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Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem Introduction to algorithms July 4, 2017

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Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem Overview 1 Insertion sort 2 Asymptotic notations 3 Decision-trees and complexity 4 Divide and conquer Merge sort Recurrence relations and the master theorem
Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem The sorting problem Notations: We denote the set { 1 , 2 , 3 , ..., n } by [ n ]. We denote the set of all permutations of [ n ] by S n (this is actually a group). Given a sequence of real numbers a = ( a 1 , a 2 , ..., a n ) we would like to reorder them from the smallest to the largest. The Sorting problem: Given a sequence of (different) real numbers a = ( a 1 , a 2 , ..., a n ), find the (unique) permutation π S n such that: a π (1) a π (2) ... a π ( n )

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Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem The sorting problem Notations: We denote the set { 1 , 2 , 3 , ..., n } by [ n ]. We denote the set of all permutations of [ n ] by S n (this is actually a group). Given a sequence of real numbers a = ( a 1 , a 2 , ..., a n ) we would like to reorder them from the smallest to the largest. The Sorting problem: Given a sequence of (different) real numbers a = ( a 1 , a 2 , ..., a n ), find the (unique) permutation π S n such that: a π (1) a π (2) ... a π ( n )
Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem The sorting problem Notations: We denote the set { 1 , 2 , 3 , ..., n } by [ n ]. We denote the set of all permutations of [ n ] by S n (this is actually a group). Given a sequence of real numbers a = ( a 1 , a 2 , ..., a n ) we would like to reorder them from the smallest to the largest. The Sorting problem: Given a sequence of (different) real numbers a = ( a 1 , a 2 , ..., a n ), find the (unique) permutation π S n such that: a π (1) a π (2) ... a π ( n )

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Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem The sorting problem We will consider sorting algorithms that uses only comparisons (and not algebraic properties of numbers for example). Such algorithms do not assume anything other then the existence of an order relation so they can be applied for objects other than numbers. We will take a brief look at algorithms that use other properties and see the differences.
Introduction to algorithms Insertion sort Asymptotic notations Decision-trees and complexity Divide and conquer Merge sort Recurrence relations and the master theorem The sorting problem

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