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Second Midterm Exam, CMPSC 465, Spring 2009
- This is open book exam, you can also have notes (several sheets or one notebook). The use of communication equipment and calculators is not permitted the exam.
- When you describe an algorithm, you should also justify its correctness and your estimate of the running time.
- If you use the extra space on page 4 and 8, indicate what problem and subproblem it is.
- Treaps, 5 pt.
Draw a treap for the following set of keys with priorities (the largest priority should be at the root): key 1 2 3 4 5 6 7 8 9 10 11 priority 5 2 7 0 20 6 2 15 4 1 12
- B-trees, 10 pt.
a. Draw a 2-3 tree that results from the tree below if we delete I. M
G
C I
P
N O T U
M
C G
P
N O T U
M
C G N O
P
T U
b. Draw a 2-3 tree that results from the tree below if we insert Q.
M
G
C I
P
N O T U
M
G
C I
P
N O Q T U
M
G
C I
P
N 0 Q
T
U
- Problem Reduction, 10 pt.
Given a number array A[n] we can in time O(n) compute array S[n] that allows to answer the the following queries: what is the value of
∑b i=a A[i]?^ (We can have^ S[−1] = 0, S[i] = S[i − 1] + A[i], and the query answer is S[b] − S[a − 1].) Use this to solve the following problem in time O(n^2 ): find i, j such that 0 ≤ i < j < n and the following quantity is maximal:
∑^ j
k=i
A[k]
2 −
∑^ j
k=i
A[k]^2
To achieve running tim e O (n^2 ) it suffices to evaluate the above expression for every possible
pair of i and j, 0 ≤ i ≤ j ≤ n and record w hen w e get the largest value. B ecause there are
n^2 such pairs, the only problem is to m ake each evaluation in tim e O (1).
The idea that w e can use is to pre-com pute partial sum s
S [ 0 ] = 0
for (i = 0; i < n; i++)
S [ i+1 ] = S [ i ] +A [ i ]
Then w e get
∑j
k=i A[i]^ as S [ j+1 ] -S [ i ].
W e also have sum of squares. A gain, w e pre-com pute partial sum s
SQ [ 0 ] = 0
for (i = 0; i < n; i++)
SQ [ i+1 ] = SQ [ i ] +A [ i ] *A [ i ]
N ow the expression can be evaluated as follow s
expr = S [ j+1 ] -S [ i ]
expr *= expr
expr += SQ [ j+1 ] -SQ [ i ]
- Radix sort, 10 pt.
Sort given sequence of words alphabetically using Radix sort: ACB, B, CCC, ACA, C, AB, CA, BC, CAC, BCA, CB Show the sequence that results from each round. If your method does not check all words in each round, indicate which words were checked in a round.
W e sort first by length: B , C, A B , CA , B C, CB , A CB , CCC, A CA , CA C, B CA.
N ext, w e list sequences of length 2, follow ed by those of length 3 that have A /B /C on the
third position:
[length 2] A B , CA , B C, CB , [A ] A CA , B CA , [B ] A CB , [C] CCC, CA C,
N ext, w e list sequences of length 1, follow ed by those of length 2/3 that have A /B /C on
the second position (from the last list):
[length 1] B , C, [A ] CA , CA C, [B ] A B , CB , [C] B C, A CA , B CA , A CB , CCC,
Lastly, w e form and joint lists according to the fist position:
[A ] A B , A CA , A CB , [B ] B , B C, B CA , [C] C, CA , CA C, CB , CCC
- Hashing, 5 pt.
Draw the array of chain headers and the chains that form a hash table with hash function h(x) = x mod 11 and which stores the following keys: 3, 5, 9, 12, 14, 17, 21, 33, 42, 50, 55, 62.
I guess you know how to do it, but here is the picture:
0 33 55 1 12 2 3 3 14 4 5 5 6 17 50 7 62 8 9 9 42 10 21
