CSC 204 – Programming I

Searching & Sorting

13.4 Sorting One of the most fundamental operations

on data is sorting: putting a list of items in order.

Examples of sorted lists:Numbers: –53 –38 9 16 45 66 89Characters: 0 1 A B a bStrings: aardvark ant anteater bat cat

In Java, the ordering of characters is based on their Unicode values.

Strings are normally sorted into lexicographic order.

Ascending vs. Descending Data items can be sorted into ascending

order (with smaller values coming before larger ones), or descending order:Ascending order: –53 –38 9 16 45 66 89Descending order: 89 66 45 16 9 –38 –53

Sorting can be very time consuming if not done properly.

Fortunately, a number of good sorting algorithms are known.

Inserting into a Sorted Array An algorithm for inserting items into an

empty array, keeping the items in sorted order at all times:1. For all positions in the array, from 0 to n – 1 (where n is the number of items stored in the array), test whether the item to be inserted is less than the one stored at position i. If so, break out of the loop.2. Insert the new item at position i after moving the items at positions i through n – 1 down one position.

Inserting into a Sorted Array The insertion algorithm expressed in Java:

int i;for (i = 0; i < n; i++) if (itemToInsert < a[i]) break;for (int j = n; j > i; j--) a[j] = a[j-1];a[i] = itemToInsert;n++;

This algorithm works even if the new item is larger than all the ones already in the array, or if the array contains no items at all.

Inserting into a Sorted Array If the array contains values other than

numbers, the condition in the if statement will need to be changed.

The test condition if a contains strings: itemToInsert.compareTo(a[i]) < 0

Inserting into a Sorted Array The algorithm for inserting items into an

array illustrates an important design principle:Design an algorithm to work in the typical case. Then go back and check to see if it correctly handles the special cases. If it doesn’t, add additional tests for those cases.

Inserting into a Sorted Array Snapshots of an array as the numbers 83,

24, 56, 90, and 17 are inserted:

Insertion Sort The idea of inserting a new item into a set

of items that’s already in order can be used to sort an array even if all the items are already stored in the array prior to sorting.

An array a can be sorted by inserting the element at position i into the ones at positions 0 through i – 1 (which will already be in order).

Insertion Sort In other words:

Insert the element at position 1 into the elements at positions 0 through 0.

Insert the element at position 2 into the elements at positions 0 through 1.

Insert the element at position 3 into the elements at positions 0 through 2, and so on.

This sorting algorithm is known as insertion sort.

Insertion Sort The insertion sort algorithm is simpler if

the element to be inserted (the one at position i) is compared with the elements at positions i – 1, i – 2, …, 0, working backward instead of forward.

If a comparison indicates that the element to be inserted is smaller than the one at position j, then the element at position j can be moved down one place in the array.

Insertion Sort An example that uses insertion sort to sort

an array of five integers:

Shaded elements are not yet in sorted order.

Insertion Sort A Java version of insertion sort:

for (int i = 1; i < a.length; i++) { int itemToInsert = a[i]; int j = i - 1; while (j >= 0 && itemToInsert < a[j]) { a[j+1] = a[j]; j--; } a[j+1] = itemToInsert;}

Insertion Sort If the array contains values other than

numbers, the condition in the while statement will need to be changed.

The while statement if a contains strings:while (j >= 0 && itemToInsert.compareTo(a[j]) < 0) { …}

Insertion Sort Insertion sort is easy to implement but not

very efficient. In the worst case, it performs

1 + 2 + 3 + … + (n – 1) = n(n – 1)/2comparisons, where n is the number of elements in the array.

This number is approximately n2/2, so the time required to sort grows rapidly as n increases.

Insertion Sort Comparisons performed by insertion sort for

various array sizes: Array Number of Size Comparisons

10 45100 4,950

1,000 499,50010,000 49,995,000

100,000 4,999,950,000 For small arrays, insertion sort is usually fast

enough. For larger arrays, better algorithms are

necessary.

Program: Sorting a Series of Lines The SortLines program will sort a series

of lines entered by the user. When the user enters a blank line, SortLines will display the lines entered by the user, sorted into lexicographic order.

User Interface An example:

Enter lines to be sorted, ending with a blank line.To be, or not to be: that is the question:Whether 'tis nobler in the mind to sufferThe slings and arrows of outrageous fortune,Or to take arms against a sea of troubles,And by opposing end them? To die: to sleep;No more; and, by a sleep to say we endThe heartache and the thousand natural shocksThat flesh is heir to, 'tis a consummationDevoutly to be wish'd. To die, to sleep;To sleep: perchance to dream: ay, there's the rub;For in that sleep of death what dreams may come user enters a blank line

User Interface An example (continued):

And by opposing end them? To die: to sleep;Devoutly to be wish'd. To die, to sleep;For in that sleep of death what dreams may comeNo more; and, by a sleep to say we endOr to take arms against a sea of troubles,That flesh is heir to, 'tis a consummationThe heartache and the thousand natural shocksThe slings and arrows of outrageous fortune,To be, or not to be: that is the question:To sleep: perchance to dream: ay, there's the rub;Whether 'tis nobler in the mind to suffer

Program Design: SortLines Because the number of lines isn’t known in

advance, a vector would be a good data structure.

The lines are read one by one, so they can be inserted into the vector so that it remains sorted at all times.

The insertElementAt method will automatically shift existing elements to make room for the new one.

SortLines.java// Sorts a series of lines entered by the user

import java.util.*;import jpb.*;

public class SortLines { public static void main(String[] args) { // Prompt user to enter lines System.out.println("Enter lines to be sorted, " + "ending with a blank line.");

// Create vector to store lines Vector v = new Vector();

// Read lines and insert into vector; elements will be // kept in order at all times while (true) {

// Read a line; exit loop if line is empty String inputLine = SimpleIO.readLine(); if (inputLine.length() == 0) break;

// Determine where line should be stored in vector int i; for (i = 0; i < v.size(); i++) if (inputLine.compareTo((String) v.elementAt(i)) < 0) break;

// Insert line at this position. Existing elements will // be moved to make room for the new element. v.insertElementAt(inputLine, i); } // Display elements of vector for (int i = 0; i < v.size(); i++) System.out.println(v.elementAt(i)); }}

try { FileReader fileIn = new FileReader(filename); BufferedReader in = new BufferedReader(fileIn);  int index = 0; String line; while ( index < A.length ){ line = in.readLine(); if (line == null) break;  lines[index++] = line; }

in.close();} catch (IOException e){ System.out.println(e.getMessage());}

Reading lines from a file

On-line Resources The Java Tutorials at SUN’s website

http://java.sun.com/docs/books/tutorial/essential/io/overview.html

13.5 Searching If the elements of an array (or vector)

aren’t in any particular order, there is no better way to search for a particular element than sequential search.

If the elements are sorted, a more efficient strategy, binary search, becomes possible.

Binary Search Suppose that a is a sorted array and key is

the value to be located in the array. The binary search algorithm compares key

with the element in the middle of a. If key is smaller than the middle element, the

algorithm can limit its search to the first half of a.

If key is larger, the algorithm searches only the last half of a.

This process is repeated until the key is found or there are no elements left to search.

Binary Search The binary search algorithm will use

variables named low and high that keep track of which array elements are still under consideration.

A third variable, mid, will indicate the halfway point between low and high.

Binary Search A Java version of the binary search

algorithm:int low = 0;int high = a.length - 1;

while (low < high) { int mid = (low + high) / 2; if (a[mid] < key) low = mid + 1; else high = mid;}

Binary Search If a contains values other than numbers,

the condition in the if statement will need to be changed.

If the elements of a are strings, the statement becomesif (a[mid].compareTo(key) < 0) …

Binary Search If the key is present in the array, the final

value of low indicates its position in the array.

If the key isn’t known to be present in the array, an additional test will be needed after the while loop terminates:if (a[low] == key) // Key was foundelse // Key was not found

Binary Search A binary search for the number 40:

Binary Search Binary search works by repeatedly dividing

the array in half, so it is extremely efficient.

In the worst case, it performs log2 n comparisons, rounded up to the nearest integer, where n is the number of elements in the array.

Binary Search The number of comparisons performed by

binary search remains small even for very large values of n, because log2 n is such a slow-growing function. Array Number of Size Comparisons

10 4100 7

1,000 1010,000 14

100,000 17

Program: Determining Air Mileage The AirMileage program will display the

air mileage from New York City to an international city specified by the user:This program finds the air mileage betweenNew York and major international cities.

Enter city name: OsloOslo is 3671 miles from New York City.

Program Design: AirMileage The AirMileage program will store the

names of 36 cities in an array. A parallel array will store the distances

from New York to each of the cities in the first array.

Using binary search to locate the city name entered by the user will be more efficient than using sequential search.

AirMileage.javaimport jpb.*;

public class AirMileage { // Names of international cities private static final String[] CITY_NAMES = {"acapulco", "amsterdam", "antigua", "aruba", "athens", "barbados", "bermuda", "bogota", "brussels", "buenos aires", "caracas", "copenhagen", "curacao", "frankfurt", "geneva", "glasgow", "hamburg", "kingston", "lima", "lisbon", "london", "madrid", "manchester", "mexico City", "milan", "nassau", "oslo", "paris", "reykjavik", "rio de janeiro", "rome", "san juan", "santo domingo", "st. croix", "tel aviv", "zurich"};

// Distances from New York to other cities private static final int[] DISTANCES = {2260, 3639, 1783, 1963, 4927, 2100, 771, 2487, 3662, 5302, 2123, 3849, 1993, 3851, 3859, 3211, 3806, 1583, 3651, 3366, 3456, 3588, 3336, 2086, 4004, 1101, 3671, 3628, 2600, 4816, 4280, 1609, 1560, 1680, 5672, 3926};

public static void main(String[] args) { // Display initial message System.out.println( "This program finds the air mileage between\n" + "New York and major international cities.\n");

// Prompt user for city name SimpleIO.prompt("Enter city name: "); String cityName = SimpleIO.readLine().trim();

// Use binary search to locate name in CITY_NAMES array int i = binarySearch(CITY_NAMES, cityName.toLowerCase());

// If name was found in array, display distance from New // York to chosen city if (cityName.equalsIgnoreCase(CITY_NAMES[i])) System.out.println(cityName + " is " + DISTANCES[i] + " miles from New York City."); else System.out.println(cityName + " wasn't found."); }

// Uses the binary search algorithm to locate key in the // array a. Returns the index of key if it is found in a. private static int binarySearch(String[] a, String key) { int low = 0; int high = a.length - 1;

while (low < high) { int mid = (low + high) / 2; if (a[mid].compareTo(key) < 0) low = mid + 1; else high = mid; } return low; }}