Data Structures and algorithems, Lecture notes of Data Structures and Algorithms

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Introduction: Analysis of

Algorithms, Insertion Sort,

Merge Sort

Lecture 1

Analysis of algorithms

The theoretical study of computer-program

performance and resource usage.

What’s more important than performance?

• modularity
• correctness
• maintainability
• functionality
• robustness
  • user-friendliness
  • programmer time
  • simplicity
  • extensibility
  • reliability

The problem of sorting

Input: sequence  a 1 , a 2 , …, an  of numbers.

Example: Input: 8 2 4 9 3 6

Output: 2 3 4 6 8 9

Output: permutation  a' 1 , a' 2 , … , a'n  such

that a' 1  a' 2 …^  a'n.

Insertion Sort

INSERTION-SORT ( A , n ) ⊳ A [1.. n ] for j ← 1 to n do key ← A [ j ] i ← j – 1 while i > 0 and A [ i ] > key do A [ i+ 1] ← A [ i ] i ← i – 1 A [ i+ 1] = key

“pseudocode”

i j

key sorted

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

Example of Insertion Sort

2 3 4 6 8 9 done

Running time

• The running time depends on the input: an already sorted sequence is easier to sort.
• Parameterize the running time by the size of the input, since short sequences are easier to sort than long ones.
• Generally, we seek upper bounds on the running time, because everybody likes a guarantee.

Machine-independent time

What is insertion sort’s worst-case time?

• It depends on the speed of our computer:
  • relative speed (on the same machine),
  • absolute speed (on different machines).

BIG IDEA:

• Ignore machine-dependent constants.
• Look at growth of T ( n ) as n → ∞.

“Asymptotic Analysis”

Q -notation

• Drop low-order terms; ignore leading constants.
• Example: 3 n^3 + 90 n^2 – 5 n + 6046 = Q( n^3 )

Math: Q( g ( n )) = { f ( n ) : there exist positive constants c 1 , c 2 , and

n 0 such that 0  c 1 g ( n )  f ( n )  c 2 g ( n ) for all n  n 0 }

Engineering: