Sequential search. Building Java Programs Chapter 13. Sequential search. Sequential search

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1 Sequental search Buldng Java Programs Chapter 13 Searchng and Sortng sequental search: Locates a target value n an array/lst by examnng each element from start to fnsh. How many elements wll t need to examne? Example: Searchng the array below for the value 42: Copyrght (c) Pearson All rghts reserved. Notce that the array s sorted. Could we take advantage of ths? 2 Sequental search Sequental search search(a[], thngtofnd) start at the begnnng for each tem n A: f the tem s what we're lookng for: return ts locaton f we got ths far wthout returnng already, what we're lookng for sn't found return falure publc statc nt search(nt A[], nt thngtofnd) { for (nt =0; <A.length; ++) { f (A[]==thngToFnd) { return ; /* we ddn t fnd t. return falure */ return -1; 3 4

2 When Array Is Sorted When Array Is Sorted Ths array s already sorted Do we really need to go through entre thng before quttng? Suppose we re searchng for 38: we know once we ve reached A[10] that we ddn t fnd t /* verson 2. ONLY WORKS f A[] IS SORTED */ search(a[], thngtofnd) start at the begnnng for each tem n A <= thngtofnd: f the tem s what we're lookng for: return ts locaton 5 f we got ths far wthout returnng already, what we're lookng for sn't found. return falure 6 When Array Is Sorted Searchng a Sorted Lst What f we re orderng a pzza (and t s 1998)? Lookng for phone number of Sammy s Pzza Start wth the A s, then the B s, etc.? /* verson 2. ONLY WORKS f A[] IS SORTED */ publc statc nt search(nt A[], nt thngtofnd) { for (nt =0; <A.length && A[]<=thngToFnd; ++) { f (A[]==thngToFnd) { return ; return -1; 7 8

3 Bnary search (13.1) bnary search: Locates a target value n a sorted array/lst by successvely elmnatng half of the array from consderaton. How many elements wll t need to examne? Example: Searchng the array below for the value 42: mn md max The Arrays class Class Arrays n java.utl has many useful array methods: Method name bnarysearch(array, value) bnarysearch(array, mnindex, maxindex, value) copyof(array, length) equals(array1, array2) fll(array, value) sort(array) tostrng(array) Descrpton returns the ndex of the gven value n a sorted array (or < 0 f not found) returns ndex of gven value n a sorted array between ndexes mn /max - 1 (< 0 f not found) returns a new reszed copy of an array returns true f the two arrays contan same elements n the same order sets every element to the gven value arranges the elements nto sorted order returns a strng representng the array, such as "[10, 30, -25, 17]" 9 Syntax: Arrays.methodName(parameters) 10 Arrays.bnarySearch Usng bnarysearch // searches an entre sorted array for a gven value // returns ts ndex f found; a negatve number f not found // Precondton: array s sorted Arrays.bnarySearch(array, value) // searches gven porton of a sorted array for a gven value // examnes mnindex (nclusve) through maxindex (exclusve) // returns ts ndex f found; a negatve number f not found // Precondton: array s sorted Arrays.bnarySearch(array, mnindex, maxindex, value) // ndex nt[] a = {-4, 2, 7, 9, 15, 19, 25, 28, 30, 36, 42, 50, 56, 68, 85, 92; nt ndex = Arrays.bnarySearch(a, 0, 16, 42); // ndex1 s 10 nt ndex2 = Arrays.bnarySearch(a, 0, 16, 21); // ndex2 s -7 bnarysearch returns the ndex where the value s found f the value s not found, bnarysearch returns: -(nsertonpont + 1) The bnarysearch method n the Arrays class searches an array very effcently f the array s sorted. You can search the entre array, or just a range of ndexes (useful for "unflled" arrays such as the one n ArrayIntLst) where nsertonpont s the ndex where the element would have been, f t had been n the array n sorted order. To nsert the value nto the array, negate nsertonpont + 1 nt ndextoinsert21 = -(ndex2 + 1); // 6 If the array s not sorted, you may need to sort t frst 11 12

4 Bnary search code // Returns the ndex of an occurrence of target n a, // or a negatve number f the target s not found. // Precondton: elements of a are n sorted order publc statc nt bnarysearch(nt[] a, nt target) { nt mn = 0; nt max = a.length - 1; whle (mn <= max) { nt md = (mn + max) / 2; f (a[md] < target) { mn = md + 1; else f (a[md] > target) { max = md - 1; else { return md; // target found Recursve bnary search (13.3) Wrte a recursve bnarysearch method. If the target value s not found, return ts negatve nserton pont. nt ndex = bnarysearch(data, 42); // 10 nt ndex2 = bnarysearch(data, 66); // -14 return -(mn + 1); // target not found Exercse soluton // Returns the ndex of an occurrence of the gven value n // the gven array, or a negatve number f not found. // Precondton: elements of a are n sorted order publc statc nt bnarysearch(nt[] a, nt target) { return bnarysearch(a, target, 0, a.length - 1); // Recursve helper to mplement search behavor. prvate statc nt bnarysearch(nt[] a, nt target, nt mn, nt max) { f (mn > max) { return -1; // target not found else { nt md = (mn + max) / 2; f (a[md] < target) { // too small; go rght return bnarysearch(a, target, md + 1, max); else f (a[md] > target) { // too large; go left return bnarysearch(a, target, mn, md - 1); else { return md; // target found; a[md] == target 15 Bnary search and objects Can we bnarysearch an array of Strngs? Operators lke < and > do not work wth Strng objects. But we do thnk of strngs as havng an alphabetcal orderng. natural orderng: Rules governng the relatve placement of all values of a gven type. comparson functon: Code that, when gven two values A and B of a gven type, decdes ther relatve orderng: A < B, A == B, A > B 16

5 The compareto method (10.2) Runtme Effcency (13.2) The standard way for a Java class to defne a comparson functon for ts objects s to defne a compareto method. Example: n the Strng class, there s a method: publc nt compareto(strng other) A call of A.compareTo(B) wll return: a value < 0 f A comes "before" B n the orderng, a value > 0 f A comes "after" B n the orderng, or 0 f A and B are consdered "equal" n the orderng. effcency: A measure of the use of computng resources by code. can be relatve to speed (tme), memory (space), etc. most commonly refers to run tme Assume the followng: Any sngle Java statement takes the same amount of tme to run. A method call's runtme s measured by the total of the statements nsde the method's body. A loop's runtme, f the loop repeats N tmes, s N tmes the runtme of the statements n ts body Effcency examples Effcency examples 2 statement1; statement2; statement3; 3 for (nt = 1; <= N; ++) { statement4; for (nt = 1; <= N; ++) { statement5; statement6; statement7; N 3N 4N + 3 for (nt = 1; <= N; ++) { for (nt j = 1; j <= N; j++) { statement1; for (nt = 1; <= N; ++) { statement2; statement3; statement4; statement5; N 2 4N N 2 + 4N 19 How many statements wll execute f N = 10? If N = 1000? 20

6 Algorthm growth rates (13.2) We measure runtme n proporton to the nput data sze, N. growth rate: Change n runtme as N changes. Say an algorthm runs 0.4N N 2 + 8N + 17 statements. Consder the runtme when N s extremely large. We gnore constants lke 25 because they are tny next to N. The hghest-order term (N 3 ) domnates the overall runtme. We say that ths algorthm runs "on the order of" N 3. or O(N 3 ) for short ("Bg-Oh of N cubed") Complexty classes complexty class: A category of algorthm effcency based on the algorthm's relatonshp to the nput sze N. Class Bg-Oh If you double N,... Example constant O(1) unchanged 10ms logarthmc O(log 2 N) ncreases slghtly 175ms lnear O(N) doubles 3.2 sec log-lnear O(N log 2 N) slghtly more than doubles 6 sec quadratc O(N 2 ) quadruples 1 mn 42 sec cubc O(N 3 ) multples by 8 55 mn exponental O(2 N ) multples drastcally 5 * years Bnary search (13.1, 13.3) bnary search successvely elmnates half of the elements. Algorthm: Examne the mddle element of the array. If t s too bg, elmnate the rght half of the array and repeat. If t s too small, elmnate the left half of the array and repeat. Else t s the value we're searchng for, so stop. Whch ndexes does the algorthm examne to fnd value 22? What s the runtme complexty class of bnary search? ndex value Bnary search runtme For an array of sze N, t elmnates ½ untl 1 element remans. N, N/2, N/4, N/8,..., 4, 2, 1 How many dvsons does t take? Thnk of t from the other drecton: How many tmes do I have to multply by 2 to reach N? 1, 2, 4, 8,..., N/4, N/2, N Call ths number of multplcatons "x". 2 x = N x = log 2 N Bnary search s n the logarthmc complexty class. 24

7 Sortng sortng: Rearrangng the values n an array or collecton nto a specfc order (usually nto ther "natural orderng"). one of the fundamental problems n computer scence can be solved n many ways: there are many sortng algorthms some are faster/slower than others some use more/less memory than others some work better wth specfc knds of data some can utlze multple computers / processors,... Sortng methods n Java The Arrays class n java.utl has a statc method sort that sorts the elements of an array Strng[] words = {"foo", "bar", "baz", "ball"; Arrays.sort(words); System.out.prntln(Arrays.toStrng(words)); // [ball, bar, baz, foo] comparson-based sortng : determnng order by comparng pars of elements: <, >, compareto, Selecton sort selecton sort: Orders a lst of values by repeatedly puttng the smallest or largest unplaced value nto ts fnal poston. The algorthm: Look through the lst to fnd the smallest value. Swap t so that t s at ndex 0. Look through the lst to fnd the second-smallest value. Swap t so that t s at ndex Repeat untl all values are n ther proper places. Intal array: Selecton sort example value After 1st, 2nd, and 3rd passes: value value value

8 Selecton sort code // Rearranges the elements of a nto sorted order usng // the selecton sort algorthm. publc statc vod selectonsort(nt[] a) { for (nt = 0; < a.length - 1; ++) { // fnd ndex of smallest remanng value nt mn = ; for (nt j = + 1; j < a.length; j++) { f (a[j] < a[mn]) { mn = j; Selecton sort runtme (Fg. 13.6) What s the complexty class (Bg-Oh) of selecton sort? // swap smallest value ts proper place, a[] swap(a,, mn); Merge sort Merge sort example sort: Repeatedly dvdes the data n half, sorts each half, and combnes the sorted halves nto a sorted whole. ndex value The algorthm: Dvde the lst nto two roughly equal halves. Sort the left half. Sort the rght half. Merge the two sorted halves nto one sorted lst Often mplemented recursvely An example of a "dvde and conquer" algorthm. Invented by John von Neumann n

9 Mergng sorted halves Merge halves code // Merges the left/rght elements nto a sorted result. // Precondton: left/rght are sorted publc statc vod (nt[] result, nt[] left, nt[] rght) { nt 1 = 0; // ndex nto left array nt 2 = 0; // ndex nto rght array 33 for (nt = 0; < result.length; ++) { f (2 >= rght.length (1 < left.length && left[1] <= rght[2])) { result[] = left[1]; 1++; else { result[] = rght[2]; 2++; // take from left // take from rght 34 Merge sort code // Rearranges the elements of a nto sorted order usng // the sort algorthm. publc statc vod Sort(nt[] a) { // array nto two halves nt[] left = Arrays.copyOfRange(a, 0, a.length/2); nt[] rght = Arrays.copyOfRange(a, a.length/2, a.length); // sort the two halves... // the sorted halves nto a sorted whole (a, left, rght); Merge sort code 2 // Rearranges the elements of a nto sorted order usng // the sort algorthm (recursve). publc statc vod Sort(nt[] a) { f (a.length >= 2) { // array nto two halves nt[] left = Arrays.copyOfRange(a, 0, a.length/2); nt[] rght = Arrays.copyOfRange(a, a.length/2, a.length); // sort the two halves Sort(left); Sort(rght); // the sorted halves nto a sorted whole (a, left, rght); 35 36

10 Merge sort runtme What s the complexty class (Bg-Oh) of sort? 37

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