Searching – Linear and Binary Searches

Comparing Algorithms Should we use Program 1 or Program 2? Is Program 1 “fast”? “Fast enough”? P1P2

You and Igor: the empirical approach Implement each candidate Run itTime it That could be lots of work – also error-prone. Which inputs? What machine/OS ?

Toward an analytic approach … How to solve “which algorithm” problems without machines, test data, or Igor!

The Big Picture Input (n = 3) Output Input (n = 4) Output Input (n = 8) Output How long does it take for the algorithm to finish? Algorithm

Primitives Primitive operations – x = 4assignment –... x arithmetic – if (x < y)...comparison – x[4]index an array – *xdereference (C) – x.foo( )calling a method Others – new/mallocmemory usage

How many foos? for (j = 1; j <= N; ++j) { foo( ); } ΣN j = 1 1 = N

How many foos? for (j = 1; j <= N; ++j) { for (k = 1; k <= M; ++k) { foo( ); } ΣN j = 1 1 = NM ΣM k = 1

How many foos? for (j = 1; j <= N; ++j) { for (k = 1; k <= j; ++k) { foo( ); } ΣN j = 1 1 = Σj k = 1 ΣN j = 1 j = N (N + 1) 2

How many foos? for (j = 0; j < N; ++j) { for (k = 0; k < j; ++k) { foo( ); } for (j = 0; j < N; ++j) { for (k = 0; k < M; ++k) { foo( ); } N(N + 1)/2 NM

How many foos? void foo(int N) { if(N <= 2) return; foo(N / 2); } T(0) = T(1) = T(2) = 1 T(n) = 1 + T(n/2) if n > 2 T(n)= 1 + (1 + T(n/4)) = 2 + T(n/4) = 2 + (1 + T(n/8)) = 3 + T(n/8) = 3 + (1 + T(n/16)) = 4 + T(n/16) … ≈ log 2 n

What is Big-O? Big-O refers to the order of an algorithm runtime growth in relation to the number of items Focus on dominant terms and ignore less significant terms and constants as n grows SortingBigOh

What is Big-O? Big-O refers to the order of an algorithm runtime growth in relation to the number of items I. O(l) - constant time (Push and pop elements on a stack) II. O(n) - linear time The algorithm requires a number of steps proportional to the size of the task. (Finding the minimum of a list) III. O(n 2 ) - quadratic time The number of operations is proportional to the size of the task squared. (Selection and Insertion sort) IV. O(log n) - logarithmic time (Binary search on a sorted list) V. O(n log n) - "n log n " time (Merge sort and quicksort) SortingBigOh

How would you find the first locker with a chinchilla in it?

It’s the 6 th one from the left

Linear search: also called sequential search Examines all values in an array until it finds a match or reaches the end Number of visits for a linear search of an array of n elements: – The average search visits n/2 elements – The maximum visits is n A linear search locates a value in an array in O(n) steps Searching

Pick a number between 1 and 100. You want to determine my number in the least amount of guesses possible. How do you do it? Pick a Number

Locates a value in a sorted array by – Determining whether the value occurs in the first or second half – Then repeating the search in one of the halves Binary Search

15 ≠ 17: we don't have a match

Count the number of visits to search an sorted array of size n – We visit one element (the middle element) then search either the left or right subarray – Thus: T(n) = T(n/2) + 1 where T(n) is time to search an array of size n If n is n/2, then T(n/2) = T(n/4) + 1 Substituting into the original equation: T(n) = T(n/4) + 2 This generalizes to: T(n) = T(n/2 k ) + k Binary Search

Assume n is a power of 2, n = 2 m where m = log 2 (n) Then: T(n) = 1 + log 2 (n) Binary search is an O(log(n)) algorithm Binary Search

Comparison of Sequential and Binary Searches

Assignment Program a linear/sequential search of an array of Strings – Output the index where the element was found. Output must be user friendly – “Your value was found at index 13”; – “Your value was found at index -1” or “Your value was not found” Program a binary search of an ArrayList of Integer – Recursive – “Your value was found in the array” or “Your value was not found in the array”