Merge Sort Algorithm: Analysis, Recurrences, and Solving Techniques, Slides of Computer Science

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Algorithms

Merge Sort

Solving Recurrences

The Master Theorem

Administrative Question

• Who here cannot make Monday-Wednesday

office hours at 10 AM?

• If nobody, should we change class time?

Review: Asymptotic Notation

• Upper Bound Notation:
  • f(n) is O(g(n)) if there exist positive constants c

and n 0 such that f(n) ≤ c ⋅ g(n) for all n ≥ n (^0)

• Formally, O(g(n)) = { f(n): ∃ positive constants c

and n 0 such that f(n) ≤ c ⋅ g(n) ∀ n ≥ n 0

• Big O fact:
  • A polynomial of degree k is O( nk )

Review: Asymptotic Notation

• Asymptotic lower bound:
  • f(n) is Ω (g(n)) if ∃ positive constants c and n 0 such

that 0 ≤ c⋅g(n) ≤ f(n) ∀ n ≥ n 0

• Asymptotic tight bound:
  • f(n) is Θ (g(n)) if ∃ positive constants c 1 , c 2 , and n 0

such that c 1 g(n) ≤ f(n) ≤ c 2 g(n) ∀ n ≥ n 0

• f(n) = Θ(g(n)) if and only if

f(n) = O(g(n)) AND f(n) = Ω(g(n))

Merge Sort

MergeSort(A, left, right) {

if (left < right) { mid = floor((left + right) / 2); MergeSort(A, left, mid); MergeSort(A, mid+1, right); Merge(A, left, mid, right); }

}

// Merge() takes two sorted subarrays of A and

// merges them into a single sorted subarray of A

// (how long should this take?)

Merge Sort: Example

• Show MergeSort() running on the array

A = {10, 5, 7, 6, 1, 4, 8, 3, 2, 9};

Recurrences

• The expression:

is a recurrence.

• Recurrence: an equation that describes a function

in terms of its value on smaller functions

cn n

n T

c n

T n

Recurrence Examples

n

n

c s n

s n 

n s n n

n s n

c n

n T

c n

T n

cn n b

n aT

c n

T n

Solving Recurrences

• The substitution method (CLR 4.1)
  • A.k.a. the “making a good guess method”
  • Guess the form of the answer, then use induction

to find the constants and show that solution works

• Examples:
  • T(n) = 2T(n/2) + Θ(n)  T(n) = Θ(n lg n)
  • T(n) = 2T(n/2) + n  ???

Solving Recurrences

• The substitution method (CLR 4.1)
  • A.k.a. the “making a good guess method”
  • Guess the form of the answer, then use induction

to find the constants and show that solution works

• Examples:
  • T(n) = 2T(n/2) + Θ(n)  T(n) = Θ(n lg n)
  • T(n) = 2T(n/2) + n  T(n) = Θ(n lg n)
  • T(n) = 2T(n/2 )+ 17) + n  ???

Solving Recurrences

• Another option is what the book calls the

“iteration method”

• Expand the recurrence
• Work some algebra to express as a summation
• Evaluate the summation
• We will show several examples

c s n n

n s n

• s(n) =

c + s(n-1)

c + c + s(n-2)

2c + s(n-2)

2c + c + s(n-3)

3c + s(n-3)

kc + s(n-k) = ck + s(n-k)

c s n n

n s n

• So far for n >= k we have
  • s(n) = ck + s(n-k)
• What if k = n?
  • s(n) = cn + s(0) = cn
• So
• Thus in general
  • s(n) = cn

c s n n

n s n

n s n n

n s n

• s(n)

= n + s(n-1)

= n + n-1 + s(n-2)

= n + n-1 + n-2 + s(n-3)

= n + n-1 + n-2 + n-3 + s(n-4)

= …

= n + n-1 + n-2 + n-3 + … + n-(k-1) + s(n-k)