Searching and Sorting Arrays
Searching in ordered and unordered
arrays
Find the minimal element in an
unordered array.

Steps:
1.
Initially, let the 0th element be the
minimal element.
2.
Then sweep through the array and
see if we find something better.
Find the minimal element in an
unordered array.
//Initially, let the 0th element be
// the minimal element.
int whereSmallest = 0;
//Then sweep through the array
// and see if we find something better.
How?
Find the minimal element in an
unordered array.
//Initially, let the 0th element be
// the minimal element.
int whereSmallest = 0;
//Then sweep through the array
// and see if we find something better.
for (int i=1; i<A.length; i++) {
//check for something better
How?
}
Find the minimal element in an
unordered array.
//Initially, let the 0th element be
// the minimal element.
int whereSmallest = 0;
//Then sweep through the array
// and see if we find something better.
for (int i=1; i<A.length; i++) {
//check for something better
if (A[i]<A[whereSmallest]) {
whereSmallest = i;
}
}
//at the end of the above loop, A[whereSmallest] is
// the smallest element of A
Find the maximal element in an
unordered array. What needs to be
changed?
//Initially, let the 0th element be
// the minimal element.
int whereSmallest = 0;
//Then sweep through the array
// and see if we find something better.
for (int i=1; i<A.length; i++) {
//check for something better
if (A[i]<A[whereSmallest]) {
whereSmallest = i;
}
}
//at the end of the above loop, A[whereSmallest] is
// the smallest element of A
Find the maximal element in an
unordered array. What needs to be
changed?
//Initially, let the 0th element be
// the minimal element.
int whereLargest = 0;
//Then sweep through the array
// and see if we find something better.
for (int i=1; i<A.length; i++) {
//check for something better
>
if (A[i] A[whereLargest]) {
whereLargest = i;
}
}
//at the end of the above loop, A[whereLargest] is
// the largest element of A
What if the array is already sort?

If the array is sorted in ascending
order, where is the minimal
element?


We could search it as before but there
is a better way.
If the array is sorted in ascending
order, where is the maximal
element?
What if the array is already sort?

If the array is sorted in descending
order, where is the minimal
element?


We could search it as before but there
is a better way.
If the array is sorted in descending
order, where is the maximal
element?
Statistical median

From wikipedia:
“In probability theory and statistics, a median is
a number dividing the higher half of a sample,
a population, or a probability distribution from
the lower half. The median of a finite list of
numbers can be found by arranging all the
observations from lowest value to highest
value and picking the middle one. If there are
an even number of observations, one often
takes the mean of the two middle values.”

Given a sorted array, we can write a
function that determines the median.
Statistical median
public static double median ( int[] A ) {
//determine if the array length is
// odd of even
double result = 0;
…
return result;
}
Statistical median
public static double median ( int[] A ) {
//determine if the array length is
// odd of even
double result = 0;
if ((A.length%2)==0) {
//is this the odd or even case?
…
} else {
…
}
return result;
}
Statistical median
public static double median ( int[] A ) {
//determine if the array length is
// odd of even
double result = 0;
if ((A.length%2)==0) {
//even case so calc mean of middle 2
…
} else {
//odd case so pick middle one
…
This case is easier.
}
return result;
}
Statistical median
public static double median ( int[] A ) {
//determine if the array length is
// odd of even
double result = 0;
if ((A.length%2)==0) {
//even case so calc mean of middle 2
…
} else {
//odd case so pick middle one
result = A[ A.length/2 ];
}
return result;
}
Statistical median
public static double median ( int[] A ) {
//determine if the array length is
// odd of even
double result = 0;
if ((A.length%2)==0) {
//even case so calc mean of middle 2
result = ( A[ A.length/2-1 ] + A[ A.length/2 ] )
/ 2.0;
} else {
//odd case so pick middle one
result = A[ A.length/2 ];
}
return result;
}
Recap

So far we’ve:
1.
2.
3.
Found min/max elements in unsorted
and sorted arrays.
Calculated median of sorted arrays.
What if we would like to check
whether or not an array contains a
specified value?

What type of thing should this function
return?
Searching an unordered (unsorted)
array for a specific element.
Unsorted search for specified element
public static boolean unsortedSearch
( int[] A, int what )
{
…
}
Unsorted search for specified element
public static boolean unsortedSearch
( int[] A, int what )
{
boolean found = false;
//now search A for what
…
return found;
}
Unsorted search for specified element
public static boolean unsortedSearch
( int[] A, int what )
{
boolean found = false;
//now search A for what
for (int i=0; i<A.length; i++) {
…
}
return found;
}
Unsorted search for specified element
public static boolean unsortedSearch
( int[] A, int what )
{
boolean found = false;
//now search A for what
for (int i=0; i<A.length; i++) {
if (A[i]==what) {
found = true;
}
}
return found;
}
Unsorted search for specified element:
another slightly more efficient way
public static boolean unsortedSearch
( int[] A, int what )
{
//search A for what
for (int i=0; i<A.length; i++) {
if (A[i]==what) {
return true;
}
}
return false;
}
An analysis of these two methods

Let’s count the number of
comparisons performed in:

the best case

the worst case

the average case
An analysis of these two methods
Method A
Method B
public static boolean unsortedSearch
( int[] A, int what )
{
boolean found = false;
//now search A for what
for (int i=0; i<A.length; i++) {
if (A[i]==what) {
found = true;
}
}
return found;
}
public static boolean unsortedSearch
( int[] A, int what )
{
//search A for what
for (int i=0; i<A.length; i++) {
if (A[i]==what) {
return true;
}
}
return false;
}
What are the number of
comparisons for the best, worse,
and average cases for each
method?
An analysis of these two methods
Method A
Method B
public static boolean unsortedSearch
( int[] A, int what )
{
boolean found = false;
//now search A for what
for (int i=0; i<A.length; i++) {
if (A[i]==what) {
found = true;
}
}
return found;
}
public static boolean unsortedSearch
( int[] A, int what )
{
//search A for what
for (int i=0; i<A.length; i++) {
if (A[i]==what) {
return true;
}
}
return false;
}
Best = N; worst = N; average = N.
Best = 1; worst = N; average = N/2.
Searching a sorted array for a specified
element.


We could treat the array as
unsorted and search it but that
would be inefficient.
So let’s introduce the
search method.
binary
Binary search
first
middle
last
Searching a sorted array for a specified
element.
public static boolean sortedSearch
( int[] A, int what, int first, int last )
{
boolean foundIt = false;
…
return foundIt;
}
Searching a sorted array for a specified
element.
public static boolean sortedSearch
( int[] A, int what, int first, int last )
{
boolean foundIt = false;
//base case
…
return foundIt;
}
Searching a sorted array for a specified
element.
public static boolean sortedSearch
( int[] A, int what, int first, int last )
{
boolean foundIt = false;
//base case
if (first>=last) {
if (what==A[first])
foundIt = true;
} else {
…
}
return foundIt;
}
Searching a sorted array for a specified
element.
public static boolean sortedSearch
( int[] A, int what, int first, int last )
{
boolean foundIt = false;
//base case
if (first>=last) {
if (what==A[first])
foundIt = true;
} else {
int middle = (first+last) / 2;
…
}
return foundIt;
}
Searching a sorted array for a specified
element.
public static boolean sortedSearch ( int[] A, int what, int first, int last )
{
boolean foundIt = false;
//base case
if (first>=last) {
if (what==A[first])
foundIt = true;
} else {
int middle = (first+last) / 2;
if (what==A[middle])
…
else if (what<A[middle])
…
else
…
}
return foundIt;
}
Searching a sorted array for a specified
element.
public static boolean sortedSearch ( int[] A, int what, int first, int last )
{
boolean foundIt = false;
//base case
if (first>=last) {
if (what==A[first])
foundIt = true;
} else {
int middle = (first+last) / 2;
if (what==A[middle])
foundIt = true;
else if (what<A[middle])
foundIt = sortedSearch( A, what, first, middle-1 );
else
foundIt = sortedSearch( A, what, middle+1, last );
}
return foundIt;
An example of a
}
recursive function (a
function that may call
itself).
Searching a sorted array for a specified
element.
private static boolean sortedSearch ( int[] A, int what, int first, int last )
{
}
boolean foundIt = false;
An example of a “helper”
//base case
function (a function that helps
if (first>=last) {
get the recursion started; not
if (what==A[first])
an “official” term).
foundIt = true;
} else {
int middle = (first+last) / 2; Also an example of function
if (what==A[middle])
overloading (an “official” term).
foundIt = true;
else if (what<A[middle])
foundIt = sortedSearch( A, what, first, middle-1 );
else
foundIt = sortedSearch( A, what, middle+1, last );
}
return foundIt;
public static boolean sortedSearch ( int[] A, int what ) {
return sortedSearch( A, what, 0, A.length-1 );
}
Searching a sorted array for a specified
element.
private static boolean sortedSearch ( int[] A, int what, int first, int last )
{
//base case
Arguably simpler with returns.
if (first>=last) {
if (what==A[first])
return true;
} else {
int middle = (first+last) / 2;
if (what==A[middle])
return true;
else if (what<A[middle])
return sortedSearch( A, what, first, middle-1 );
else
return sortedSearch( A, what, middle+1, last );
}
return false;
}
public static boolean sortedSearch ( int[] A, int what ) {
return sortedSearch( A, what, 0, A.length-1 );
}
Searching a sorted array for a specified
element.
public static boolean sortedSearch ( int[] A, int what )
{
if (A.length==0) return false; //empty!
}
int first = 0, last = A.length - 1;
while (first<last) {
int middle = (first+last) / 2;
if (what == A[middle])
return true;
if (what < A[middle])
last = middle - 1;
else
first = middle + 1;
}
if (what==A[first])
return true;
return false;
Non recursive version.
Summary


Searching in an unordered array (of
length N) requires us to search N
elements in the worst case.
Searching in an ordered array (also
of length N) requires us to search
log2 elements in the worst case.
Summary

Searching in an unordered array (of
length N) requires us to search N
elements in the worst case.


So searching 1,000,000,000 things requires
searching 1,000,000,000 things.
Searching in an ordered array (also of
length N) requires us to search log2
elements in the worst case.

So searching 1,000,000,000 things requires
searching only 30 things! At most!
Sorting an array: the selection sort
Sorting


An arrangement or permutation of
data
May be either:


ascending (non decreasing)
descending (non increasing)
Selection sort


Based on the idea of repeatedly
finding the minimal elements.
But first, how can we find the
(single, most) minimal element in
an array?
Selection sort
How can we find the (single, most) minimal element in an array?
…
//let 0 be the location of the smallest element so far
int whereSmallest = 0;
for (int i=1; i<A.length; i++) {
if (A[i]<A[whereSmallest]) {
whereSmallest = i;
}
}
System.out.println( "the smallest is " + A[whereSmallest]
+ " which was located at position " + whereSmallest + "." );
…
Selection sort
Idea:






Find the smallest in A[0]..A[ A.length-1 ].
Put that in A[0].
Then find the smallest in A[1]..A[ A.length-1 ].
Put that in A[1].
…
But first, let’s develop a swapPairs function that
swaps a pair of elements denoted by a and b in
some array, A.

<-1, 2, 4, -5, 12>

<-5, 2, 4, -1, 12>
Selection sort
public static void swapPair ( … ) {
…
}
What do we need in here to
do the job (function
parameters)?
Selection sort
public static void swapPair ( int[] A, int a, int b )
{
…
}
What do we need in here to
do the job (function
parameters)?
Selection sort
public static void swapPair ( int[] A, int a, int b )
{
int temp = A[a];
A[a] = A[b];
A[b] = temp;
}
Selection sort
public static void swapPair ( int[] A, int a, int b )
{
int temp = A[a];
A[a] = A[b];
A[b] = temp;
}
Why doesn’t this work?
public static void swapPair ( int a, int b )
{
int temp = a;
a = b;
b = temp;
}
Selection sort
Idea:




Find the smallest in A[0]..A[ A.length-1 ].
Put that in A[0].
Then find the smallest in A[1]..A[ A.length-1 ].
Put that in A[1].
…
//let 0 be the location of the smallest element so far
int whereSmallest = 0;
for (int i=1; i<A.length; i++) {
if (A[i]<A[whereSmallest]) {
whereSmallest = i;
}
}
swapPairs( A, 0, whereSmallest );
Selection sort
Idea:




Find the smallest in A[0]..A[ A.length-1 ].
Put that in A[0].
Then find the smallest in A[1]..A[ A.length-1 ].
Put that in A[1].
…
for (int j=0; j<A.length; j++) {
//let j be the location of the smallest element so far
int whereSmallest = j;
for (int i=j+1; i<A.length; i++) {
if (A[i]<A[whereSmallest]) {
whereSmallest = i;
}
}
swap( A, j, whereSmallest );
}