ADS 1 Algorithms and Data Structures 1 Syllabus

Asymptotical notation(Binary trees,) AVL trees, Red-Black treesB-treesHashingGraph alg: searching, topological sorting, strongly connected components

Minimal spaning tree (d.s. Union-Find)Divide et Impera methodSorting: low approximation of sorting problem, average case of Quicksort, randomization of Quicksort, linear sorting alg.

Algebraic alg. (LUP decomposition)

LiteratureT.H. Cormen, Ch.E. Leiserson, R.L. Rivest,

Introduction to Algorithms, MIT Press, 1991

Organization Lecture Seminar

Comparing of algorithms

Measures: Time complexity, in (elementary) steps Space complexity, in words/cells Communication complexity, in packets/bytes

How it is measured: Worst case, average case (wrt probability distribution) Usually approximation: upper bound

Using functions depending on size of input data We abstract from particular data to data size We compare functions

Size of data

Q: How to measure the size of (input) data? Formally: number of bits of data Ex: Input are (natural) numbers ,

size D of input data is Time complexity: a function f: N->N, such that f(|

D|) gives number of algorithm steps depending on data of the size |D|

Intuitively: Asymptotical behaviour: exact graph of function f does not matter (ignoring additive and multiplicative constants), a class of f matters (linear, quadratic, exponential)

Step of algorithm

In theory: Based on a abstract machine: Random Access Machine (RAM), Turing m. Informally: algorithm step = operation executable in

constant time (independent of data size) RAM, operations:

Arithmetical: +, -, *, mod, <<, && … Comparision of two numbers Assignment of basic data types (not for arrays)

Numbers have fixed maximal size Ex: sorting of n numbers: |D| = n (Contra)Ex: test for zero vector

Why to measure time comlexity Why sometimes more quickly machine doesn't help Difference between polynomial and worse algs.

Asymptotical complexity

Measures a behaviour of the algorithm on „big“ data It ignores a finite number of exceptions

It supress additive and multiplicative constants Abstracts from processor, language, (Moore law)

Classifies algs to categories: linear, quadratic, logarithmic, exponential, constant ... Compares functions

(Big) O notation, definitions

f(n) is asymptotically less or equal g(n), notation iff f(n) is asymptotically greater or equal g(n),notation , „big Omega“iff f(n) is asymptotically equal g(n), notation , „big Theta“iff

O-notation, def. II

f(n) is asymptotically strictly less than g(n),notation , „small o“iff f(n) is asymptotically strictly greater than g(n),notation , „small omega“iff Examples of classes: O(1), log log n, log n, n, n log n, n^2, n^3, 2^n, 2^2n, n!, n^n, 2^(2^n),... Some functions are incomparable

Exercise

Notation: sometimes is used f=O(g) Prove: max(f,g) ɛ Θ(f+g) Prove: if c,d>0, g(n)=c.f(n)+d then g ɛ O(f) Ex.: if f ɛ O(h), g ɛ O(h) then f+g ɛ O(h)

Appl: A bound to sequence of commands Compare n+100 to n^2, 2^10 n to n^2

Simple algorithms (with low overhead) are sometimes better for small data

Dynamic sets

Data structures for remembering some data Dynamic structure: changes in time An element of a dynamic d.s. is accesible

through a pointer and has1. A key, usually from some (lineary) ordered set2. Pointer(s) to other elements, or parts of d.s.3. Other (user) data (!)

Operations S is a dynamic set, k is a key, x is a pointer to a

element Operations

Find(S,k) – returns a pointer to a element with the key k or NIL

Insert(S,x) – Inserts to S an element x Delete(S, x) – Deletes from S an element x Min(S) – returns a pointer to an element with

minimal key in S Succ(S,x) – returms a pointer to an element next to

x (wrt. linear ordering) Max(S), Predec(S,x) – analogy to Min, Succ

Binary search trees

Dynamic d.s. which supports all operations