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n2 become relatively insignificant Imagine the worst case of sorting a million numbers The n2 term will overwhelm the other terms of the equation Since one usually reports the order of growth for an algorithm as the worst-case order of growth, the insertion sort has a theta of n2, or (n2) If one computes the average case order of growth for the insertion sort, one also finds a quadratic equation; it s just somewhat smaller, since on average each new element will be compared with only half of the elements already sorted So we say the performance of the insertion sort is (n2) Merge sort An example of order of growth of n(lg n) Q(n lg n) Another algorithm for sorting numbers uses recursion, a technique we will discuss in more detail shortly, to divide the problem into many smaller problems before recombining the elements of the full solution.

Java is a strongly typed language, which means that every variable must be declared to have a type which determines how that variable can be used. A char cannot be used where a boolean is expected, and a Ratio object cannot be used where a Person object is expected. But there are some situations where a variable of one type can be used where another type is expected. This is called polymorphism (literally, many forms ) because the object appears to have more than one type. There are several kinds of polymorphism. The simplest kind is called inclusion polymorphism. Illustrated in Example 1.8, it refers to the ability of an object to invoke a method that it inherits. EXAMPLE 1.8 Inclusion Polymorphism

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First, this solution requires a routine to combine two sets of sorted numbers into a single set Imagine two piles of playing cards, each sorted from smallest to largest, with the cards face up in two piles, and the two smallest cards showing The merge routine compares the two cards that are showing, and places the smaller card face down in what will be the merged pile Then the routine compares the two cards showing after the first has been put face down on the merged pile Again, the routine picks up the smaller card, and puts it face down on the merged pile The merge routine continues in this manner until all the cards have been moved into the sorted merged pile Here is pseudocode for the merge routine.

public class TestRatio { public static void main(String[] args) { Ratio x = new Ratio(22, 7); System.out.println("x.hashCode(): " + x.hashCode()); } }

The output is:

It expects to work on two previously sorted lists of numbers, and it merges the two lists into one sorted list, which it returns The variable index keeps track of where it is working in sorted_list The routine compares the first (top) numbers in the two original lists, and puts the smaller of the two into sorted_list Then it discards the number from the original list, which means that the number that used to be the second one in the original list becomes the first number in that list Again the routine compares the first numbers in the two lists, and again it moves the smaller to sorted_list The routine continues this way until one of the original lists becomes empty At that point, it adds the remaining numbers (which were in sorted order originally, remember) to sorted_list, and returns sorted_list.

x.hashCode(): 1671711

I have to say I don t see a difference between denouncing and rejecting. There s no formal offer of help from Minister Farrakhan that would involve me rejecting it. But if the word reject Senator Clinton feels is stronger than the word denounce, then I m happy to concede the point, and I would reject and denounce. The audience burst into applause. They recognized that Obama had just doused a potential firestorm. Keeping his eyes on his goals, Obama had spoken the right words in a firm manner and quelled a controversy that could have lingered for weeks and done considerable damage to his campaign. An important lesson: before facing controversy, be clear about your goals; while addressing the difficult situation, align your actions, words, and behavior in ways that are consistent with your goals.

At line 4, the Ratio object x invokes the hashCode() method that it inherits from the Object class.

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