Searching and Sorting Arrays

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.


//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?


//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 betterHow?

}



//check for something betterif (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?


//check for something betterif (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 evendouble result = 0;…return result;

}

Statistical median

public static double median ( int[] A ) {//determine if the array length is// odd of evendouble result = 0;if ((A.length%2)==0) {

//is this the odd or even case?…

} else {…

}return result;

}

Statistical median


//even case so calc mean of middle 2…

} else {//odd case so pick middle one… This case is easier.

}return result;

}

Statistical median


//even case so calc mean of middle 2…

} else {//odd case so pick middle oneresult = A[ A.length/2 ];

}return result;

}

Statistical median


//even case so calc mean of middle 2result = ( A[ A.length/2-1 ] + A[ A.length/2 ] )

/ 2.0;} else {

//odd case so pick middle oneresult = A[ A.length/2 ];

}return result;

}

Recap

So far we’ve:1. Found min/max elements in unsorted

and sorted arrays.2. Calculated median of sorted arrays.3. 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 )

{…

}



{boolean found = false;//now search A for what…return found;

}



{boolean found = false;//now search A for whatfor (int i=0; i<A.length; i++) {

…}return found;

}


public static boolean unsortedSearch( int[] A, int what )

{boolean found = false;//now search A for whatfor (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


{//search A for whatfor (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


Method Apublic static boolean unsortedSearch

( int[] A, int what ){

boolean found = false;//now search A for whatfor (int i=0; i<A.length; i++) {


}}return found;

}

Method Bpublic static boolean unsortedSearch


//search A for whatfor (int i=0; i<A.length; i++) {


}}return false;

}

What are the number of comparisons for the best, worse, and average cases for each method?


Method Apublic static boolean unsortedSearch


boolean found = false;//now search A for whatfor (int i=0; i<A.length; i++) {


}}return found;

}

Best = N; worst = N; average = N.

Method Bpublic static boolean unsortedSearch


//search A for whatfor (int i=0; i<A.length; i++) {


}}return false;

}

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 binary search method.

Binary searchfirst middle last


public static boolean sortedSearch( int[] A, int what, int first, int last )

{boolean foundIt = false;…return foundIt;

}



{boolean foundIt = false;//base case…return foundIt;

}



{boolean foundIt = false;//base caseif (first>=last) {

if (what==A[first])foundIt = true;

} else {…

}return foundIt;

}



{boolean foundIt = false;//base caseif (first>=last) {


} else {int middle = (first+last) / 2;…

}return foundIt;

}


public static boolean sortedSearch ( int[] A, int what, int first, int last ){

boolean foundIt = false;//base caseif (first>=last) {


} else {int middle = (first+last) / 2;if (what==A[middle])

…else if (what<A[middle])

…else

…}return foundIt;

}


public static boolean sortedSearch ( int[] A, int what, int first, int last ){




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).


private static boolean sortedSearch ( int[] A, int what, int first, int last ){




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 );

}

An example of a “helper” function (a function that helps get the recursion started; not an “official” term).

Also an example of function overloading (an “official” term).


private static boolean sortedSearch ( int[] A, int what, int first, int last ){

//base caseif (first>=last) {

if (what==A[first])return true;


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 );

}

Arguably simpler with returns.


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 sortHow can we find the (single, most) minimal element in an array?

…//let 0 be the location of the smallest element so farint 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


int temp = A[a];A[a] = A[b];A[b] = temp;

}

Selection sort


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 sortIdea:

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 farint whereSmallest = 0;for (int i=1; i<A.length; i++) {

if (A[i]<A[whereSmallest]) {whereSmallest = i;

}}swapPairs( A, 0, whereSmallest );

Selection sortIdea:

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 farint whereSmallest = j;for (int i=j+1; i<A.length; i++) {

if (A[i]<A[whereSmallest]) {whereSmallest = i;

}}swap( A, j, whereSmallest );

}

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Searching and Sorting Arrays