Binary Search, Hashing, Tree Insertion And Search, Longest Common Subsequence, Pattern Matching, Transitive Closure Of A Directed Graph
Iterative and Recursive Binary Search with Pseudo-Code
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, mid-1) //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
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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
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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
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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
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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 matrix-chain <7, 10, 5, 18, 20, 40, 10, 15>,
A1 7 × 10
A2 10 × 5
A3 5 × 18
A4 18 × 20
A5 20 × 40
A6 40 × 10
A7 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
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A) Brute-force algorithm
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Found pattern at index: 16
B) Boyer-Moore Algorithm
Indexes:
Formula for BAD-MATCH TABLE values = length-index-1
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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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