COMP 171 Data Structures and Algorithms

1
COMP 171Data Structures and Algorithms

• Tutorial 2
• Analysis of algorithms

2
?-notation

• Big-Oh
• f(n) ?(g(n))
• ?(g(n)) f(n) there exist positive constants
  c and n0 such that 0?f(n)?cg(n) for all n?n0
• Upper bound
• Worst-case running time

3
?-notation

• Little-Oh
• f(n) ?(g(n))
• ?(g(n)) f(n) for any positive constants c
  and n0 such that 0?f(n)ltcg(n) for all n?n0
• Non-tight upper bound

4
O-notation

• Big-Omega
• f(n) O(g(n))
• O(g(n)) f(n) there exist positive constants
  c and n0 such that 0 ?cg(n)?f(n) for all n?n0
• Lower bound
• Best-case running time

5
?-notation

• Little-Omega
• f(n) ?(g(n))
• ?(g(n)) f(n) for any positive constants c
  and n0 such that 0 ?cg(n)ltf(n) for all n?n0
• Non-tight lower bound

6
T-notation

• Theta
• f(n) T(g(n))
• T(g(n)) f(n) there exist positive constants
  c1, c2 and n0 such that 0?c1g(n)?f(n)?c2g(n) for
  all n?n0
• Tight bound

7
Summary
Notation Constants ?n?n0
?(g(n)) ?c, n0, both gt 0 0 ? f(n) ? cg(n)
O(g(n)) ?c, n0, both gt 0 0 ? f(n) lt cg(n)
?(g(n)) ?c, n0, both gt 0 0 ? cg(n) ? f(n)
?(g(n)) ?c, n0, both gt 0 0 ? cg(n) lt f(n)
T(g(n)) ?c1, c2, n0, all gt 0 0 ? c1g(n) ? f(n) ? c2g(n)
8
Transitivity

• f(n)T(g(n)), g(n)T(h(n)) ?f(n)T(h(n))
• f(n)?(g(n)), g(n)?(h(n)) ?f(n)?(h(n))
• f(n)O(g(n)), g(n)O(h(n)) ?f(n)O(h(n))
• f(n)?(g(n)), g(n)?(h(n)) ?f(n)?(h(n))
• f(n)?(g(n)), g(n)?(h(n)) ?f(n)?(h(n))

9
Reflexivity, Symmetry Transpose Symmetry

• f(n)T(f(n))
• f(n)?(f(n))
• f(n)O(f(n))
• f(n)T(g(n)) if and only if g(n)T(f(n))
• f(n)?(g(n)) if and only if g(n)O(f(n))
• f(n)?(g(n)) if and only if g(n)?(f(n))

10
Selection Sort

• Input Array A of Size n
• Output A sorted array A
• Algorithm Find the smallest element of A and
  exchanging it with the element in A1. Then find
  the second smallest element of A and exchange it
  with A2. Continue for the first n-1 elements in
  A.

11

• e.g. 5, 2, 4, 7, 3
• Input 5, 2, 4, 7, 3
• 1st iteration 2, 5, 4, 7, 3
• 2nd iteration 2, 3, 4, 7, 5
• 3rd iteration 2, 3, 4, 7, 5
• 4th iteration 2, 3, 4, 5, 7
• Output 2, 3, 4, 5, 7

12

• 1 Find the smallest(m) in the unsorted part
• 2 Swap with h
• 3 Put m into the sorted part
• 4 Back to 1 until unsorted part is size 1

Sorted Part
Unsorted Part
m
h
m
h
m
13

• for i ? range 1
• min value
• min_pos value
• for j ? range 2
• find min
• end for j
• swap(value, value)
• end for i

14
for i ? 1 to n-1 min infinity min_pos
0 for j ? i to n if Aj lt min then min
Aj min_pos j end if end for
j swap(Ai, Amin_pos) end for i
15
for i ? 1 to n-1 min infinity min_pos
0 for j ? i to n if Aj lt min then min
Aj min_pos j end if end for
j swap(Ai, Amin_pos) end for i
O(1)
O(n)
O(n)
O(1)
O(1)
16

• ?(n2)
• O(n2)?
• T(n2)?
• In class exercise
• Improve the algorithm so it can achieve
• ?(n2)
• O(n)
• Given a sorted input sequence, which sorting
  algorithm(s) can achieve O(n)?

17
Binary Search
..
..
..
..
..
..
18

• If tree height is k
• The number of elements is 2k1-1n
• The number of comparison is at most k1
• k1 log22k1 log2 ( n1 )
• ?(? n)