- 67 Views
- Uploaded on
- Presentation posted in: General

Chapter 14: Sorting and Searching

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

- 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.

- 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.

- 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.

- Example: to order a List<Student> need

- 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.

public boolean inOrder (Student first, Student second)

- 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.

- 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.

- 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.

- A list is ordered:
- s1<s2, thens1 comes before s2 on the list:

- Or

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)).

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

- Find the smallest.
- Interchange it with the first.
- Find the next smallest.
- Interchange it with the second.

- Find the next smallest.
- Interchange it with the third.
- Find the next smallest.
- Interchange it with the fourth.

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

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

/**

* Sort the specified List<Student> using selection sort.

* @ensure

*for all indexes i, j:

*inOrder(list.get(i),list.get(j)) implies i < j.

*/

publicvoid sort (List<Student> list) {

int first;// index of first element to consider on this step

int last;// index of last element to consider on this step

int 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;

}

}

/**

* Index of the smallest of

* list.get(first) through list.get(last)

*/

privateint 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;

}

/**

* 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)

*/

privatevoid interchange (List<Student> list,

int i, int j) {

Student temp = list.get(i);

list.set(i, list.get(j));

list.set(j, temp);

}

- 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).

- 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.

Pass 4

// Sort specified List<Student> using bubble sort.

publicvoid sort (List<Student> list) {

int last;// index of last element to position on this pass

last = list.size() - 1;

while (last > 0) {

makePassTo(list, last);

last = last-1;

}

}

// Make a pass through the list, bubbling an element to position last.

privatevoid 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;

}

}

- 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.

- Sorting algorithms are independent of:
- the method inOrder, as long as it satisfies ordering requirements.
- The elements in the list being sorted.

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

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

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

- 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.

- In the method definition:

Method type parameter

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

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);

}

- 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.

- 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.

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

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

/**

* transitive, and anti-symmetric order on Element instances

*/

publicinterface Order<Element> {

boolean inOrder (Element e1, Element e2);

}

- 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 finalGrade

class GradeOrder implements Order<Student> {

publicboolean inOrder (Student s1, Student s2) {

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

}

}

- Define the class and instantiate it in one expression.
- For example,

- This expression
- defines an anonymous class implementing interface Order<Student>, and
- creates an instance of the class.

new Order<Student>() {

boolean inOrder(Student s1, Student s2) {

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

}

}

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

publicclass Sorts {

publicstatic <Element> void selectionSort (

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

publicstatic <Element> void bubbleSort (

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

}

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

privatestatic <Element> int smallestOf (

List<Element> list, int first, int last,

Order<Element> order ) {…}

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

- Or, using anonymous classes:

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

Sorts.selectionSort(roll,

new Order<Student>() {

boolean inOrder(Student s1, Student s2) {

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

}

}

);

- wrap sort algorithm and ordering in the same object.
- Define interface Sorter :

//A sorter for a List<Element>.

publicinterface Sorter<Element> {

//e1 precedes e2 in the sort ordering.

publicboolean inOrder (Element e1, Element e2);

//Sort specified List<Element> according to this.inOrder.

publicvoid sort (List<Element> list);

}

- Provide 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

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

class GradeSorter extends SelectionSorter<Student> {

publicboolean inOrder (Student s1, Student s2){

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

}

}

- Instantiate the class to get an object that can sort:

- Using an anonymous class,

GradeSorter gradeSorter = new GradeSorter();

gradeSorter.sort(roll);

SelectionSorter<Student> gradeSorter =

new SelectionSorter<Student>() {

publicboolean inOrder (Student s1, Student s2){

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

}

};

gradeSorter.sort(roll);

- 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.

- 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);

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

publicvoid add (Element element)

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

- 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.

list.get(7) < 42

No need to look below 8

list.get(11) > 42

No need to look above 10

list.get(9)<42

No 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.

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)

- 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) {

int low; // the lowest index being examined

int 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;

}

low

high

item

42

low

0

high

14

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

mid

?

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

item

42

low

8

high

14

s

s

s

s

s

s

s

28

?

?

?

?

?

?

?

mid

7?

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

high

item

42

low

8

high

10

s

s

s

s

s

s

s

28

?

?

?

56

g

g

g

high

mid

11

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

item

42

low

8

high

10

s

s

s

s

s

s

s

28

s

33

?

56

g

g

g

high

mid

10

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

item

42

low

11

high

10

s

s

s

s

s

s

s

28

s

33

40

56

g

g

g

high

mid

10

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

42 is not found using itemIndex algorithm

low

high

item

12

low

0

high

14

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

mid

?

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

item

12

low

0

high

6

?

?

?

?

?

?

?

28

g

g

g

g

g

g

g

mid

7

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

high

item

12

low

0

high

2

?

?

?

12

g

g

g

28

g

g

g

g

g

g

g

mid

3

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

high

item

12

low

2

high

2

s

5

?

12

g

g

g

28

g

g

g

g

g

g

g

mid

1

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

low

high

item

12

low

3

high

2

s

5

?

12

g

g

g

28

g

g

g

g

g

g

g

mid

2

(0)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

high

low

12 found in list at index 3

/**

* 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;

}

public int indexOf (Element element) {

int i = 0;// index of the next element to examine

while (i < this.size() && !this.get(i).equals(element))

i = i+1;

if (i < this.size())

return i;

else

return -1;

}

- 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

- 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.

- loop invariant:
- it is true at the start of execution of a loop;
- remains true no matter how many times loop body is executed.

- private <Element> int itemIndex (Element item,List<Element> list, Order<Element> order) {
- int low = 0;
- int 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;
- }

- 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 implies

order.inOrder(list.get(i),item)

for all indexes i: i >= low implies

!order.inOrder(list.get(i),item)

- 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).

- We are guaranteed that
for 0 <= i < mid

order.inOrder(list.get(i), item)

- After the assignment, low equals mid+1 and so
for 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)

- If loop body is not executed at all, and point (d) is reached withlow == 0andhigh == -1.
- If the loop body is performed, at line 6, low <= mid <= high.
- low <= highbecomes false only if
- mid == highandlowis set tomid + 1or
- low == midandhighis set tomid - 1
- In each case,low == high + 1when loop is exited.

- 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)

- 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)

- When the loop is executed, midwill be set to a value between highand 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.

- 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.

- 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.

- 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.

- 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.

- 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.