(a) Write pseudocode for an iterative binary search algorithm, looking for v. Show that the running time of your algorithm is Θ(lg n).
(b) Write pseudocode for a recursive binary search algorithm, looking for v. Show that the running time of your algorithm is Θ(lg n).
2. Draw the 13entry hash table that results from using the hash function h(i) = (3i + 5) mod 13 to hash the keys 14, 42, 15, 81, 27, 92, 3, 39, 20, 16, and 5, assuming collisions are handled by:
(a) separate chaining;
(c) quadratic probing (up to the point where the method fails).
(d) Using the secondary hash function h
3. Insert entries with keys, 2, 7, 3, 12, 5, 20, 14, 6, 11, 8, 15, 17, 1, 19, 23, 14 (in this order), into an empty:
(a) heap.
(b) binary search tree.
(c) AVL tree.
(d) (2, 4) tree.
4. Consider the set of keys, 2, 15, 7, 33, 21, 5, 14, 17, 19, 11, 9, 41, 31, 71, 52, 67, 29, 27, 49. Using pruneandsearch paradigm by picking the first element as pivot, determine the 9th element of these keys when they are sorted in ascending order.
5. Find an optimal parenthesization of a matrixchain product whose sequence of dimensions is (7, 10, 5, 18,
20, 40, 10, 15). That is:
matrix dimension
———– —————
A1 7 × 10
A2 10 × 5
A3 5 × 18
A4 18 × 20
A5 20 × 40
A6 40 × 10
A7 10 × 15
6. Determine an LCS (Longest Common Subsequence) of sequences A = (1, 1, 1, 0, 0, 0, 1, 0, 1) and B =(0, 1, 0, 1, 0, 0, 1, 1).
7. Show all the steps for performing any of the following algorithms for matching the pattern ‘belabela’ in
the text ‘bellisalabelabelbelabela’.
(a) bruteforce
(b) BoyerMoore
(c) KnuthMorrisPratt
8. Consider a directed graph G, where its adjucency list is: Adjacent Vertices
————– ————————–
A B, C, D, E
B D, E
C A, B, E
D B, C, F
E A, C, F
F A, C, E
(a) Determine whether or not the graph G is strongly connected.
(b) Draw the transitive closure of graph G (show graphs GA, GB, GC, GD, GE, GF after each step of the FloydWarshall algorithm).
Iterative and Recursive Binary Search with PseudoCode
BinarySearch(A[1..n], v)
//considering the array is sorted in increasing order
low = 0
high = n  1
while low <= high
mid = (low + high) / 2
if A[mid] > v
high = mid  1
else if A[mid] < v
low = mid + 1
else
return mid
return v_not_found
Complexity: O(log n)
b) Iterative Binary Search
BinarySearch(A[1..n], v, low, high)
//considering the array is sorted in increasing order
if high < low
return v_not_found
mid = (low + high) / 2
if A[mid] > v
return BinarySearch(A[], v, low, mid1) //recursive call with first half of array
else if A[mid] < v
return BinarySearch(A[], v, mid+1, high) //recursive call with last half of array
else
return mid
Complexity: O(log n)
For Binary Search, T(N) = T(N/2) + O(1) This is the recurrence relation.
Applying Masters Theorem to compute Run time complexity of recurrence relation:
T(N) = aT(N/b) + f(N)
Now,
a = 1 and b = 2
=> log (a base b) = 1
f(N) = n^c log^k(n) //k = 0 & c = log (a base b)
So, T(N) = O(N^c log^(k+1)N)
= O(log(N))
Question 2
a) Separate Chaining
Hash function h(i) = (3i + 5) mod 13
Set of keys= 14, 42, 15, 81, 27, 92, 3, 39, 20, 16, and 5
0 
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81 
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92 

16 
Linear Probing
Hash function h(i) = (3i + 5) mod 13
Set of keys= 14, 42, 15, 81, 27, 92, 3, 39, 20, 16, and 5
0 
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92 
15 
Quadratic Probing
Hash function h(i) = (3i + 5) mod 13
Set of keys= 14, 42, 15, 81, 27, 92, 3, 39, 20, 16, and 5
0 
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81 
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15 
Secondary Hashing
Hash function h(i) = (3i + 5) mod 13
Hash function h’(i) = 11 − (i mod 11)
Set of keys= 14, 42, 15, 81, 27, 92, 3, 39, 20, 16, and 5
0 
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81 
14 
15 
Method failed with the attempt to insert 92.
Question 3
Keys: 2, 7, 3, 12, 5, 20, 14, 6, 11, 8, 15, 17, 1, 19, 23, 14
Question 4
A = (1, 1, 1, 0, 0, 0, 1, 0, 1)
B = (0, 1, 0, 1, 0, 0, 1, 1)
LCS of A and B is (1, 0, 0, 0, 1, 1)
Question 5
Using the given matrixchain <7, 10, 5, 18, 20, 40, 10, 15>,
A_{1} 7 × 10
A_{2} 10 × 5
A_{3} 5 × 18
A_{4} 18 × 20
A_{5} 20 × 40
A_{6} 40 × 10
A_{7} 10 × 15
p0=7, p1=10, p2=5, p3=18, p4=20, p5=40, p6=10, p7=15
m[i, j] = 0, if i = j,
m[i,j]= {min {m[i,k] + m[k+1, j] + pi –1pkpj}}, if i < j
[1,1] = m[2,2] = m[3,3] = m[4,4] = m[5,5] = m[6,6] = m[7,7] = 0
m[1,2] = p0xp1xp2 = 7x10x5 = 350
m[2,3] = p1xp2xp3 = 10x5x18 = 900
m[3,4] = p2xp3xp4 = 5x18x20 = 1800
m[4,5] = p3xp4xp5 = 18x20x40 = 14400
m[5,6] = p4xp5xp6 = 20x40x10 = 8000
m[6,7] = p4xp5xp6 = 40x10x15 = 6000
m 
1 
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3 
4 
5 
6 
7 
1 
0 
350 

2 
0 
900 

3 
0 
1800 

4 
0 
14400 

5 
0 
8000 

6 
0 
6000 

7 
0 
A) Bruteforce algorithm
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Found pattern at index: 16
B) BoyerMoore Algorithm
Indexes:
Formula for BADMATCH TABLE values = lengthindex1
Letter 
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Value 
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Match Found
Build Comparisons= 7
Search Comparisons= 27
Match Found
Question 8
 a) Yes, graph G is a strongly connected graph as all the nodes are reachable from every other node in the graph as visible from the transitive closure chart below.
 b) Transitive Closure

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