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We usually sort a list of numbers, characters, strings and records similar tocollege information about their students, library information and company information ischosen for guiding the sorting technique. Such pieces of informationis called a key. The most important when we use the searching of records.
There aredifferent types of sorting algorithms. There are some algorithms that sort an arbitraryof size n using nlog2n comparisons, On the other hand, no algorithm that sorts by keycomparisons can do better than that.
Although some algorithms are better than others,there is no algorithm that would be the best in all situations. Some algorithms are simplebut relatively slow while others are faster but more complex.
Some are suitable only forlists residing in the fast memory while others can be adapted for sorting large filesstored on a disk, and so on.
There are two important properties. The first is called stable, if it preserves therelative order of any two equal elements in its input.
For example, if we sort the studentlist based on their GPA and if two students GPA are the same, then the elements arestored or sorted based on its position. There are some sorting algorithms that are in place and thosethat are not. The searching problem deals with finding a given value, called a search key, in agiven set. The searching can be either a straightforward algorithm or binary searchalgorithm which is a different form.
These algorithms play a important role in real-lifeapplications because they are used for storing and retrieving information from largedatabases. Some algorithms work faster but require more memory, some are very fastbut applicable only to sorted arrays.
Searching, mainly deals with addition and deletionS. In such cases, the data structures and algorithms are chosen to balanceamong the required set of operations. String processing: A String is a sequence of characters. It is mainly used in string handlingalgorithms. Most common ones are text strings, which consists of letters, numbers andspecial characters. Bit strings consist of zeroes and ones. The most important problemis the string matching, which is used for searching a given word in a text.
Graph problems: One of the interesting area in algorithmic is graph algorithms. A graph is acollection of points called vertices which are connected by line segments called edges.
Graphs are used for modeling a wide variety of real-life applications such astransportation and communication networks. It includes graph traversal, shortest-path and topological sorting algorithms. Some graph problems are very hard, only very small instances of the problems can besolved in realistic amount of time even with fastest computers. There are two commonproblems: The graph-coloring problem is to assign the smallestnumber of colors to vertices of a graph so that no two adjacent vertices are of thesame color.
It arises in event-scheduling problem, where the events are represented byvertices that are connected by an edge if the corresponding events cannot be scheduledin the same time, a solution to this graph gives an optimal schedule. Combinatorial problems: The traveling salesman problem and the graph-coloring problem are examples ofcombinatorial problems. These are problems that ask us to find a combinatorial objectsuch as permutation, combination or a subset that satisfies certain constraints and hassome desired e.
These problems are difficult to solve for the following facts. Second, there are noknown algorithms, which are solved in acceptable amount of time. Geometric problems: Geometric algorithms deal with geometric objects such as points, lines and polygons. It also includes various geometric shapes such as triangles, circles etc. The applicationsfor these algorithms are in computer graphic, robotics etc.
The convex-hull problem is to find thesmallest convex polygon that would include all the points of a given set. Numerical problems: This is another large special area of applications, where the problems involvemathematical objects of continuous nature: These problems can be solved onlyapproximately. These require real numbers, which can be represented in a computer onlyapproximately.
If can also lead to an accumulation of round-off errors. The algorithmsdesigned are mainly used in scientific and engineering applications.
Data structure play an important role in designing of algorithms, since it operates ondata. A data structure can be defined as a particular scheme of organizing related dataitems. The data items range from elementary data types to data structures. The two most important elementary data structure are the array and the linked list.
Item  item - - - item[n-1] Array of n elements. The index is an integer ranges from 0 to n Each and every element in the array takesthe same amount of time to access and also it takes the same amount of computerstorage. Arrays are also used for implementing other data structures. One among is the string: Strings composed of zeroes and ones are called binary strings or bit strings. Operations performed on strings are: A pointer called null is used to represent no more nodes.
In a singly linked list, eachnode except the last one contains a single pointer to the next element. To access a particular node, we start with the first node and traverse the pointerchain until the particular node is reached. The time needed to access depends on wherein the list the element is located. There are various forms of linked list. One is, we can start a linked list with aspecial node called the header.
This contains information about the linked list such asits current length, also a pointer to the first element, a pointer to the last element.
Another form is called the doubly linked list, in which every node, except thefirst and the last, contains pointers to both its success or and its predecessor.
The another more abstract data structure called a linear list or simply a list. Alist is a finite sequence of data items, i. The basic operations performed are searching for, inserting anddeleting on element. Two special types of lists, stacks and queues. A stack is a list in which insertionsand deletions can be made only at one end.
This end is called the top. The two operationsdone are: Its used in recursive algorithms,where the last- in- first- out LIFO fashion is used.
The last inserted will be the firstone to be removed. A queue, is a list for, which elements are deleted from one end of the structure,called the front this operation is called dequeue , and new elements are added to theother end, called the rear this operation is called enqueue.
It operates in a first- in-first-out basis. Its having many applications including the graph problems.
A priority queue is a collection of data items from a totally ordered universe. Theprincipal operations are finding its largest elements, deleting its largest element andadding a new element.
A better implementation is based on a data structure called aheap.
A graph is informally thought of a collection of points in a plane called vertices ornodes, some of them connected by line segments called edges or arcs. If these pairs of vertices are unordered, i. Directed graphs are alsocalled digraphs.
The inequality for the number of edges E possible in anundirected graph with v vertices and no loops is: A graph with every pair of its vertices connected by an edge is called complete.
Notation with V vertices is K V. A graph with relatively few possible edges missing iscalled dense; a graph with few edges relative to the number of its vertices is calledsparse.
Graph representation ii. Weighted graphs and iii. Paths and cycles. Graphs for computer algorithms can be represented in two ways: The adjacency matrixfor the undirected graph is given below: The adjacency matrix of an undirected graph is symmetric. A weighted graph is a graph or digraph with numbers assigned to its edges.
Thesenumbers are weights or costs. The real-life applications are traveling salesman problem,Shortest path between two points in a transportation or communication network. The adjacency matrix. The adjacency linked list consists of the nodes name and also the weight ofthe edges. Two properties: Connectivity and acyclicity are important for various applications,which depends on the notion of a path.
A path from vertex v to vertex u of a graph Gcan be defined as a sequence of adjacent vertices that starts with v and ends with u. Ifall edges of a path are distinct, the path is said to be simple. The length of a path is thetotal number of vertices in a vertex minus one.
A graph is said to be connected if for every pair of its vertices u and v there is apath from u to v.
If a graph is not connected, it will consist of several connected piecesthat are called connected components of the graph. A connected component is themaximal subgraph of a given graph.
The graphs a and b represents connected and notconnected graph.
A graph with no cycles is said to be acyclic. A tree is a connected acyclic graph. A graph that has no cycles but is notnecessarily connected is called a forest: For every two vertices in a tree there always exists exactly one simple path fromone of these vertices to the other. For this, select an arbitrary vertex in a free treeand consider it as the root of the so-called rooted tree.
Rooted trees plays an importantrole in various applications with the help of state-space-tree which leads to twoimportant algorithm design techniques: The rootstarts from level 0 and the vertices adjacent to the root below is level 1 etc. The set of ancestors that excludes the vertexitself is referred to as proper ancestors. A vertex with nochildren is called a leaf; a vertex with at least one child is called parental. All thevertices for which a vertex v is an ancestor are said to be descendants of v.
A vertex vwith all its descendants is called the sub tree of T rooted at that vertex. The depth of a vertex v is the length of the simple path from the root to v. Theheight of the tree is the length of the longest simple path from the root to a leaf. An ordered tree is a rooted tree in which all the children of each vertex areordered. A binary tree can be defined as an ordered tree in which every vertex has nomore than two children and each child is a left or right child of its parent.
The sub treewith its root at the left right child of a vertex is called the left right sub tree ofthat vertex. A number assigned to each parental vertex is larger than all the numbers in itsleft sub tree and smaller than all the numbers in its right sub tree.
Such trees arecalled Binary search trees. Binary search trees can be more generalized to formmultiway search trees, for efficient storage of very large files on disks. The binary search tree can be represented with the help of linked list: The left pointer point to the first child and the right pointer pointsto the next sibling. This representation is called the first child-next siblingrepresentation. Thus all the siblings of a vertex are linked in a singly linked list, with thefirst element of the list pointed to by the left pointer of their parent.
A set can be described as an unordered collection of distinct items calledelements of the set. A specific set is defined either by an explicit listing of itselements or by specifying a set of property.
Sets can be implemented in computer applications in two ways. The first considersonly sets that are subsets of some large set U called the universal set. If set U has nelements, then any subset S of U can be represented by a bit string of size n, called abit vector, in which the ith element is 1 iff the ith element of U is included in set S. Bit string operations are faster but consume a large amount ofstorage.
A multiset or bag is an unordered collection of items that are not necessarilydistinct. Note, changing the order of the set elements does not change the set, whereasthe list is just opposite. A set cannot contain identical elements, a list can. The operation that has to be performed in a set is searching for a given item,adding a new item, and deletion of an item from the collection.
A data structure thatimplements these three operations is called the dictionary. A number of applications incomputing require a dynamic partition of some n-element set into a collection of disjointsubsets. After initialization, it performs a sequence of union and search operations.
Thisproblem is called the set union problem. These data structure play an important role in algorithms efficiency, which leadsto an abstract data type ADT: Fundamentals of the Analysis of Algorithm Efficiency.
This chapter deals with analysis of algorithms. Efficiency is studied first inquantitative terms unlike simplicity and generality. The mathematical analysis shows the frameworksystematically applied to analyzing the efficiency of nonrecursive algorithms.
Time efficiency indicates how fast an algorithm in question runs; spaceefficiency deals with the extra space the algorithm requires. The space requirement isnot of much concern, because now we have the fast main memory, cache memory etc. Almost all algorithms run longer on larger inputs. For example, it takes to sortlarger arrays, multiply larger matrices and so on. For example, it will be the size of the list for problems of sorting, searching etc.
The size also be influenced by the operations of the algorithm. We can use some standard unit of time to measure the running time of a programimplementing the algorithm. The drawbacks to such an approach are: The drawback to such an approach are: Here, we do not consider these extraneous factors for simplicity. The simple way, is to identify the most important operation ofthe algorithm, called the basic operation, the operation contributing the most to thetotal running time and compute the umber of times the basic operation is executed.
For example, most sorting algorithm works by comparingelements keys , of a list being sorted with each other; for such algorithms, the basicoperation is the key comparison. Then we can estimate the running time, T n as: Theconstant Cop is also an approximation whose reliability is not easy to assess. The answer is four times longer. This is mainly considered for large input size. Contact Details. Rent and save from the world's largest eBookstore.
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