![22
Algorithm Analysis: Example
Alg.: MIN (a[1], …, a[n])
m ← a[1];
for i ← 2 to n
if a[i] < m
then m ← a[i];](https://image.slidesharecdn.com/tvudsunit1-240206160527-794a60f6/85/Data-Structures-unit-I-Introduction-data-types-22-320.jpg)
![ Running time:
the number of primitive operations (steps) executed
before termination
T(n) =1 [first step] + (n) [for loop] + (n-1) [if
condition] +
(n-1) [the assignment in then] = 3n - 1](https://image.slidesharecdn.com/tvudsunit1-240206160527-794a60f6/85/Data-Structures-unit-I-Introduction-data-types-23-320.jpg)
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Data Structures unit I Introduction - data types
- 1. PROGRAMMING AND DATA STRUCTURES IN C Ms. UMA. S Assistant Professor Computer Science/Computer Applications Thiruvalluvar University College of Arts and Science, Tirupattur
- 2. OBJECTIVES Learning program independent view of data structures, including its representation and operations performed on them, which are then linked to sorting, searching and indexing methods to increase the knowledge of usage of data structures in algorithmic perspective.
- 3. UNIT-I Abstract Data Types – Asymptotic Notations: Big-Oh, Omega and Theta – Best, Worst and Average case Analysis: Definition and an example – Arrays and its representations – Stacks and Queues – Linked lists – Linked list based implementation of Stacks and Queues – Evaluation of Expressions – Linked list based polynomial addition.
- 4. UNIT-II Trees – Binary Trees – Binary tree representation and traversals – Threaded binary trees – Binary tree representation of trees – Application of trees: Set representation and Union- Find operations – Graph and its representations – Graph Traversals – Connected components
- 5. UNIT-III AVL Trees – Red-Black Trees – Splay Trees – Binary Heap – Leftist Heap UNIT–IV Insertion sort – Merge sort – Quick sort – Heap sort – Sorting with disks – k- way merging – Sorting with tapes – Polyphase merge. UNIT-V Linear Search – Binary Search - Hash tables – Overflow handling – Cylinder Surface Indexing – Hash Index – B-Tree Indexing.
- 6. TEXT BOOK 1. Ellis Horowitz and Sartaj Sahni, Fundamentals of Data Structures, Galgotia Book Sorce, Gurgaon, 1993. 2. Gregory L. Heilman, Data Structures, Algorithms and Object Oriented Programming, Tata Mcgraw-Hill, New Delhi, 2002. REFERENCES 1. Jean-Paul Tremblay and Paul G. Sorenson, An Introduction to Data Structures with Applications, Second Edition, Tata McGraw-Hill, New Delhi, 1991. 2. Alfred V. Aho, John E. Hopcroft and Jeffry D. Ullman, Data Structures and Algorithms, Pearson Education, New Delhi, 2006
- 7. Data Structure Data structure is a representation of data and the operations allowed on that data. A data structure is a way to store and organize data in order to facilitate the access and modifications. Data Structure are the method of representing of logical relationships between individual data elements related to the solution of a given problem.
- 8. Basic Data Structure Basic Data Structures Linear Data Structures Non-Linear Data Structures Arrays Linked Lists Stacks Queues Trees Graphs Hash Tables
- 10. Selection of Data Structure The choice of particular data model depends on two consideration: It must be rich enough in structure to represent the relationship between data elements The structure should be simple enough that one can effectively process the data when necessary
- 11. Types of Data Structure Linear: In Linear data structure, values are arrange in linear fashion. Array: Fixed-size Linked-list: Variable-size Stack: Add to top and remove from top Queue: Add to back and remove from front Priority queue: Add anywhere, remove the highest priority
- 12. Types of Data Structure Non-Linear: The data values in this structure are not arranged in order. Hash tables: Unordered lists which use a ‘hash function’ to insert and search Tree: Data is organized in branches. Graph: A more general branching structure, with less strict connection conditions than for a tree
- 13. Type of Data Structures Homogenous: In this type of data structures, values of the same types of data are stored. Array Non-Homogenous: In this type of data structures, data values of different types are grouped and stored. Structures Classes
- 14. Abstract Data Type and Data Structure Definition:- Abstract Data Types (ADTs) stores data and allow various operations on the data to access and change it. A mathematical model, together with various operations defined on the model An ADT is a collection of data and associated operations for manipulating that data
- 15. Data Structures Physical implementation of an ADT data structures used in implementations are provided in a language (primitive or built-in) or are built from the language constructs (user-defined) Each operation associated with the ADT is implemented by one or more subroutines in the implementation
- 16. Abstract Data Type ADTs support abstraction, encapsulation, and information hiding. Abstraction is the structuring of a problem into well- defined entities by defining their data and operations. The principle of hiding the used data structure and to only provide a well-defined interface is known as encapsulation.
- 17. Core Operations of ADT Every Collection ADT should provide a way to: add an item remove an item find, retrieve, or access an item Other possibilities is the collection empty make the collection empty give me a sub set of the collection
- 18. • No single data structure works well for all purposes, and so it is important to know the strengths and limitations of several of them
- 19. 19 Analyzing Algorithms Predict the amount of resources required: • memory: how much space is needed? • computational time: how fast the algorithm runs? FACT: running time grows with the size of the input
- 20. Input size (number of elements in the input) Size of an array, polynomial degree, # of elements in a matrix, # of bits in the binary representation of the input, vertices and edges in a graph
- 21. Def: Running time = the number of primitive operations (steps) executed before termination Arithmetic operations (+, -, *), data movement, control, decision making (if, while), comparison
- 22. 22 Algorithm Analysis: Example Alg.: MIN (a[1], …, a[n]) m ← a[1]; for i ← 2 to n if a[i] < m then m ← a[i];
- 23. Running time: the number of primitive operations (steps) executed before termination T(n) =1 [first step] + (n) [for loop] + (n-1) [if condition] + (n-1) [the assignment in then] = 3n - 1
- 24. Order (rate) of growth: The leading term of the formula Expresses the asymptotic behavior of the algorithm
- 25. 25 Typical Running Time Functions 1 (constant running time): Instructions are executed once or a few times logN (logarithmic) A big problem is solved by cutting the original problem in smaller sizes, by a constant fraction at each step N (linear) A small amount of processing is done on each input element N logN A problem is solved by dividing it into smaller problems, solving them independently and combining the solution
- 26. 26 Typical Running Time Functions N2 (quadratic) Typical for algorithms that process all pairs of data items (double nested loops) N3 (cubic) Processing of triples of data (triple nested loops) NK (polynomial) 2N (exponential) Few exponential algorithms are appropriate for practical use
- 31. Complexity Graphs (log scale) n10 n20 nn 1.1n 2n 3n
- 32. Algorithm Complexity Worst Case Complexity: the function defined by the maximum number of steps taken on any instance of size n Best Case Complexity: the function defined by the minimum number of steps taken on any instance of size n Average Case Complexity: the function defined by the average number of steps taken on any instance of size n
- 33. Best, Worst, and Average Case Complexity Worst Case Complexity Average Case Complexity Best Case Complexity Number of steps N (input size)
- 34. Doing the Analysis It’s hard to estimate the running time exactly Best case depends on the input Average case is difficult to compute So we usually focus on worst case analysis Easier to compute Usually close to the actual running time
- 35. Strategy: find a function (an equation) that, for large n, is an upper bound to the actual function (actual number of steps, memory usage, etc.)
- 36. Upper bound Lower bound Actual function
- 37. Motivation for Asymptotic Analysis An exact computation of worst-case running time can be difficult Function may have many terms: 4n2 - 3n log n + 17.5 n - 43 n⅔ + 75 An exact computation of worst-case running time is unnecessary Remember that we are already approximating running time by using RAM model