Chapter 14: Sorting and Searching

An Introduction to Programming and Object

Oriented Design using Java2nd Edition. May 2004

Jaime NiñoFrederick Hosch

Chapter 14: Sorting and Searching

2May 2004 NH-Chapter 14

Objectives After studying this chapter you should understand the

following: orderings and the ordering of list elements; the simple sorting algorithms selection sort and bubble

sort; how to generalize sort methods. the binary search algorithm. the notion of a loop invariant, and its role in reasoning

about methods.

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Objectives Also, you should be able to:

trace a selection sort and bubble sort with a specific list of values;

sort a list by instantiating an ordering and using a predefined sort method;

trace a binary search with a specific list of values; state and verify key loop invariants.

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Ordering lists To order a list, there must be an order on the element class.

We’ll assume There is a boolean method inOrder defined for the class whose

instances we want to order.

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Ordering lists

Thus if s1 and s2 are Student objects, inOrder(s1,s2) true: s1 comes before s2. inOrder(s1,s2) false: s1 need not come

before s2.

Example: to order a List<Student> need

public boolean inOrder (Student first, Student second)

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Ordering lists Ordering alphabetically by name, inOrder(s1,s2) is true

if s1’s name preceded s2’s name lexicographically.

Ordering by decreasing grade, inOrder(s1,s2) is true if s1’s grade was greater than s2’s.

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Order properties We write

s1 < s2 when inOrder(s1,s2) == true s1 >= s2 when inOrder(s1,s2)== false

An ordering is antisymmetric: it cannot be the case that both s1 < s2 and s2 < s1.

An ordering is transitive. That is, if s1 < s2 and s2 < s3 for objects s1, s2, and s3, then s1 < s3.

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Order properties Equivalence of objects: neither inOrder(s1,s2) nor inOrder(s2,s1) is true for objects s1 and s2.

Two equivalent objects do not have to be equal.

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Ordered list

Or

A list is ordered: s1 < s2, then s1 comes before s2 on the list:

for all indexes i, j:inOrder(list.get(i),list.get(j)) implies i < j.

for all indexes i and j, i < j implies!inOrder(list.get(j),list.get(i)).

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Design: Find the smallest element in the list, and put it in as first.

Find the second smallest and put it as second, etc.

Selection Sort

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Selection Sort (cont.) Find the smallest.

Interchange it with the first.

Find the next smallest.

Interchange it with the second.

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Selection Sort (cont.) Find the next smallest.

Interchange it with the third.

Find the next smallest.

Interchange it with the fourth.

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To interchange items, we must store one of the variables temporarily.

Selection sort

While making list.get(0) refer to list.get(2), loose reference to original entry referenced by list.get(0).

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Selection sort algorithm/** * Sort the specified List<Student> using selection sort. * @ensure * for all indexes i, j: * inOrder(list.get(i),list.get(j)) implies i < j. */public void sort (List<Student> list) {

int first; // index of first element to consider on this stepint last; // index of last element to consider on this stepint small; // index of smallest of list.get(first)...list.get(last)last = list.size() - 1;first = 0;while (first < last) {

small = smallestOf(list,first,last);interchange(list,first,small);first = first+1;

}}

15May 2004 NH-Chapter 14

Selection sort algorithm/** * Index of the smallest of * list.get(first) through list.get(last)*/private int smallestOf (List<Student> list, int first, int last) {

int next; // index of next element to examine.int small; // index of the smallest of get(first)...get(next-1)small = first;next = first+1;while (next <= last) {

if (inOrder(list.get(next),list.get(small)))small = next;

next = next+1;}return small;

}

16May 2004 NH-Chapter 14

Selection sort algorithm/** * Interchange list.get(i) and list.get(j) * require * 0 <= i < list.size() && 0 <= j < list.size() * ensure * list.old.get(i) == list.get(j) * list.old.get(j) == list.get(i) */private void interchange (List<Student> list,

int i, int j) {Student temp = list.get(i);list.set(i, list.get(j));list.set(j, temp);

}

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If there are n elements in the list, the outer loop is performed n-1 times. The inner loop is performed n-first times. i.e. time= 1, n-1 times; time=2, n-2 times; … time=n-2, 1 times.

(n-1)x(n-first) = (n-1)+(n-2)+…+2+1 = (n2-n)/2 As n increases, the time to sort the list goes up by this factor

(order n2).

Analysis of Selection sort

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Bubble sort Make a pass through the list comparing pairs of

adjacent elements.

If the pair is not properly ordered, interchange them.

At the end of the first pass, the last element will be in its proper place.

Continue making passes through the list until all the elements are in place.

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Pass 1

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Pass 2

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Pass 3

Pass 4

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Bubble sort algorithm// Sort specified List<Student> using bubble sort.public void sort (List<Student> list) {

int last; // index of last element to position on this passlast = list.size() - 1;while (last > 0) { makePassTo(list, last); last = last-1;}

}

// Make a pass through the list, bubbling an element to position last.private void makePassTo (List<Student> list, int last) {

int next; // index of next pair to examine.next = 0;while (next < last) { if (inOrder(list.get(next+1),list.get(next))) interchange(list, next, next+1);

next = next+1;}

}

23May 2004 NH-Chapter 14

Fine-tuning bubble sort algorithm Making pass through list no elements interchanged

then the list is ordered.

If list is ordered or nearly so to start with, can complete sort in fewer than n-1 passes.

With mostly ordered lists, keep track of whether or not any elements have been interchanged in a pass.

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Generalizing the sort methods Sorting algorithms are independent of:

the method inOrder, as long as it satisfies ordering requirements.

The elements in the list being sorted.

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Generalizing the sort methods

Thus: Need to learn about generic methods. Need to make the inOrder method part of a class.

Want to generalize the sort to List<Element> instances with the following specification:

public <Element> void selectionSort (List<Element> list, Order<Element> order)

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Generic methods

Can define a method with types as parameters.

Method type parameters are enclosed in angles and appear before the return type in the method heading.

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Generic methods

swap is now a generic method: it can swap to list entries of any given type.

In the method definition: Method type parameter

public <Element> void swap (List<Element> list, int i, int j) {Element temp = list.get(i);list.set(i,list.get(j));list.set(j,temp);

}

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Generic swap When swap is invoked, first argument will be a List

of some type of element, and local variable temp will be of that type.

No special syntax required to invoke a generic method.

When swap is invoked, the type to be used for the type parameter is inferred from the arguments.

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Generic swap For example, if roll is a List<Student>,

List<Student> roll = …

And the method swap is invoked as

swap(roll,0,1); Type parameter Element is Student, inferred from roll.

The local variable temp will be of type Student.

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inOrder as function object

A concrete order will implement this interface for some particular Element.

Wrap up method inOrder in an object to pass it as an argument to sort.

Define an interface/** * transitive, and anti-symmetric order on Element instances */public interface Order<Element> {

boolean inOrder (Element e1, Element e2);}

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Implementing Order interface To sort a list of Student by grade, define a class

(GradeOrder) implementing the interface, and then instantiated the class to obtain the required object.

//Order Students by decreasing finalGradeclass GradeOrder implements Order<Student> {

public boolean inOrder (Student s1, Student s2) {return s1.finalGrade() > s2.finalGrade();

}}

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Anonymous classes

This expression defines an anonymous class implementing interface

Order<Student>, and creates an instance of the class.

Define the class and instantiate it in one expression. For example, new Order<Student>() {

boolean inOrder(Student s1, Student s2) {return s1.finalGrade() > s2.finalGrade();

}}

33May 2004 NH-Chapter 14

Generalizing sort using generic methods

Generalized sort methods have both a list and an order as parameters.

public class Sorts {public static <Element> void selectionSort (

List<Element> list, Order<Element> order) {…}

public static <Element> void bubbleSort (List<Element> list, Order<Element> order) {… }

}

34May 2004 NH-Chapter 14

Generalizing sort using generic methods

The order also gets passed to auxiliary methods. The selection sort auxiliary method smallestOf will be defined as follows:

private static <Element> int smallestOf (List<Element> list, int first, int last,Order<Element> order ) {…}

35May 2004 NH-Chapter 14

Sorting a roll by grade

Or, using anonymous classes:

If roll is a List<Student>, to sort it invoke:

Sorts.selectionSort(roll, new GradeOrder());

Sorts.selectionSort(roll,new Order<Student>() {

boolean inOrder(Student s1, Student s2) {return s1.finalGrade() > s2.finalGrade();

}}

);

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Sorts as generic objects wrap sort algorithm and ordering in the same object. Define interface Sorter :

//A sorter for a List<Element>.public interface Sorter<Element> {

//e1 precedes e2 in the sort ordering.public boolean inOrder (Element e1, Element e2);

//Sort specified List<Element> according to this.inOrder.public void sort (List<Element> list);

}

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Sorts as generic objectsProvide specific sort algorithms in abstract classes, leaving the ordering abstract.

public abstract class SelectionSorter<Element> implements Sorter<Element> { // Sort the specified List<Element> using selection sort.public void sort (List<Element> list) { … }

}Selection sort algorithm

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Sorts as generic objects

To create a concrete Sorter, we extend the abstract class and furnish the order:

class GradeSorter extends SelectionSorter<Student> {public boolean inOrder (Student s1, Student s2){

return s1.finalGrade() > s2.finalGrade();}

}

39May 2004 NH-Chapter 14

Sorts as generic objects

Using an anonymous class,

Instantiate the class to get an object that can sort:

GradeSorter gradeSorter = new GradeSorter();gradeSorter.sort(roll);

SelectionSorter<Student> gradeSorter =new SelectionSorter<Student>() {

public boolean inOrder (Student s1, Student s2){return s1.finalGrade() > s2.finalGrade();

}};

gradeSorter.sort(roll);

40May 2004 NH-Chapter 14

Ordered Lists Typically need to maintain lists in specific order.

We treat ordered and unordered lists in different ways. may add an element to the end of an unordered list but want to

put the element in the “right place” when adding to an ordered list.

Interface OrderedList ( does not extend List)public interface OrderedList<Element>

A finite ordered list.

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Ordered Lists OrderedList shares features from List, but does not include

those that may break the ordering, such as public void add(int index, Element element); public void set( List<Element> element, int i, int j);

OrderedList invariant: for all indexes i, j:ordering().inOrder(get(i),get(j)) implies i < j.

OrderedList add method is specified as:public void add (Element element)

Add the specified element to the proper place in this OrderedList.

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Binary Search Assumes an ordered list.

Look for an item in a list by first looking at the middle element of the list.

Eliminate half the list.

Repeat the process.

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Binary Search for 42list.get(7) < 42No need to look below 8

list.get(11) > 42No need to look above 10

list.get(9)<42No need to look below 10

Down to one element, at position 10; this isn’t what we’re looking for, so we can conclude that 42 is not in the list.

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Generic search method itemIndex

It returns an index such that all elements prior to that index are smaller than

item searched for, and all of items from the index to end of list are not.

private <Element> int itemIndex (Element item, List<Element> list, Order<Element> order)

Proper place for item on list found using binary search.

require:list is sorted according to order.

ensure: 0 <= result && result <= list.size() for all indexes i: i < result implies order.inOrder(list.get(i),item) for all indexes i: i >= result implies !order.inOrder(list.get(i),item)

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Implementation of itemIndexprivate <Element> int itemIndex (Element item, List<Element> list, Order<Element> order) {

int low; // the lowest index being examinedint high; // the highest index begin examined

// for all indexes i: i < low implies order.inOrder(list.get(i),item)

// for all indexes i: i > high implies !order.inOrder(list.get(i),item)

int mid; // the middle item between low and high. mid == (low+high)/2

low = 0;high = list.size() - 1;while (low <= high) {

mid = (low+high)/2;if (order.inOrder(list.get(mid),item))

low = mid+1;else

high = mid-1;}return low;

}

46May 2004 NH-Chapter 14

Searching for 42 in

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47May 2004 NH-Chapter 14

Searching for 42

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Searching for 42

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49May 2004 NH-Chapter 14

Searching for 42

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50May 2004 NH-Chapter 14

Searching for 42

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42 is not found using itemIndex algorithm

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Searching for 12

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Searching for 12

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53May 2004 NH-Chapter 14

Searching for 12

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54May 2004 NH-Chapter 14

Searching for 12

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55May 2004 NH-Chapter 14

Searching for 12

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12 found in list at index 3

56May 2004 NH-Chapter 14

indexOf using binary search/** * Uses binary search to find where and if an element is in a list. * require: item != null * ensure: * if item == no element of list indexOf(item, list) == -1 * else item == list.get(indexOf(item, list)), * and indexOf(item, list) is the smallest value for which this is true */

public <Element> int indexOf (Element item, List<Element> list, Order<Element> order) {int i = itemIndex(item, list, order);if (i < list.size() && list.get(i).equals(item))

return i;else

return -1;}

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Recall sequential (linear) searchpublic int indexOf (Element element) {

int i = 0; // index of the next element to examinewhile (i < this.size() && !this.get(i).equals(element))i = i+1;

if (i < this.size())return i;elsereturn -1;

}

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Relative algorithm efficiency Number of steps required by the algorithm with a

list of length n grows in proportion to Selection sort: n2

Bubble sort: n2

Linear search: n

Binary search: log2n

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Loop invariant Loop invariant: condition that remains true as we repeatedly

execute loop body; it captures the fundamental intent in iteration.

Partial correctness: assertion that loop is correct if it terminates.

Total correctness: assertion that loop is both partially correct, and terminates.

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Loop invariant loop invariant:

it is true at the start of execution of a loop;

remains true no matter how many times loop body is executed.

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Correctness of itemIndex algorithm1. private <Element> int itemIndex (Element item,

List<Element> list, Order<Element> order) {2. int low = 0;3. int high = list.size() - 1;4. while (low <= high) {5. mid = (low+high)/2;6. if (order.inOrder(list.get(mid),item))7. low = mid+1;8. else9. high = mid-1;10. }11. return low;12.}

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Key invariant Purpose of method is to find index of first list element

greater than or equal to a specified item.

Since method returns value of variable low, we want low to satisfy this condition when the loop terminates

for all indexes i: i < low impliesorder.inOrder(list.get(i),item)

for all indexes i: i >= low implies!order.inOrder(list.get(i),item)

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Key invariant This holds true at all four key places (a, b, c, d).

It’s vacuously true for indexes less than low or greater than high (a) We assume it holds after merely testing the condition (b) and (d) If condition holds before executing the if statement and list is sorted

in ascending order, it will remain true after executing the if statement (condition c).

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Key invariant We are guaranteed that

for 0 <= i < mid order.inOrder(list.get(i), item)

After the assignment, low equals mid+1 and sofor 0 <= i < low order.inOrder( list.get(i), item)

This is true before the loop body is done:for high < i < list.size( !order.inOrder( list.get(i), item)

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Partial correctness If loop body is not executed at all, and point (d) is reached

with low == 0 and high == -1.

If the loop body is performed, at line 6, low <= mid <= high.

low <= high becomes false only if mid == high and low is set to mid + 1 or low == mid and high is set to mid - 1 In each case, low == high + 1 when loop is exited.

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Partial correctness The following conditions are satisfied on loop exit:

low == high+1 for all indexes i: i < low implies order.inOrder(list.get(i),item) for all indexes i: i > high implies !order.inOrder(list.get(i),item)

which imply for all indexes i: i < low implies order.inOrder(list.get(i),item) for all indexes i: i >= low implies !order.inOrder(list.get(i),item)

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Loop termination When the loop is executed, mid will be set to a value

between high and low.

The if statement will either cause low to increase or high to decrease.

This can happen only a finite number of times before low becomes larger than high.

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Summary Sorting and searching are two fundamental list operations.

Examined two simple sort algorithms, selection sort and bubble sort. Both of these algorithms make successive passes through the list,

getting one element into position on each pass. They are order n2 algorithms: time required for the algorithm to sort a

list grows as the square of the length of the list.

We also saw a simple modification to bubble sort that improved its performance on a list that was almost sorted.

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Summary Considered how to generalize sorting algorithms so that they

could be used for any type list and for any ordered.

We proposed two possible homes for sort algorithms: static generic methods, located in a utility class; abstract classes implementing a Sorter interface.

With later approach, we can dynamically create “sorter objects” to be passed to other methods.

Introduced Java’s anonymous class construct. in a single expression we can create and instantiate a nameless class

that implements an existing interface or extends an existing class.

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Summary Considered OrderedList container.

Developed binary search: search method for sorted lists. At each step of the algorithm, the middle of the remaining

elements is compared to the element being searched for. Half the remaining elements are eliminated from

consideration.

Major advantage of binary search: it looks at only log2n elements to find an item on a list of length n.

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Summary Two steps were involved in verifying the correctness

of the iteration in evaluating the correctness of binary search algorithm:

First, demonstrated partial correctness: iteration is correct if it terminates. found a key loop invariant that captured the essential

behavior of the iteration.

Second, showed that iteration always terminates.

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Summary A loop invariant is a condition that remains true no

matter how many times the loop body is performed.

The key invariant insures that when the loop terminates it has satisfied its purpose.

Verification of the key invariant provides a demonstration of partial correctness.