ADS 1 Algorithms and Data Structures 1 Syllabus Asymptotical notation (Binary trees,) AVL trees, Red-Black trees B-trees Hashing Graph alg: searching, topological sorting, strongly connected components Minimal spaning tree (d.s. Union-Find) Divide et Impera method Sorting: low approximation of sorting problem, average case of Quicksort, randomization of Quicksort, linear sorting alg. Algebraic alg. (LUP decomposition)

Literature T.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 has 1. A key, usually from some (lineary) ordered set 2. 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