Kk20503 1 introduction

Introductions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

About The Course ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Course Delivery ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Lectures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Textbook ,[object Object],[object Object],[object Object]

Dr James Mountstephens 1: Introduction

Contents ,[object Object],[object Object],[object Object],[object Object]

What is an Algorithm? ,[object Object],[object Object],“ computer” problem algorithm input output solution

What is an Algorithm? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

What is an Algorithm? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

What is an Algorithm? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example Problem: Greatest Common Divisor ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example: Euclid’s Algorithm for GCD ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],m mod n is the remainder of m/n

Two Descriptions of Euclid’s algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],Step 1 If n = 0, return m and stop; otherwise go to Step 2 Step 2 Divide m by n and assign the value of the remainder to r Step 3 Assign the value of n to m and the value of r to n. Go to Step 1. ,[object Object]

Other Algorithms for GCD ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Other Algorithms for GCD ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Other Algorithms for GCD ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],48 180 gcd = 12

Other Algorithms for GCD ,[object Object],[object Object],[object Object],[object Object]

Other Algorithms for GCD ,[object Object],[object Object],[object Object],[object Object],[object Object],Prime Factorisation Input: Integer x ≥ 2, Output: List F of prime factors of x P ← Sieve(x) while n > 1 do while n mod P[i] = 0 do F ← F + P[i] x ← x / P[i] i ← i + 1 Sieve of Eratosthenes Input: Integer x ≥ 2 Output: List of primes less than or equal to x for p ← 2 to x do A [ p ] ← p for p ← 2 to x do if A [ p ]  0 j ← p * p while j ≤ x do A [ j ] ← 0 j ← j + p ,[object Object],[object Object]

Fundamentals of Algorithmic Problem Solving ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Algorithm Design Techniques ,[object Object],[object Object],[object Object],[object Object],[object Object]

Analysing an Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Important Problem Types ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sorting Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sorting Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Sorting Problems SelectionSort(A[0..n-1]) //Input: An array A[0..n-1] of orderable elements //Output: Array A[0..n-1] sorted in ascending order for i ← 0 to n – 2 do min ← i for j ← i + 1 to n – 1 do if A[j] < A[min] min ← j swap A[i] and A[min] ,[object Object]

Searching Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Searching Problems ,[object Object],[object Object],[object Object],Binary Search // Input: sorted array a_i < … < a_j and key x; m  (i+j)/2; while i < j and x != a_m do if x < a_m then j  m-1 else i  m+1; if x = a_m then output a_m;

String Processing Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Graph Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 2 3 4

Graph Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Combinatorial Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Combinatorial Problems ,[object Object],[object Object]

Geometric Problems ,[object Object],[object Object],[object Object],[object Object]

Numerical Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Fundamental Data Structures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Linear Data Structures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Linear Data Structures ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Stacks and Queues ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Priority Queues and Heaps ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],9 6 8 5 2 3 9 6 5 8 2 3

Graphs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Complete, undirected graph Incomplete, directed graph Both graphs here are dense 1 2 3 4 1 2 3 4

Graph Representation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 Incomplete, directed graph G Adjacency Matrix for G Adjacency Linked List for G 2 3 4 4 4 1 2 3 4

Weighted Graphs ,[object Object],[object Object],6 8 5 7 9 1 2 3 4

Graph Properties -- Paths and Connectivity ,[object Object],[object Object],[object Object],[object Object],Simple Path from 4 to 3 (4,1,2,3) Path Length 3 Path from 4 to 3 (4,1,2,5,4,1,2,3) Path Length 7 1 2 4 5 3 1 2 4 5 3

Graph Properties -- Paths and Connectivity ,[object Object],[object Object],[object Object],[object Object],Connected Graph G1 Non-Connected Graph G2 1 2 4 5 3 1 2 4 5 3

Graph Properties -- Acyclicity ,[object Object],[object Object],[object Object],[object Object],[object Object],Directed Acyclic Graph Cycle (1,2,5,4,1) 1 2 3 4 1 2 4 5 3

Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Free Tree Free Tree that is also a forest 1 3 2 4 5 1 3 2 4 5 6 7

Rooted Trees ,[object Object],[object Object],[object Object],Free Tree Rooted Tree with 2 levels 1 3 2 4 5 1 3 2 4 5

Rooted Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 3 2 4 5 7 6 8 9

Rooted Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 3 2 4 5 7 6 8 9

Ordered Trees ,[object Object],[object Object],[object Object],[object Object],Binary Tree L R L 9 6 8 5 2 3

Ordered Trees ,[object Object],[object Object],[object Object],[object Object],6 3 9 2 5 8

Sets and Dictionaries ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Kk20503 1 introduction

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Kk20503 1 introduction