LET375 V(Weiss)

compiler

A compiler is a computer program (or a set of programs) that transforms source code written in a programming language (the source language) into another computer language (the target language), with the latter often having a binary form known as object code.[1] The most common reason for converting source code is to create an executable program.

maps

An object that maps keys to values. A map cannot contain duplicate keys; each key can map to at most one value. This interface takes the place of the Dictionary class, which was a totally abstract class rather than an interface. The Map interface provides three collection views, which allow a map's contents to be viewed as a set of keys, collection of values, or set of key-value mappings. The order of a map is defined as the order in which the iterators on the map's collection views return their elements. Some map implementations, like the TreeMap class, make specific guarantees as to their order; others, like the HashMap class, do not. Note: great care must be exercised if mutable objects are used as map keys. The behavior of a map is not specified if the value of an object is changed in a manner that affects equals comparisons while the object is a key in the map. A special case of this prohibition is that it is not permissible for a map to contain itself as a key. While it is permissible for a map to contain itself as a value, extreme caution is advised: the equals and hashCode methods are no longer well defined on such a map. All general-purpose map implementation classes should provide two "standard" constructors: a void (no arguments) constructor which creates an empty map, and a constructor with a single argument of type Map, which creates a new map with the same key-value mappings as its argument. In effect, the latter constructor allows the user to copy any map, producing an equivalent map of the desired class. There is no way to enforce this recommendation (as interfaces cannot contain constructors) but all of the general-purpose map implementations in the JDK comply. The "destructive" methods contained in this interface, that is, the methods that modify the map on which they operate, are specified to throw UnsupportedOperationException if this map does not support the operation. If this is the case, these methods may, but are not required to, throw an UnsupportedOperationException if the invocation would have no effect on the map. For example, invoking the putAll(Map) method on an unmodifiable map may, but is not required to, throw the exception if the map whose mappings are to be "superimposed" is empty. Some map implementations have restrictions on the keys and values they may contain. For example, some implementations prohibit null keys and values, and some have restrictions on the types of their keys. Attempting to insert an ineligible key or value throws an unchecked exception, typically NullPointerException or ClassCastException. Attempting to query the presence of an ineligible key or value may throw an exception, or it may simply return false; some implementations will exhibit the former behavior and some will exhibit the latter. More generally, attempting an operation on an ineligible key or value whose completion would not result in the insertion of an ineligible element into the map may throw an exception or it may succeed, at the option of the implementation. Such exceptions are marked as "optional" in the specification for this interface. This interface is a member of the Java Collections Framework. Many methods in Collections Framework interfaces are defined in terms of the equals method. For example, the specification for the containsKey(Object key) method says: "returns true if and only if this map contains a mapping for a key k such that (key==null ? k==null : key.equals(k))." This specification should not be construed to imply that invoking Map.containsKey with a non-null argument key will cause key.equals(k) to be invoked for any key k. Implementations are free to implement optimizations whereby the equals invocation is avoided, for example, by first comparing the hash codes of the two keys. (The Object.hashCode() specification guarantees that two objects with unequal hash codes cannot be equal.) More generally, implementations of the various Collections Framework interfaces are free to take advantage of the specified behavior of underlying Object methods wherever the implementor deems it appropriate.

the collections api

As part of Java, a supporting library known as the Collections API is pro- vided. Most of the Collections API resides in java.util. This API provides a collection of data structures. It also provides some generic algorithms, such as sorting. The Collections API makes heavy use of inheritance.

the HashMap class

Hash table based implementation of the Map interface. This implementation provides all of the optional map operations, and permits null values and the null key. (The HashMap class is roughly equivalent to Hashtable, except that it is unsynchronized and permits nulls.) This class makes no guarantees as to the order of the map; in particular, it does not guarantee that the order will remain constant over time. This implementation provides constant-time performance for the basic operations (get and put), assuming the hash function disperses the elements properly among the buckets. Iteration over collection views requires time proportional to the "capacity" of the HashMap instance (the number of buckets) plus its size (the number of key-value mappings). Thus, it's very important not to set the initial capacity too high (or the load factor too low) if iteration performance is important. An instance of HashMap has two parameters that affect its performance: initial capacity and load factor. The capacity is the number of buckets in the hash table, and the initial capacity is simply the capacity at the time the hash table is created. The load factor is a measure of how full the hash table is allowed to get before its capacity is automatically increased. When the number of entries in the hash table exceeds the product of the load factor and the current capacity, the hash table is rehashed (that is, internal data structures are rebuilt) so that the hash table has approximately twice the number of buckets. As a general rule, the default load factor (.75) offers a good tradeoff between time and space costs. Higher values decrease the space overhead but increase the lookup cost (reflected in most of the operations of the HashMap class, including get and put). The expected number of entries in the map and its load factor should be taken into account when setting its initial capacity, so as to minimize the number of rehash operations. If the initial capacity is greater than the maximum number of entries divided by the load factor, no rehash operations will ever occur. If many mappings are to be stored in a HashMap instance, creating it with a sufficiently large capacity will allow the mappings to be stored more efficiently than letting it perform automatic rehashing as needed to grow the table. Note that this implementation is not synchronized. If multiple threads access a hash map concurrently, and at least one of the threads modifies the map structurally, it must be synchronized externally. (A structural modification is any operation that adds or deletes one or more mappings; merely changing the value associated with a key that an instance already contains is not a structural modification.) This is typically accomplished by synchronizing on some object that naturally encapsulates the map. If no such object exists, the map should be "wrapped" using the Collections.synchronizedMap method. This is best done at creation time, to prevent accidental unsynchronized access to the map: Map m = Collections.synchronizedMap(new HashMap(...)); The iterators returned by all of this class's "collection view methods" are fail-fast: if the map is structurally modified at any time after the iterator is created, in any way except through the iterator's own remove method, the iterator will throw a ConcurrentModificationException. Thus, in the face of concurrent modification, the iterator fails quickly and cleanly, rather than risking arbitrary, non-deterministic behavior at an undetermined time in the future. Note that the fail-fast behavior of an iterator cannot be guaranteed as it is, generally speaking, impossible to make any hard guarantees in the presence of unsynchronized concurrent modification. Fail-fast iterators throw ConcurrentModificationException on a best-effort basis. Therefore, it would be wrong to write a program that depended on this exception for its correctness: the fail-fast behavior of iterators should be used only to detect bugs. This class is a member of the Java Collections Framework.

greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, when at least one of them is not zero, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.[1][2]

Factory

In object-oriented programming (OOP), a factory is an object for creating other objects - formally a factory is a function or method that returns objects of a varying prototype or class[1] from some method call, which is assumed to be "new".[a] More broadly, a subroutine that returns a "new" object may be referred to as a "factory", as in factory method or factory function. This is a basic concept in OOP, and forms the basis for a number of related software design patterns.

recursion

Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other, the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, in which it refers to a method of defining functions in which the function being defined is applied within its own definition. Specifically, this defines an infinite number of instances (function values), using a finite expression that for some instances may refer to other instances, but in such a way that no loop or infinite chain of references can occur. The term is also used more generally to describe a process of repeating objects in a self-similar way.

doubly linked list

A linked list that allows bidirectional traversal by storing two links per node. (630)

Binary heap

A binary heap is a heap data structure created using a binary tree. It can be seen as a binary tree with two additional constraints:[1]

circularly linked list

A linked list in which the last cell's next link references first. This action is useful when wraparound matters. (633)

the Collection interface

A Collection represents a group of objects known as its elements. The Collection interface is used to pass around collections of objects where maximum generality is desired. For example, by convention all general-purpose collection implementations have a constructor that takes a Collection argument. This constructor, known as a conversion constructor, initializes the new collection to contain all of the elements in the specified collection, whatever the given collection's subinterface or implementation type. In other words, it allows you to convert the collection's type.

the TreeSet class

A NavigableSet implementation based on a TreeMap. The elements are ordered using their natural ordering, or by a Comparator provided at set creation time, depending on which constructor is used. This implementation provides guaranteed log(n) time cost for the basic operations (add, remove and contains). Note that the ordering maintained by a set (whether or not an explicit comparator is provided) must be consistent with equals if it is to correctly implement the Set interface. (See Comparable or Comparator for a precise definition of consistent with equals.) This is so because the Set interface is defined in terms of the equals operation, but a TreeSet instance performs all element comparisons using its compareTo (or compare) method, so two elements that are deemed equal by this method are, from the standpoint of the set, equal. The behavior of a set is well-defined even if its ordering is inconsistent with equals; it just fails to obey the general contract of the Set interface. Note that this implementation is not synchronized. If multiple threads access a tree set concurrently, and at least one of the threads modifies the set, it must be synchronized externally. This is typically accomplished by synchronizing on some object that naturally encapsulates the set. If no such object exists, the set should be "wrapped" using the Collections.synchronizedSortedSet method. This is best done at creation time, to prevent accidental unsynchronized access to the set: SortedSet s = Collections.synchronizedSortedSet(new TreeSet(...)); The iterators returned by this class's iterator method are fail-fast: if the set is modified at any time after the iterator is created, in any way except through the iterator's own remove method, the iterator will throw a ConcurrentModificationException. Thus, in the face of concurrent modification, the iterator fails quickly and cleanly, rather than risking arbitrary, non-deterministic behavior at an undetermined time in the future. Note that the fail-fast behavior of an iterator cannot be guaranteed as it is, generally speaking, impossible to make any hard guarantees in the presence of unsynchronized concurrent modification. Fail-fast iterators throw ConcurrentModificationException on a best-effort basis. Therefore, it would be wrong to write a program that depended on this exception for its correctness: the fail-fast behavior of iterators should be used only to detect bugs. This class is a member of the Java Collections Framework.

the TreeMap class

A Red-Black tree based NavigableMap implementation. The map is sorted according to the natural ordering of its keys, or by a Comparator provided at map creation time, depending on which constructor is used. This implementation provides guaranteed log(n) time cost for the containsKey, get, put and remove operations. Algorithms are adaptations of those in Cormen, Leiserson, and Rivest's Introduction to Algorithms. Note that the ordering maintained by a tree map, like any sorted map, and whether or not an explicit comparator is provided, must be consistent with equals if this sorted map is to correctly implement the Map interface. (See Comparable or Comparator for a precise definition of consistent with equals.) This is so because the Map interface is defined in terms of the equals operation, but a sorted map performs all key comparisons using its compareTo (or compare) method, so two keys that are deemed equal by this method are, from the standpoint of the sorted map, equal. The behavior of a sorted map is well-defined even if its ordering is inconsistent with equals; it just fails to obey the general contract of the Map interface. Note that this implementation is not synchronized. If multiple threads access a map concurrently, and at least one of the threads modifies the map structurally, it must be synchronized externally. (A structural modification is any operation that adds or deletes one or more mappings; merely changing the value associated with an existing key is not a structural modification.) This is typically accomplished by synchronizing on some object that naturally encapsulates the map. If no such object exists, the map should be "wrapped" using the Collections.synchronizedSortedMap method. This is best done at creation time, to prevent accidental unsynchronized access to the map: SortedMap m = Collections.synchronizedSortedMap(new TreeMap(...)); The iterators returned by the iterator method of the collections returned by all of this class's "collection view methods" are fail-fast: if the map is structurally modified at any time after the iterator is created, in any way except through the iterator's own remove method, the iterator will throw a ConcurrentModificationException. Thus, in the face of concurrent modification, the iterator fails quickly and cleanly, rather than risking arbitrary, non-deterministic behavior at an undetermined time in the future. Note that the fail-fast behavior of an iterator cannot be guaranteed as it is, generally speaking, impossible to make any hard guarantees in the presence of unsynchronized concurrent modification. Fail-fast iterators throw ConcurrentModificationException on a best-effort basis. Therefore, it would be wrong to write a program that depended on this exception for its correctness: the fail-fast behavior of iterators should be used only to detect bugs. All Map.Entry pairs returned by methods in this class and its views represent snapshots of mappings at the time they were produced. They do not support the Entry.setValue method. (Note however that it is possible to change mappings in the associated map using put.) This class is a member of the Java Collections Framework.

binary expression trees

A binary expression tree is a specific kind of a binary tree used to represent expressions. Two common types of expressions that a binary expression tree can represent are algebraic[1] and boolean. These trees can represent expressions that contain both unary and binary operators.[1] Each node of a binary tree, and hence of a binary expression tree, has zero, one, or two children. This restricted structure simplifies the processing of expression trees.

function object

A function object[a] is a computer programming construct allowing an object to be invoked or called as if it were an ordinary function, usually with the same syntax (a function parameter that can also be a function).

hash function

A hash function is any function that can be used to map data of arbitrary size to data of fixed size. The values returned by a hash function are called hash values, hash codes, hash sums, or simply hashes. One use is a data structure called a hash table, widely used in computer software for rapid data lookup. Hash functions accelerate table or database lookup by detecting duplicated records in a large file. An example is finding similar stretches in DNA sequences. They are also useful in cryptography. A cryptographic hash function allows one to easily verify that some input data maps to a given hash value, but if the input data is unknown, it is deliberately difficult to reconstruct it (or equivalent alternatives) by knowing the stored hash value. This is used for assuring integrity of transmitted data, and is the building block for HMACs, which provide message authentication.

StringBuilder class

A mutable sequence of characters. This class provides an API compatible with StringBuffer, but with no guarantee of synchronization. This class is designed for use as a drop-in replacement for StringBuffer in places where the string buffer was being used by a single thread (as is generally the case). Where possible, it is recommended that this class be used in preference to StringBuffer as it will be faster under most implementations. The principal operations on a StringBuilder are the append and insert methods, which are overloaded so as to accept data of any type. Each effectively converts a given datum to a string and then appends or inserts the characters of that string to the string builder. The append method always adds these characters at the end of the builder; the insert method adds the characters at a specified point. For example, if z refers to a string builder object whose current contents are "start", then the method call z.append("le") would cause the string builder to contain "startle", whereas z.insert(4, "le") would alter the string builder to contain "starlet". In general, if sb refers to an instance of a StringBuilder, then sb.append(x) has the same effect as sb.insert(sb.length(), x). Every string builder has a capacity. As long as the length of the character sequence contained in the string builder does not exceed the capacity, it is not necessary to allocate a new internal buffer. If the internal buffer overflows, it is automatically made larger. Instances of StringBuilder are not safe for use by multiple threads. If such synchronization is required then it is recommended that StringBuffer be used.

the pairing heap

A pairing heap is a type of heap data structure with relatively simple implementation and excellent practical amortized performance, introduced by Michael Fredman, Robert Sedgewick, Daniel Sleator, and Robert Tarjan in 1986.[1] Pairing heaps are heap-ordered multiway tree structures, and can be considered simplified Fibonacci heaps. They are considered a "robust choice" for implementing such algorithms as Prim's MST algorithm,[2] and support the following operations (assuming a min-heap): find-min: simply return the top element of the heap. merge: compare the two root elements, the smaller remains the root of the result, the larger element and its subtree is appended as a child of this root. insert: create a new heap for the inserted element and merge into the original heap. decrease-key (optional): remove the subtree rooted at the key to be decreased, replace the key with a smaller key, then merge the result back into the heap. delete-min: remove the root and merge its subtrees. Various strategies are employed. The analysis of pairing heaps' time complexity was initially inspired by that of splay trees.[1] The amortized time per delete-min is O(log n), and the operations find-min, merge, and insert run in O(1) amortized time.[3]

Primality test

A primality test is an algorithm for determining whether an input number is prime. Amongst other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller-Rabin prove that a number is composite. Therefore, the latter might be called compositeness tests instead of primality tests.

random number generators

A random number generator (RNG) is a computational or physical device designed to generate a sequence of numbers or symbols that can not be reasonably predicted better than by a random chance.

randomized algorithms

A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally, the algorithm's performance will be a random variable determined by the random bits; thus either the running time, or the output (or both) are random variables.

red-black trees

A red-black tree is a kind of self-balancing binary search tree. Each node of the binary tree has an extra bit, and that bit is often interpreted as the color (red or black) of the node. These color bits are used to ensure the tree remains approximately balanced during insertions and deletions.[2] Balance is preserved by painting each node of the tree with one of two colors (typically called 'red' and 'black') in a way that satisfies certain properties, which collectively constrain how unbalanced the tree can become in the worst case. When the tree is modified, the new tree is subsequently rearranged and repainted to restore the coloring properties. The properties are designed in such a way that this rearranging and recoloring can be performed efficiently. The balancing of the tree is not perfect but it is good enough to allow it to guarantee searching in O(log n) time, where n is the total number of elements in the tree. The insertion and deletion operations, along with the tree rearrangement and recoloring, are also performed in O(log n) time.[3] Tracking the color of each node requires only 1 bit of information per node because there are only two colors. The tree does not contain any other data specific to its being a red-black tree so its memory footprint is almost identical to a classic (uncolored) binary search tree. In many cases the additional bit of information can be stored at no additional memory cost.

the skew heap

A skew heap (or self-adjusting heap) is a heap data structure implemented as a binary tree. Skew heaps are advantageous because of their ability to merge more quickly than binary heaps. In contrast with binary heaps, there are no structural constraints, so there is no guarantee that the height of the tree is logarithmic. Only two conditions must be satisfied: The general heap order must be enforced Every operation (add, remove_min, merge) on two skew heaps must be done using a special skew heap merge. A skew heap is a self-adjusting form of a leftist heap which attempts to maintain balance by unconditionally swapping all nodes in the merge path when merging two heaps. (The merge operation is also used when adding and removing values.) With no structural constraints, it may seem that a skew heap would be horribly inefficient. However, amortized complexity analysis can be used to demonstrate that all operations on a skew heap can be done in O(log n).[1]

sorting

A sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order. Efficient sorting is important for optimizing the use of other algorithms (such as search and merge algorithms) which require input data to be in sorted lists; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions: The output is in nondecreasing order (each element is no smaller than the previous element according to the desired total order); The output is a permutation (reordering) of the input.

splay trees

A splay tree is a self-adjusting binary search tree with the additional property that recently accessed elements are quick to access again. It performs basic operations such as insertion, look-up and removal in O(log n) amortized time. For many sequences of non-random operations, splay trees perform better than other search trees, even when the specific pattern of the sequence is unknown. The splay tree was invented by Daniel Dominic Sleator and Robert Endre Tarjan in 1985.[1]

aa-trees

An AA tree in computer science is a form of balanced tree used for storing and retrieving ordered data efficiently. AA trees are named for Arne Andersson, their inventor. AA trees are a variation of the red-black tree, a form of binary search tree which supports efficient addition and deletion of entries. Unlike red-black trees, red nodes on an AA tree can only be added as a right subchild. In other words, no red node can be a left sub-child. This results in the simulation of a 2-3 tree instead of a 2-3-4 tree, which greatly simplifies the maintenance operations. The maintenance algorithms for a red-black tree need to consider seven different shapes to properly balance the tree: Red Black Shape Cases.svg An AA tree on the other hand only needs to consider two shapes due to the strict requirement that only right links can be red: AA Tree Shape Cases.svg

the ListIterator interface

An iterator for lists that allows the programmer to traverse the list in either direction, modify the list during iteration, and obtain the iterator's current position in the list. A ListIterator has no current element; its cursor position always lies between the element that would be returned by a call to previous() and the element that would be returned by a call to next(). An iterator for a list of length n has n+1 possible cursor positions, as illustrated by the carets (^) below: Element(0) Element(1) Element(2) ... Element(n-1) cursor positions: ^ ^ ^ ^ ^ Note that the remove() and set(Object) methods are not defined in terms of the cursor position; they are defined to operate on the last element returned by a call to next() or previous(). This interface is a member of the Java Collections Framework.

Iterator interface

An iterator over a collection. Iterator takes the place of Enumeration in the Java Collections Framework. Iterators differ from enumerations in two ways: Iterators allow the caller to remove elements from the underlying collection during the iteration with well-defined semantics. Method names have been improved. This interface is a member of the Java Collections Framework.

the List interface

An ordered collection (also known as a sequence). The user of this interface has precise control over where in the list each element is inserted. The user can access elements by their integer index (position in the list), and search for elements in the list. Unlike sets, lists typically allow duplicate elements. More formally, lists typically allow pairs of elements e1 and e2 such that e1.equals(e2), and they typically allow multiple null elements if they allow null elements at all. It is not inconceivable that someone might wish to implement a list that prohibits duplicates, by throwing runtime exceptions when the user attempts to insert them, but we expect this usage to be rare. The List interface places additional stipulations, beyond those specified in the Collection interface, on the contracts of the iterator, add, remove, equals, and hashCode methods. Declarations for other inherited methods are also included here for convenience. The List interface provides four methods for positional (indexed) access to list elements. Lists (like Java arrays) are zero based. Note that these operations may execute in time proportional to the index value for some implementations (the LinkedList class, for example). Thus, iterating over the elements in a list is typically preferable to indexing through it if the caller does not know the implementation. The List interface provides a special iterator, called a ListIterator, that allows element insertion and replacement, and bidirectional access in addition to the normal operations that the Iterator interface provides. A method is provided to obtain a list iterator that starts at a specified position in the list. The List interface provides two methods to search for a specified object. From a performance standpoint, these methods should be used with caution. In many implementations they will perform costly linear searches. The List interface provides two methods to efficiently insert and remove multiple elements at an arbitrary point in the list. Note: While it is permissible for lists to contain themselves as elements, extreme caution is advised: the equals and hashCode methods are no longer well defined on such a list. Some list implementations have restrictions on the elements that they may contain. For example, some implementations prohibit null elements, and some have restrictions on the types of their elements. Attempting to add an ineligible element throws an unchecked exception, typically NullPointerException or ClassCastException. Attempting to query the presence of an ineligible element may throw an exception, or it may simply return false; some implementations will exhibit the former behavior and some will exhibit the latter. More generally, attempting an operation on an ineligible element whose completion would not result in the insertion of an ineligible element into the list may throw an exception or it may succeed, at the option of the implementation. Such exceptions are marked as "optional" in the specification for this interface. This interface is a member of the Java Collections Framework.

backtracking

Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons each partial candidate c ("backtracks") as soon as it determines that c cannot possibly be completed to a valid solution.[1][2]

dynamic equivalence

Dynamic equivalence and formal equivalence are two dissimilar translation techniques used to achieve differing levels of literalness between the original and target languages of a text. Both of these techniques are used in biblical translation. The two terms have often been understood fundamentally as sense-for-sense translation (translating the meanings of phrases or whole sentences) and word-for-word translation (translating the meanings of words and phrases in a more literal method).

preorder traversal

Display the data part of the root (or current node). Traverse the left subtree by recursively calling the pre-order function. Traverse the right subtree by recursively calling the pre-order function.

LinkedList class

Doubly-linked list implementation of the List and Deque interfaces. Implements all optional list operations, and permits all elements (including null). All of the operations perform as could be expected for a doubly-linked list. Operations that index into the list will traverse the list from the beginning or the end, whichever is closer to the specified index. Note that this implementation is not synchronized. If multiple threads access a linked list concurrently, and at least one of the threads modifies the list structurally, it must be synchronized externally. (A structural modification is any operation that adds or deletes one or more elements; merely setting the value of an element is not a structural modification.) This is typically accomplished by synchronizing on some object that naturally encapsulates the list. If no such object exists, the list should be "wrapped" using the Collections.synchronizedList method. This is best done at creation time, to prevent accidental unsynchronized access to the list: List list = Collections.synchronizedList(new LinkedList(...)); The iterators returned by this class's iterator and listIterator methods are fail-fast: if the list is structurally modified at any time after the iterator is created, in any way except through the Iterator's own remove or add methods, the iterator will throw a ConcurrentModificationException. Thus, in the face of concurrent modification, the iterator fails quickly and cleanly, rather than risking arbitrary, non-deterministic behavior at an undetermined time in the future. Note that the fail-fast behavior of an iterator cannot be guaranteed as it is, generally speaking, impossible to make any hard guarantees in the presence of unsynchronized concurrent modification. Fail-fast iterators throw ConcurrentModificationException on a best-effort basis. Therefore, it would be wrong to write a program that depended on this exception for its correctness: the fail-fast behavior of iterators should be used only to detect bugs. This class is a member of the Java Collections Framework.

double-ended queues

In computer science, a double-ended queue (dequeue, often abbreviated to deque, pronounced deck) is an abstract data type that generalizes a queue, for which elements can be added to or removed from either the front (head) or back (tail).[1] It is also often called a head-tail linked list, though properly this refers to a specific data structure implementation (see below).

divide-and-conquer algorithms

In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type (divide), until these become simple enough to be solved directly (conquer). The solutions to the sub-problems are then combined to give a solution to the original problem.

Heapsort

In computer science, heapsort is a comparison-based sorting algorithm. Heapsort can be thought of as an improved selection sort: like that algorithm, it divides its input into a sorted and an unsorted region, and it iteratively shrinks the unsorted region by extracting the largest element and moving that to the sorted region. The improvement consists of the use of a heap data structure rather than a linear-time search to find the maximum.[2]

Iterator

In computer programming, an iterator is an object that enables a programmer to traverse a container, particularly lists.[1][2][3] Various types of iterators are often provided via a container's interface. Though the interface and semantics of a given iterator are fixed, iterators are often implemented in terms of the structures underlying a container implementation and are often tightly coupled to the container to enable the operational semantics of the iterator. Note that an iterator performs traversal and also gives access to data elements in a container, but does not perform iteration (i.e., not without some significant liberty taken with that concept or with trivial use of the terminology). An iterator is behaviorally similar to a database cursor. Iterators date to the CLU programming language in 1974.

b-trees

In computer science, a B-tree is a self-balancing tree data structure that keeps data sorted and allows searches, sequential access, insertions, and deletions in logarithmic time. The B-tree is a generalization of a binary search tree in that a node can have more than two children (Comer 1979, p. 123). Unlike self-balancing binary search trees, the B-tree is optimized for systems that read and write large blocks of data. B-trees are a good example of a data structure for external memory. It is commonly used in databases and filesystems.

Container

In computer science, a container is a class, a data structure,[1][2] or an abstract data type (ADT) whose instances are collections of other objects. In other words, they store objects in an organized way that follows specific access rules. The size of the container depends on the number of objects (elements) it contains. Underlying implementation of various container types may vary in space and time complexity, which provides flexibility in choosing the right implementation for a given scenario.

Data structure

In computer science, a data structure is a particular way of organizing data in a computer so that it can be used efficiently.[1][2] Data structures can implement one or more particular abstract data types (ADT), which specify the operations that can be performed on a data structure and the computational complexity of those operations. In comparison, a data structure is a concrete implementation of the specification provided by an ADT.

the disjoint set class

In computer science, a disjoint-set data structure, also called a union-find data structure or merge-find set, is a data structure that keeps track of a set of elements partitioned into a number of disjoint (nonoverlapping) subsets. It supports two useful operations: Find: Determine which subset a particular element is in. Find typically returns an item from this set that serves as its "representative"; by comparing the result of two Find operations, one can determine whether two elements are in the same subset. Union: Join two subsets into a single subset. The other important operation, MakeSet, which makes a set containing only a given element (a singleton), is generally trivial. With these three operations, many practical partitioning problems can be solved (see the Applications section). In order to define these operations more precisely, some way of representing the sets is needed. One common approach is to select a fixed element of each set, called its representative, to represent the set as a whole. Then, Find(x) returns the representative of the set that x belongs to, and Union takes two set representatives as its arguments.

Heap

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: If A is a parent node of B then the key of node A is ordered with respect to the key of node B with the same ordering applying across the heap. A heap can be classified further as either a "max heap" or a "min heap". In a max heap, the keys of parent nodes are always greater than or equal to those of the children and the highest key is in the root node. In a min heap, the keys of parent nodes are less than or equal to those of the children and the lowest key is in the root node. Heaps are crucial in several efficient graph algorithms such as Dijkstra's algorithm, and in the sorting algorithm heapsort. A common implementation of a heap is the binary heap, in which the tree is a complete binary tree (see figure). In a heap, the highest (or lowest) priority element is always stored at the root. A heap is not a sorted structure and can be regarded as partially ordered. As visible from the heap-diagram, there is no particular relationship among nodes on any given level, even among the siblings. When a heap is a complete binary tree, it has a smallest possible height—a heap with N nodes always has log N height. A heap is a useful data structure when you need to remove the object with the highest (or lowest) priority. Note that, as shown in the graphic, there is no implied ordering between siblings or cousins and no implied sequence for an in-order traversal (as there would be in, e.g., a binary search tree). The heap relation mentioned above applies only between nodes and their parents, grandparents, etc. The maximum number of children each node can have depends on the type of heap, but in many types it is at most two, which is known as a binary heap. The heap is one maximally efficient implementation of an abstract data type called a priority queue, and in fact priority queues are often referred to as "heaps", regardless of how they may be implemented. A heap data structure should not be confused with the heap which is a common name for the pool of memory from which dynamically allocated memory is allocated. The term was originally used only for the data structure.

Linked list

In computer science, a linked list is a linear collection of data elements, called nodes pointing to the next node by means of a pointer. It is a data structure consisting of a group of nodes which together represent a sequence. Under the simplest form, each node is composed of data and a reference (in other words, a link) to the next node in the sequence; more complex variants add additional links. This structure allows for efficient insertion or removal of elements from any position in the sequence.

priority queues

In computer science, a priority queue is an abstract data type which is like a regular queue or stack data structure, but where additionally each element has a "priority" associated with it. In a priority queue, an element with high priority is served before an element with low priority. If two elements have the same priority, they are served according to their order in the queue.

Queue

In computer science, a queue (/ˈkjuː/ kew) is a particular kind of abstract data type or collection in which the entities in the collection are kept in order and the principal (or only) operations on the collection are the addition of entities to the rear terminal position, known as enqueue, and removal of entities from the front terminal position, known as dequeue. This makes the queue a First-In-First-Out (FIFO) data structure. In a FIFO data structure, the first element added to the queue will be the first one to be removed. This is equivalent to the requirement that once a new element is added, all elements that were added before have to be removed before the new element can be removed. Often a peek or front operation is also entered, returning the value of the front element without dequeuing it. A queue is an example of a linear data structure, or more abstractly a sequential collection.

randomization

In computer science, a randomization function or randomizing function is an algorithm or procedure that implements a randomly chosen function between two specific sets, suitable for use in a randomized algorithm.

Set

In computer science, a set is an abstract data type that can store certain values, without any particular order, and no repeated values. It is a computer implementation of the mathematical concept of a finite set. Unlike most other collection types, rather than retrieving a specific element from a set, one typically tests a value for membership in a set.

Stack

In computer science, a stack is an abstract data type that serves as a collection of elements, with two principal operations: push, which adds an element to the collection, and pop, which removes the most recently added element that was not yet removed. The order in which elements come off a stack gives rise to its alternative name, LIFO (for last in, first out). Additionally, a peek operation may give access to the top without modifying the stack.

amortized analysis

In computer science, amortized analysis is a method for analyzing a given algorithm's time complexity, or how much of a resource, especially time or memory in the context of computer programs, it takes to execute. The motivation for amortized analysis is that looking at the worst-case run time per operation can be too pessimistic.[1] While certain operations for a given algorithm may have a significant cost in resources, other operations may not be as costly. Amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm. This may include accounting for different types of input, length of the input, and other factors that affect its performance.[2]

avl trees

In computer science, an AVL tree (Georgy Adelson-Velsky and Evgenii Landis' tree, named after the inventors) is a self-balancing binary search tree. It was the first such data structure to be invented.[2] In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations. The AVL tree is named after its two Soviet inventors, Georgy Adelson-Velsky and Evgenii Landis, who published it in their 1962 paper "An algorithm for the organization of information".[3] AVL trees are often compared with red-black trees because both support the same set of operations and take O(log n) time for the basic operations. For lookup-intensive applications, AVL trees are faster than red-black trees because they are more rigidly balanced.[4] Similar to red-black trees, AVL trees are height-balanced. Both are in general not weight-balanced nor μ-balanced for any \scriptstyle \mu\leq\tfrac12;[5] that is, sibling nodes can have hugely differing numbers of descendants.

binary search

In computer science, binary search, also known as half-interval search[1] or logarithmic search[2], is a search algorithm that finds the position of a target value, whether alone or part of a record, within a sorted array.[3][4] It works by comparing the target value to the middle element of the array; if they are not equal, the lower or upper half of the array is eliminated depending on the result and the search is repeated until the position of the target value is found.

sequential/linear search

In computer science, linear search or sequential search is a method for finding a particular value in a list that checks each element in sequence until the desired element is found or the list is exhausted.[1] The list need not be ordered. Linear search is the simplest search algorithm; it is a special case of brute-force search. Its worst case cost is proportional to the number of elements in the list. Its expected cost is also proportional to the number of elements if all elements are searched equally. If the list has more than a few elements and is searched often, then more complicated search methods such as binary search or hashing may be appropriate. Those methods have faster search times but require additional resources to attain that speed.

mergesort

In computer science, merge sort (also commonly spelled mergesort) is an efficient, general-purpose, comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945.[1] A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.[2]

quickselect

In computer science, quickselect is a selection algorithm to find the kth smallest element in an unordered list. It is related to the quicksort sorting algorithm. Like quicksort, it was developed by Tony Hoare, and thus is also known as Hoare's selection algorithm.[1] Like quicksort, it is efficient in practice and has good average-case performance, but has poor worst-case performance. Quickselect and variants is the selection algorithm most often used in efficient real-world implementations.

algorithm analysis

In computer science, the analysis of algorithms is the determination of the amount of resources (such as time and storage) necessary to execute them. Most algorithms are designed to work with inputs of arbitrary length. Usually, the efficiency or running time of an algorithm is stated as a function relating the input length to the number of steps (time complexity) or storage locations (space complexity).

Tree traversal

In computer science, tree traversal (also known as tree search) is a form of graph traversal and refers to the process of visiting (checking and/or updating) each node in a tree data structure, exactly once. Such traversals are classified by the order in which the nodes are visited. The following algorithms are described for a binary tree, but they may be generalized to other trees as well.

Hash Table

In computing, a hash table (hash map) is a data structure used to implement an associative array, a structure that can map keys to values. A hash table uses a hash function to compute an index into an array of buckets or slots, from which the desired value can be found.

encryption & decryption

In cryptography, encryption is the process of encoding messages or information in such a way that only authorized parties can read it. Encryption does not of itself prevent interception, but denies the message content to the interceptor. In an encryption scheme, the intended communication information or message, referred to as plaintext, is encrypted using an encryption algorithm, generating ciphertext that can only be read if decrypted. For technical reasons, an encryption scheme usually uses a pseudo-random encryption key generated by an algorithm. It is in principle possible to decrypt the message without possessing the key, but, for a well-designed encryption scheme, large computational resources and skill are required. An authorized recipient can easily decrypt the message with the key provided by the originator to recipients, but not to unauthorized interceptors.

Tree rotation

In discrete mathematics, tree rotation is an operation on a binary tree that changes the structure without interfering with the order of the elements. A tree rotation moves one node up in the tree and one node down. It is used to change the shape of the tree, and in particular to decrease its height by moving smaller subtrees down and larger subtrees up, resulting in improved performance of many tree operations. There exists an inconsistency in different descriptions as to the definition of the direction of rotations. Some say that the direction of a rotation depends on the side which the tree nodes are shifted upon whilst others say that it depends on which child takes the root's place (opposite of the former). This article takes the approach of the side where the nodes get shifted to.

algorithm

In mathematics and computer science, an algorithm (Listeni/ˈælɡərɪðəm/ al-gə-ri-dhəm) is a self-contained step-by-step set of operations to be performed. Algorithms exist that perform calculation, data processing, and automated reasoning.

multiplicative inverses

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).

equivalence relations

In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes.

dynamic programming

In mathematics, management science, economics, computer science, and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions - ideally, using a memory-based data structure. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of a (hopefully) modest expenditure in storage space. (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) The technique of storing solutions to subproblems instead of recomputing them is called "memoization".

modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below, 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 is congruent to 0 modulo 12.

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120. \ The value of 0! is 1, according to the convention for an empty product.[1]

Logarithm

In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. In simple cases the logarithm counts repeated multiplication. For example, the base 10 logarithm of 1000 is 3, as 10 to the power 3 is 1000 (1000 = 10 × 10 × 10 = 103); the multiplication is repeated three times. More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm can be calculated for any two positive real numbers b and x where b is not equal to 1. The logarithm of x to base b, denoted logb(x), is the unique real number y such that by = x. For example, as 64 = 26, we have log2(64) = 6

permutation

In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting. These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters. In this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics.

Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.[1]

the iterator pattern

In object-oriented programming, the iterator pattern is a design pattern in which an iterator is used to traverse a container and access the container's elements. The iterator pattern decouples algorithms from containers; in some cases, algorithms are necessarily container-specific and thus cannot be decoupled.

order statistics

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value.[1] Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and (with some qualifications discussed below) the sample median and other sample quantiles. When using probability theory to analyze order statistics of random samples from a continuous distribution, the cumulative distribution function is used to reduce the analysis to the case of order statistics of the uniform distribution.

topological sorting

In the field of computer science, a topological sort (sometimes abbreviated toposort[1]) or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time.

Separate chaining

In the method known as separate chaining, each bucket is independent, and has some sort of list of entries with the same index. The time for hash table operations is the time to find the bucket (which is constant) plus the time for the list operation. In a good hash table, each bucket has zero or one entries, and sometimes two or three, but rarely more than that. Therefore, structures that are efficient in time and space for these cases are preferred. Structures that are efficient for a fairly large number of entries per bucket are not needed or desirable. If these cases happen often, the hashing is not working well, and this needs to be fixed.

Insertion sort

Insertion sort is a simple sorting algorithm that builds the final sorted array (or list) one item at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages: Simple implementation: Bentley shows a three-line C version, and a five-line optimized version[1]:116 Efficient for (quite) small data sets, much like other quadratic sorting algorithms More efficient in practice than most other simple quadratic (i.e., O(n2)) algorithms such as selection sort or bubble sort Adaptive, i.e., efficient for data sets that are already substantially sorted: the time complexity is O(nk) when each element in the input is no more than k places away from its sorted position Stable; i.e., does not change the relative order of elements with equal keys In-place; i.e., only requires a constant amount O(1) of additional memory space Online; i.e., can sort a list as it receives it When people manually sort cards in a bridge hand, most use a method that is similar to insertion sort.[2]

interpolation search

Interpolation search is an improved variant of binary search. This search algorithm works on the probing position of required value. For this algorithm to work properly the data collection should be in sorted form and equally distributed. Binary search has huge advantage of time complexity over linear search. Linear search has worst-case complexity of Ο(n) whereas binary search has Ο(log n). There are cases where the location of target data may be known in advance. For example, in case of telephone directory, if we want to search telephone number of Morphius. Here, linear search and even binary search will seem slow as we can directly jump to memory space where names start from 'M' are stored.

Linear probing

Linear probing is a scheme in computer programming for resolving collisions in hash tables, data structures for maintaining a collection of key-value pairs and looking up the value associated with a given key. It was invented in 1954 by Gene Amdahl, Elaine M. McGraw, and Arthur Samuel and first analyzed in 1963 by Donald Knuth. Along with quadratic probing and double hashing, linear probing is a form of open addressing. In these schemes, each cell of a hash table stores a single key-value pair. When the hash function causes a collision by mapping a new key to a cell of the hash table that is already occupied by another key, linear probing searches the table for the closest following free location and inserts the new key there. Lookups are performed in the same way, by searching the table sequentially starting at the position given by the hash function, until finding a cell with a matching key or an empty cell. As Thorup & Zhang (2012) write, "Hash tables are the most commonly used nontrivial data structures, and the most popular implementation on standard hardware uses linear probing, which is both fast and simple."[1] Linear probing can provide high performance because of its good locality of reference, but is more sensitive to the quality of its hash function than some other collision resolution schemes. It takes constant expected time per search, insertion, or deletion when implemented using a random hash function, a 5-independent hash function, or tabulation hashing.

Comparator function objects

Many Collections API classes and routines require the ability to order objects. There are two ways to do this. One possibility is that the objects implement the Comparable interface and provide a compareTo method. The other possibility is that the comparison function is embedded as the compare method in an object that implements the Comparator interface. Comparator is defined in java.util; a sam- ple implementation was shown in Figure 4.39 and is repeated in Figure 6.12.

modular exponentiation

Modular exponentiation is a type of exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography. The operation of modular exponentiation calculates the remainder when an integer b (the base) raised to the eth power (the exponent), be, is divided by a positive integer m (the modulus). In symbols, given base b, exponent e, and modulus m, the modular exponentiation c is: c ≡ be (mod m). For example, given b = 5, e = 3 and m = 13, the solution c = 8 is the remainder of dividing 53 = 125 by 13. Given integers b and e, and a positive integer m, a unique solution c exists with the property 0 ≤ c < m. Modular exponentiation can be performed with a negative exponent e by finding the modular multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: c ≡ be ≡ d−e mod m where e < 0 and b ⋅ d ≡ 1 mod m. Modular exponentiation similar to the one described above are considered easy to compute, even when the numbers involved are enormous. On the other hand, computing the discrete logarithm - that is, the task of finding the exponent e when given b, c, and m - is believed to be difficult. This one-way function behavior makes modular exponentiation a candidate for use in cryptographic algorithms.

nonuniform random numbers

Pseudo-random number sampling or non-uniform pseudo-random variate generation is the numerical practice of generating pseudo-random numbers that are distributed according to a given probability distribution.

quadratic probing

Quadratic probing is an open addressing scheme in computer programming for resolving collisions in hash tables—when an incoming data's hash value indicates it should be stored in an already-occupied slot or bucket. Quadratic probing operates by taking the original hash index and adding successive values of an arbitrary quadratic polynomial until an open slot is found.

quicksort

Quicksort (sometimes called partition-exchange sort) is an efficient sorting algorithm, serving as a systematic method for placing the elements of an array in order. Developed by Tony Hoare in 1959,[1] with his work published in 1961,[2] it is still a commonly used algorithm for sorting. When implemented well, it can be about two or three times faster than its main competitors, merge sort and heapsort.[3] Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined. In efficient implementations it is not a stable sort, meaning that the relative order of equal sort items is not preserved. Quicksort can operate in-place on an array, requiring small additional amounts of memory to perform the sorting. Mathematical analysis of quicksort shows that, on average, the algorithm takes O(n log n) comparisons to sort n items. In the worst case, it makes O(n2) comparisons, though this behavior is rare.

the rsa cryptosystem

RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and differs from the decryption key which is kept secret. In RSA, this asymmetry is based on the practical difficulty of factoring the product of two large prime numbers, the factoring problem. RSA is made of the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who first publicly described the algorithm in 1977. Clifford Cocks, an English mathematician working for the UK intelligence agency GCHQ, had developed an equivalent system in 1973, but it was not declassified until 1997.[1]

shellsort

Shellsort, also known as Shell sort or Shell's method, is an in-place comparison sort. It can be seen as either a generalization of sorting by exchange (bubble sort) or sorting by insertion (insertion sort).[2] The method starts by sorting pairs of elements far apart from each other, then progressively reducing the gap between elements to be compared. Starting with far apart elements can move some out-of-place elements into position faster than a simple nearest neighbor exchange. Donald Shell published the first version of this sort in 1959.[3][4] The running time of Shellsort is heavily dependent on the gap sequence it uses. For many practical variants, determining their time complexity remains an open problem.

generic algorithms

The Collections API provides a few general purpose algorithms that operate on all of the containers. These are static methods in the Collections class (note that this is a different class than the Collection interface). There are also some static methods in the Arrays class that manipulate arrays (sorting, searching, etc.). Most of those methods are overloaded—a generic version, and once for each of the primitive types (except boolean). We examine only a few of the algorithms, with the intention of showing the general ideas that pervade the Collections API, while documenting the specific algorithms that will be used in Part Three. Some of the algorithms make use of function objects. Consequently, the material in Section 4.8 is an essential prerequisite to this section.

nested classes

The Java programming language allows you to define a class within another class. Such a class is called a nested class and is illustrated here: class OuterClass { ... class NestedClass { ... } } Terminology: Nested classes are divided into two categories: static and non-static. Nested classes that are declared static are called static nested classes. Non-static nested classes are called inner classes. class OuterClass { ... static class StaticNestedClass { ... } class InnerClass { ... } } A nested class is a member of its enclosing class. Non-static nested classes (inner classes) have access to other members of the enclosing class, even if they are declared private. Static nested classes do not have access to other members of the enclosing class. As a member of the OuterClass, a nested class can be declared private, public, protected, or package private. (Recall that outer classes can only be declared public or package private.)

Pivot element

The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as multiplication by permutation matrices. However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations. Overall, pivoting adds more operations to the computational cost of an algorithm. These additional operations are sometimes necessary for the algorithm to work at all. Other times these additional operations are worthwhile because they add numerical stability to the final result.

the Collections class

This class consists exclusively of static methods that operate on or return collections. It contains polymorphic algorithms that operate on collections, "wrappers", which return a new collection backed by a specified collection, and a few other odds and ends. The methods of this class all throw a NullPointerException if the collections or class objects provided to them are null. The documentation for the polymorphic algorithms contained in this class generally includes a brief description of the implementation. Such descriptions should be regarded as implementation notes, rather than parts of the specification. Implementors should feel free to substitute other algorithms, so long as the specification itself is adhered to. (For example, the algorithm used by sort does not have to be a mergesort, but it does have to be stable.) The "destructive" algorithms contained in this class, that is, the algorithms that modify the collection on which they operate, are specified to throw UnsupportedOperationException if the collection does not support the appropriate mutation primitive(s), such as the set method. These algorithms may, but are not required to, throw this exception if an invocation would have no effect on the collection. For example, invoking the sort method on an unmodifiable list that is already sorted may or may not throw UnsupportedOperationException. This class is a member of the Java Collections Framework.

the HashSet class

This class implements the Set interface, backed by a hash table (actually a HashMap instance). It makes no guarantees as to the iteration order of the set; in particular, it does not guarantee that the order will remain constant over time. This class permits the null element. This class offers constant time performance for the basic operations (add, remove, contains and size), assuming the hash function disperses the elements properly among the buckets. Iterating over this set requires time proportional to the sum of the HashSet instance's size (the number of elements) plus the "capacity" of the backing HashMap instance (the number of buckets). Thus, it's very important not to set the initial capacity too high (or the load factor too low) if iteration performance is important. Note that this implementation is not synchronized. If multiple threads access a hash set concurrently, and at least one of the threads modifies the set, it must be synchronized externally. This is typically accomplished by synchronizing on some object that naturally encapsulates the set. If no such object exists, the set should be "wrapped" using the Collections.synchronizedSet method. This is best done at creation time, to prevent accidental unsynchronized access to the set: Set s = Collections.synchronizedSet(new HashSet(...)); The iterators returned by this class's iterator method are fail-fast: if the set is modified at any time after the iterator is created, in any way except through the iterator's own remove method, the Iterator throws a ConcurrentModificationException. Thus, in the face of concurrent modification, the iterator fails quickly and cleanly, rather than risking arbitrary, non-deterministic behavior at an undetermined time in the future. Note that the fail-fast behavior of an iterator cannot be guaranteed as it is, generally speaking, impossible to make any hard guarantees in the presence of unsynchronized concurrent modification. Fail-fast iterators throw ConcurrentModificationException on a best-effort basis. Therefore, it would be wrong to write a program that depended on this exception for its correctness: the fail-fast behavior of iterators should be used only to detect bugs. This class is a member of the Java Collections Framework.

the AbstractCollection class

This class provides a skeletal implementation of the Collection interface, to minimize the effort required to implement this interface. To implement an unmodifiable collection, the programmer needs only to extend this class and provide implementations for the iterator and size methods. (The iterator returned by the iterator method must implement hasNext and next.) To implement a modifiable collection, the programmer must additionally override this class's add method (which otherwise throws an UnsupportedOperationException), and the iterator returned by the iterator method must additionally implement its remove method. The programmer should generally provide a void (no argument) and Collection constructor, as per the recommendation in the Collection interface specification. The documentation for each non-abstract method in this class describes its implementation in detail. Each of these methods may be overridden if the collection being implemented admits a more efficient implementation. This class is a member of the Java Collections Framework.

inorder traversal

Traverse the left subtree by recursively calling the in-order function. Display the data part of the root (or current node). Traverse the right subtree by recursively calling the in-order function. In a search tree, in-order traversal retrieves data in sorted order.[4]

postorder traversal

Traverse the left subtree by recursively calling the post-order function. Traverse the right subtree by recursively calling the post-order function. Display the data part of the root (or current node). The trace of a traversal is called a sequentialisation of the tree. The traversal trace is a list of each visited root. No one sequentialisation according to pre-, in- or post-order describes the underlying tree uniquely. Given a tree with distinct elements, either pre-order or post-order paired with in-order is sufficient to describe the tree uniquely. However, pre-order with post-order leaves some ambiguity in the tree structure.[5]

level-order traversals

Trees can also be traversed in level-order, where we visit every node on a level before going to a lower level. This search is referred to as breadth-first search (BFS), as the search tree is broadened as much as possible on each depth before going to the next depth.

Big Omega notation

definition: (Big-Omega) T(N) is Ω(F(N)) if there are positive constants c and N0 such that T(N) ≥ cF(N) when N ≥ N0 . The second definition, T(N) = Ω(F(N)), called Big-Omega, says that the growth rate of T(N) is greater than or equal to that of F(N). For instance, we might say that any algorithm that works by examining every possible subse- quence in the maximum subsequence sum problem must take Ω(N2) time because a quadratic number of subsequences are possible. This is a lower-bound argument that is used in more advanced analysis. Later in the text, we will see one example of this argument and demonstrate that any general-purpose sorting algorithm requires Ω ( N log N ) time.

Big Theta notation

definition: (Big-Theta) T(N) is Θ(F(N)) if and only if T(N) is O(F(N)) and T(N) is Ω(F(N)). The third definition, T(N) = Θ(F(N)), called Big-Theta, says that the growth rate of T ( N ) equals the growth rate of F( N ) . For instance, the max- imum subsequence algorithm shown in Figure 5.5 runs in Θ( N 2) time. In other words, the running time is bounded by a quadratic function, and this bound cannot be improved because it is also lower-bounded by another qua- dratic function. When we use Big-Theta notation, we are providing not only an upper bound on an algorithm but also assurances that the analysis that leads to the upper bound is as good (tight) as possible. In spite of the addi- tional precision offered by Big-Theta, however, Big-Oh is more commonly used, except by researchers in the algorithm analysis field.

Big O notation

definition: (Big-Oh) T ( N ) is O( F( N )) if there are positive constants c and N0 such that T(N) ≤ cF(N) when N ≥ N0 . The first definition, Big-Oh notation, states that there is a point N0 such that for all values of N that are past this point, T(N) is bounded by some multiple of F ( N ) . This is the sufficiently large N mentioned earlier. Thus, if the running time T ( N ) of an algorithm is O( N 2) , then, ignoring constants, we are guaranteeing that at some point we can bound the running time by a quadratic function. Notice that if the true running time is linear, then the state- ment that the running time is O( N 2) is technically correct because the ine- quality holds. However, O( N ) would be the more precise claim. If we use the traditional inequality operators to compare growth rates, then the first definition says that the growth rate of T ( N ) is less than or equal to that of F ( N ) .

Little O notation

definition: (Little-Oh) T(N) is o(F(N)) if and only if T(N) is O(F(N)) and T ( N ) is not Θ ( F ( N ) ) .4 The final definition, T(N) = o(F(N)), called Little-Oh, says that the growth rate of T ( N ) is strictly less than the growth rate of F( N ) . This func- tion is different from Big-Oh because Big-Oh allows the possibility that the growth rates are the same. For instance, if the running time of an algorithm is o( N 2) , then it is guaranteed to be growing at a slower rate than quadratic (that is, it is a subquadratic algorithm). Thus a bound of o(N2) is a better bound than Θ( N 2) . Figure 5.9 summarizes these four definitions.