Data S tructures and Algorithms

Data StructuresandAlgorithms Course’s slides: Introduction, Basic data types www.mif.vu.lt/~algis

Motto • Data Structures and Algorithms • The key to your professional reputation • A much more dramatic effect can be made on the performance of a program by changing to a better algorithm than by • hacking • converting to assembler • Buy a text for long-term reference! • Professional software engineers have algorithms text(s) on their shelves • Hackers have user manuals for the latest software package

Content 1. Introduction, computing model, von Neuman principles, data, abstract data types, data structures, basic data types 2. Sorting, internal sorting, quicksort 3. Merge sort, von Neumansorting, external sorting 4. Abstract data types, stack, queue, programming of stack and queue 5. Heap, priority queue, priority queue by heap structure, lists, list programming, dynamic sets ADT 6. Hierarchical structures, binary search trees, tree allocation in memory 7. AVL trees, 2-3-4 trees, red-black trees

Content 8. B-trees and other similar trees, Huffman algorithm for data compression 9. Hashing idea, hashing functions and tables, hashing procedures and algorithms, extendable hashing 10. Radix search algorithms, radix trees, radix algorithms, radix search 11. Patricia trees, suffix tree 12. Text search 13. Analysis of algorithms

Introduction Informally, algorithm means is a well-defined computational procedure that takes value (set of values) as input and produces some value(set of values) as output. An algorithm is thus a sequence of computational steps that transform the input into the output. Algorithm is also viewed as a tool for solving a well-specified problem, involving computers. There exist many points of view to algorithms. A good example of this is a famous Euclid’s algorithm: for two integers x, y calculate the greatest common divisor gcd (x, y). Direct implementation of the algorithm looks like:

Introduction program euclid (input, output);varx,y: integer;function gcd(u,v: integer): integer; vart: integer; begin repeat if u<v then begin t := u; u := v; v := t end; u := u-v; until u = 0; gcd:= v end; beginwhile not eofdo begin readln(x, y); if (x>0) and (y>0) then writeln (x, y, gcd (x, y)) end; end.

Introduction • This algorithm has some exceptional features • it is applicable only to numbers; • it has to be changed every time when something of the environment changes, say if numbers are very long and does not fit into a size of variable (numbers like 1000!). • For algorithms of applications in the focus of this course, like databases, information systems, etc., they are usually understood in a slightly different way: • is repeated many times; • in a various circumstancies; • with different types of data.

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Von Neuman computing model 1943: ENIACPresper Eckert and John Mauchly -- first general electronic computer. Hard-wired program -- settings of dials and switches. 1944: Beginnings of EDVAC among other improvements, includes program stored in memory 1945: John von Neumann wrote a report on the stored program concept, known as the First Draft of a Report on EDVAC , the “von Neumann machine” (or model). a memory, containing instructions and dataa processing unit, for performing arithmetic and logical operations a control unit, for interpreting instructions

Von Neuman computing model

Sub-Components • Clearly, requiring hardware changes with each new programming operation was time-consuming, error-prone, and costly • Von Neuman’sproposal was to store the program instructions right along with the data • The stored program concept was proposed about fifty years ago; to this day, it is the fundamental architecture that fuels computers.

Von Neuman computing model 3 5 RAM 8 skaičiavimas valdymas registrai PC

Memory Types: RAM • RAM is typically volatile memory (meaning it doesn’t retain voltage settings once power is removed) • RAM is an array of cells, each with a unique address • A cell is the minimum unit of access. Originally, this was 8 bits taken together as a byte. In today’s computer, word-sized cells (16 bits, grouped in 4) are more typical. • RAM gets its name from its access performance. In RAM memory, theoretically, it would take the same amount of time to access any memory cell, regardless of its location with the memory bank (“random”access).

The ALU • The third component in the von Neumann architecture is called the Arithmetic Logic Unit. • This is the subcomponent that performs the arithmetic and logic operations for which we have been building parts. • The ALU is the “brain” of the computer.

The ALU • It houses the special memory locations, called registers, of which we have already considered. • The ALU is important enough that we will come back to it later, For now, just realize that it contains the circuitry to perform addition, subtraction,multiplication and division, as well as logical comparisons (less than, equal to and greater than).

Boolean data Data values: {false, true} In C/C++: false= 0, true= 1 (or nonzero) Could store 1 value per bit, but usually use a byte (or word)   Operations: and && or || not !

Java Character Data

Integer Data • Nonegative (unsigned) integer: • type unsigned (and variations) in C++ • Store its base-two representation in a fixed number w of bits • (e.g., w = 16 or w = 32) • 88 = 00000000010110002 • Signed integer: type int (and variations) in C++ • Store in a fixed number w of bits using one of the following representations:

Sign-magnitude representation • Save one bit (usually most significant) for sign(0 = +, 1 = – ) Use base-two representation in the other bits. • 88 ® _000000001011000 0 sign bit  1 –88 ® _000000001011000 • Cumbersome for arithmetic computations • 2 0’s in this scheme • Incrementing by one results in subtraction of one, not addition! Both 0 and -0

Complement representation Same as sign mag. For non-negative n: Use ordinary base-two representation with leading (sign) bit 0 Same as subtracting the number from 0! For negative n (–n): (1) Find w-bit base-2 representation of n (2) Complement each bit. (3) Add 1 Example: –88 1. 88 as a 16-bit base-two number 0000000001011000 2. Complement this bit string 1111111110100111 3. Add 1 1111111110101000

Good for arithmetic computation (see p. 38) These work for both + and – integers • 5 + 7: • 0000000000000101 • +0000000000000111 111¬¾¾ carry bits 0000000000001100 • 5 + –6: • 0000000000000101 • +1111111111111010 1111111111111111

Problems with Integer Representation • Limited Capacity -- a finite number of bits • Overflow and Underflow: Overflow - addition or multiplication can exceed largest value permitted by storage scheme Underflow - subtraction or multiplication can exceed smallest value permitted by storage scheme • Not a perfect representation of (mathematical) integers • can only store a finite (sub)range of them

How is Real Data represented? p. 756 double: Exp: 11 bits, bias 1023 Mant: 52 bits base Round-off Errors • Example: 22.625 = (see p.41) • Floating point form: 10110.1012 1.01101012 ´ 24 + 127

Basic data types for C C programming language: char - smallest addressable unit of the machine that can contain basic character set. It is an integer type, actual type can be either signed or unsigned depending on the implementation. signed char - same size as char, but guaranteed to be signed. unsigned char - same size as char, but guaranteed to be unsigned. shortshort intsigned shortsigned short int- shortsigned integer type, at least 16 bits in size.

Basic data types for C unsignedunsigned int- same as int, but unsigned. longlong intsigned longsigned long int- longsigned integer type, at least 32 bits in size. unsigned longunsigned long int- same as long, but unsigned. long longlong long intsigned long longsigned long long int- long long signed integer type, at least 64 bits in size

Basic data types for C • unsigned long long int- same as long long, but unsigned. • float - single precision floating-point type. • double- double precision floating-point type, actual properties unspecified (except minimum limits), however on most systems this is the IEEE 754 double-precision binary floating-point format • long double - extended precision floating-point type, actual properties unspecified. • Boolean type • Structures: • structbirthday { char name[20]; int day; int month; int year; };

Basic data types for C • Array - array of N elements of type T • int cat[10]; // array of 10 elements, each of type int • intbob[]; // array of an unspecified number of 'int' elements. • int a[10][8]; // array of 10 elements, each of type 'array of 8 int elements' • float f[][32]; // array of unspecified number of 'array of 32 float elements’ • Pointer - char *square; long *circle; • Unions - union types are special structures which allow access to the same memory using different type descriptions

Data S tructures and Algorithms