4.2 Sorting and Searching. Section 4.2

Size: px

Start display at page:

Download "4.2 Sorting and Searching. Section 4.2"

Ashley Ray
5 years ago
Views:

1 4.2 Sorting and Searching 1

2 Sequential Search Scan through array, looking for key. Search hit: return array index. Search miss: return -1. public static int search(string key, String[] a) { int N = a.length; for (int i = 0; i < a.length; i++) if (a[i].compareto(key) == 0) return i; return -1; } 2

3 Searching Challenge 1 Q. A credit card company needs to whitelist 10 million customer account numbers, processing 10,000 transactions per second. Using sequential search, what kind of computer is needed? A. Toaster. B. Cell phone. C. Your laptop. D. Supercomputer. E. Google server farm. 3

4 Twenty Questions Intuition: Find a hidden integer 4

5 Binary Search Main Idea: Sort the array (stay tuned) Play "20 questions" to determine index with given key Examples: Dictionary, phone book, index, credit card numbers,... Binary Search: Examine the middle key If it matches, return its index Otherwise, search either the left or right half 5

6 Binary Search Invariant: Algorithm maintains a[lo] key a[hi-1] public static int search(string key, String[] a) { return search(key, a, 0, a.length); } public static int search(string key, String[] a, int lo, int hi) { if (hi <= lo) return -1; int mid = lo + (hi - lo) / 2; int cmp = a[mid].compareto(key); if (cmp > 0) return search(key, a, lo, mid); else if (cmp < 0) return search(key, a, mid+1, hi); else return mid; } 6

7 Binary Search Analysis: Binary search in an array of size N One compare Binary search in array of size N/2 N N / 2 N / 4 N / 8 1 Q. How many times can you divide by 2 until you reach 1? A. log 2 N

8 Searching Challenge 2 Q. A credit card company needs to whitelist 10 million customer account numbers, processing 10,000 transactions per second. Using binary search, what kind of computer is needed? A. Toaster. B. Cell phone. C. Your laptop. D. Supercomputer. E. Google server farm. 8

9 Sorting pentrust.org 9

10 Sorting pentrust.org shanghaiscrap.org 10

11 Sorting pentrust.org shanghaiscrap.org De Beers 11

12 Sorting Sorting problem. Rearrange N items in ascending order. Applications. Statistics, databases, data compression, bioinformatics, computer graphics, scientific computing, (too numerous to list),... Hauser Hong Hsu Hayes Haskell Hanley Hornet Hanley Haskell Hauser Hayes Hong Hornet Hsu 12

13 Insertion Sort The Bridge Table 13

14 Insertion Sort Brute-force sorting solution. Move left-to-right through array. Exchange value with larger ones to left, one-by-one. 14

15 Insertion Sort Brute-force sorting solution. Move left-to-right through array. Exchange value with larger ones to left, one-by-one. 15

16 Insertion Sort public class Insertion { public static void sort(int[] a) { int N = a.length; for (int i = 1; i < N; i++) for (int j = i; j > 0; j--) if (a[j-1] < a[j]) exch(a, j-1, j); else break; } } private static void exch(int[] a, int i, int j) { int swap = a[i]; a[i] = a[j]; a[j] = swap; } 16

17 Insertion Sort: Call By Reference public class Insertion { public static void sort(int[] a) { int N = a.length; for (int i = 1; i < N; i++) for (int j = i; j > 0; j--) if (a[j-1] < a[j]) exch(a, j-1, j); else break; } } private static void exch(int[] b, int i, int j) { int swap = b[i]; b[i] = b[j]; b[j] = swap; } 17

18 Insertion Sort: Observation Observe and tabulate running time for various values of N. Data source: N random numbers between 0 and 1. Machine: imac Core i5 2.7GH, 12GB RAM. Timing: System.currentTimeMillis(). N 5,000 10,000 20,000 40,000 80,000 Comparisons 6.2 million 25 million 99 million 400 million 1600 million Time seconds seconds seconds 0.79 seconds seconds 18

19 Comparsions (millions) Empirical Analysis Observation. Number of compares depends on input family. Descending: ~ N 2 / 2 Random: ~ N 2 / 4 Ascending: ~ N Descending Random Ascending Input Size 19

20 Mathematical Analysis Worst Case. (descending) Iteration i requires i comparisons. Total = ( N-1) ~ N 2 / 2 compares. E F G H I J D C B A i Average Case. (random) Iteration i requires i / 2 comparisons on average. Total = ( N-1) / 2 ~ N 2 / 4 compares A C D F H J E B I G i 20

Mathematical Analysis Worst Case. (descending) Iteration i requires i comparisons. Total = (0 + 1 + 2 +... + N-1) ~ N 2 / 2 compares.

21 Mathematical Analysis Worst Case. (descending) Iteration i requires i comparisons. Total = ( N-1) ~ N 2 / 2 compares. E F G H I J D C B A i Average Case. (random) Iteration i requires i / 2 comparisons on average. Total = ( N-1) / 2 ~ N 2 / 4 compares A C D F H J E B I G i 21

22 Sorting Challenge 1 Q. A credit card company sorts 10 million customer account numbers, for use with binary search. Using insertion sort, what kind of computer is needed? A. Toaster B. Cell phone C. Your laptop D. Supercomputer E. Google server farm 22

23 Insertion Sort: Lesson Lesson. Supercomputer can't rescue a bad algorithm. Computer Comparisons Per Second Thousand Million Billion laptop 10 7 instant 1 day 3 centuries super instant 1 second 2 weeks 23

24 Moore's Law Moore's Law. Transistor density on a chip doubles every 2 years. Variants. Memory, disk space, bandwidth, computing power/$. 24

Moore's Law and Algorithms Quadratic algorithms do not scale with technology. New computer may be 10x as fast. But, has 10x as much memory so problem may be 10x bigger.

25 Moore's Law and Algorithms Quadratic algorithms do not scale with technology. New computer may be 10x as fast. But, has 10x as much memory so problem may be 10x bigger. With quadratic algorithm, takes 10x as long! Software inefficiency can always outpace Moore's Law. Moore's Law isn't a match for our bad coding. Jaron Lanier Lesson. Need linear or N log N (linearithmic) algorithm to keep pace with Moore's law. 25

26 Mergesort 26

27 Mergesort Algorithm: Divide array into two halves. Recursively sort each half. Merge two halves to make sorted whole. 27

28 Mergesort: Example 28

29 Merging Merging. Combine two pre-sorted lists into a sorted whole. How to merge efficiently? Use an auxiliary array. 29

30 Merging Merging. Combine two pre-sorted lists into a sorted whole. How to merge efficiently? Use an auxiliary array. int[] aux = new int[n]; // merge into auxiliary array int i = lo, j = mid; for (int k = 0; k < N; k++) { if (i == mid) aux[k] = a[j++]; else if (j == hi) aux[k] = a[i++]; else if (a[j] < a[i]) aux[k] = a[j++]; else aux[k] = a[i++]; } // copy back for (int k = 0; k < N; k++) { a[lo + k] = aux[k]; } 30

31 Mergesort: Java Implementation public class Merge { public static void sort(int[] a) { sort(a, 0, a.length); } // Sort a[lo, hi). public static void sort(int[] a, int lo, int hi) { int N = hi - lo; if (N <= 1) return; // recursively sort left and right halves int mid = lo + N/2; sort(a, lo, mid); sort(a, mid, hi); } } // merge sorted halves (see previous slide) 31

Mergesort: Mathematical Analysis Analysis. To mergesort array of size N, mergesort two subarrays of size N / 2, and merge them together using N compares.

32 Mergesort: Mathematical Analysis Analysis. To mergesort array of size N, mergesort two subarrays of size N / 2, and merge them together using N compares. T(N) N T(N / 2) T(N / 2) 2 (N / 2) T(N / 4) T(N / 4) T(N / 4) T(N / 4) 4 (N / 4) T(N / 2 k ) log 2 N... T(2) T(2) T(2) T(2) T(2) T(2) T(2) T(2) N / 2 (2) N log 2 N 32

Validation. Theory agrees with observations.

33 Mergesort: Mathematical Analysis Mathematical analysis. analysis worst average best comparisons N log 2 N N log 2 N 1/2 N log 2 N Validation. Theory agrees with observations. N 10, million 50 million actual 120 thousand 460 million 1,216 million predicted 133 thousand 485 million 1,279 million 33

34 Sorting Challenge 2 Q. A credit card company sorts 10 million customer account numbers, for use with binary search. Using mergesort, what kind of computer is needed? A. Toaster B. Cell phone C. Your laptop D. Supercomputer E. Google server farm 34

35 Mergesort: Lesson Lesson. Great algorithms can be more powerful than supercomputers. Computer Compares Per Second Insertion Mergesort laptop centuries 3 hours super weeks instant N = 1 billion 35

Sorting. 4.2 Sorting and Searching. Sorting. Sorting. Insertion Sort. Sorting. Sorting problem. Rearrange N items in ascending order.

Sorting. 4.2 Sorting and Searching. Sorting. Sorting. Insertion Sort. Sorting. Sorting problem. Rearrange N items in ascending order. 4.2 and Searching pentrust.org Introduction to Programming in Java: An Interdisciplinary Approach Robert Sedgewick and Kevin Wayne Copyright 2002 2010 23/2/2012 15:04:54 pentrust.org pentrust.org shanghaiscrap.org