show that 2
n+1 is O(2n
)Problem 4. (15 points, 3 points for each case) In each case below, demonstrate
whether the given expression is true or false.You need to provide proper reason to receive
credit.
a) 5n
3 ∈ O(n
3
)
b) 100n
2 ∈ O(n
4
)
c) log n
2 ∈ O(log n)
d) (n
2 + 7n − 10)3 ∈ O(n
6
)
e) √
n ∈ O((log n)
3
)Problem 5. (18 points, 2 points for each row) In each row of the following table,
write whether f = O(g), or f = Ω(g), or both i.e. f = Θ(g)
2
f(n) g(n) Your Answer
100 17
10200 n − 10200
n
2
log 300 n log n
n
100 2
n
n log n n − 100
√
n log n
n
1.01 n + 100
2 log n log(n
2
)
log(n
2
) (log n)
2Problem 6. (18 points) Order the following functions of n from the smallest to largest
complexity. If two have the same complexity put them in one line next to each other.
n log n, (n − 5)(n), 10 log 300, n + 7 log(n
2
),
2
n
, 10n
2 + 15n, 17, 15n
3 + n log n,
3n
8
n6
10200, n2
log 300, 2 log10 10200, n2 + 7 log(n
2
),
3
n
2
, n2 +
√
n, n2
log n, n4 + n
2 + n
2
log n, n4Problem 7. [Complexity Classes] (18 points, 2 points for each function) Order
the following functions of n from the smallest to largest complexity and also identify the
complexity class for each. If two have the same complexity put them in one line next
to each other. Briefly explain next to each line why these functions are in the same
complexity class.
10200, n2
log 300, 2 log10 10200, n2 + 7 log(n
2
),
3
n
2
, n2 +
√
n, n2
log n, n4 + n
2 + n
2
log n, n4Problem 8. (24 points, 3 points for each part) Let f(n) and g(n) by asymptotically
positive functions. Prove or disprove each of the following conjectures.
1. f(n) = O(g(n)) implies g(n) = O(f(n)).
2. f(n) + g(n) = Θ(max(f(n), g(n))).
3. f(n) = O(g(n)) implies lg(f(n)) = O(lg(g(n))), where lg(g(n)) ≥ 1 and f(n) ≥ 1
for all sufficiently large n.
4. f(n) = O(g(n)) implies 2
f(n) = O(2g(n)
).
5. f(n) = O((f(n))2
).
6. f(n) = O(g(n)) implies g(n) = Ω(f(n)).
7. f(n) = Θ(f(n/2)).
8. f(n) + Ω(f(n)) = Θ(f(n)).Problem 9. (24 points – 5 + 5 + 14) Consider the following basic problem. You are
given an array A consisting of n integers A[1], A[2], . . . A[n]. You would like to output a
two-dimensional n − by − n array B in which B[i, j] (for i < j) contains the sum of array
enries A[i] through A[j] – that is, the sum A[1] + A[2], + · · · + A[n]. (The value of array
entry B[i, j] is left unspecified whenever i ≥ j, so it does not matter what is output for
these values.)Following is a simple algorithm to solve this problem.
Algorithm 1 : Summing array entries
for i = 1 to n − 1 do
for j = i + 1 to n do
B[i, j] ←
Pj
k=i A[k]
return B1. For some function f that you should choose, give a bound of the form O(f(n))
on the running time of this algorithm on an input of size n (i.e., a bound on the
number of operations performed by the algorithm)2. For this same function f, show the running time of the algorithm on an input of
size n is also Ω(f(n)). (This shows an asymptoticaly tight bound of Θ(f(n)) on the
running time.3. Although the algorithm given above is the most natural way to solve the problem
– after all, it just iterates through the relevant entries of the array B, filling in a
value for each – it contains some highly unnecessary sources of inefficiency. Give
a different algorith to solve this problem, with an asymptotically better running
time. In other words, you should design an algorithm with running time O(g(n)),
where limn→∞
g(n)/f(n) = 0.