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Efficient Algorithms Lecture 1

Efficient Algorithms Lecture 1. Contents of today’s lecture: Introduction to the lecturers The contents of the course The process from Problem to Algorithm Algorithm Examples Introduction to efficency of algorithms Lecture: 2 x 35 minutes, followed by exercises. Lecturers.

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Efficient Algorithms Lecture 1

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  1. Efficient AlgorithmsLecture 1 • Contents of today’s lecture: • Introduction to the lecturers • The contents of the course • The process from Problem to Algorithm • Algorithm Examples • Introduction to efficency of algorithms • Lecture: 2 x 35 minutes, followed by exercises

  2. Lecturers • Assistant Professor Jens Myrup Pedersen • NetSec, Aalborg University • jens@es.aau.dk • Associate Professor Volker Krüger • Media, Aalborg University Copenhagen • vok@cvmi.aau.dk

  3. Course contents • Introduction to Analysis and Design of Algorithms • Recursive Algorithms and Recurrences • Greedy algorithms • Backtracking • Counting, Probabilities • Randomized algorithms • Sorting algorithms: Heap sort, quick sort • Hashing, hash tables • Binary search trees, red-black trees • Graph and graph algorithms • Examples from security and network planning points of view

  4. Litterature etc. • Homepage: www.control.aau.dk/~jens/teaching/EA_2009 • Updated info on lecture contents and proposals for litterature • Lecture notes, ppt’s, additional material • Book: • Cormen et al., Introduction to Algorithms, 2nd edition, MIT Press, 2003. • We suggest that you also look for additional litterature yourself.

  5. From problem to algorithm • What is an algorithm – how would you define it? • What is required from an algorithm? • Correctness – Efficiency • Consider the whole problem to be solved – usually a combination of algorithms and data structures are required for efficient solutions. • Examples of algorithms (blackboard) • Integer sorting (Insertion sort, Merge sort) • Travelling salesman • Towers of Hanoi

  6. Insertion sort

  7. Merge sort

  8. Analysis of an algorithm • Example: Insertion sort. • Two important properties: Correctness and Efficiency.

  9. Correctness • To verify correctness we need to get hold on loop invariants.... • But first: How to verify intuitively that it holds?

  10. Loop invarianter • Similar to induction – as you probably know from the Math lectures. • Loop invariant: Describes the properties, for example of a variable, an array, etc. Loop invariants must fulfill 3 properties: • Initialisation: It must be true before the first iteration of the loop. • Maintenance: If it is true before an iteration – it is also true afterwards. • Termination: When the loop terminates, the invariant must have a useful properti that helps showing the correctness of the algorithm. • Example from previous slide: Insertion sort.

  11. Efficiency of an algorithm • How can we use these calculations – and what is the price of a ”step”?

  12. A few words on efficiency • We want efficient algorithms - BUT • We also want algorithms that are easy to understand and maintain. • We also want algorithms, which can be reused (for example in other programs). • There is an important trade-off between development times and efficiency. • Conclusion: Algorithm-design is always dependend on how the algorithm is to be used.

  13. Analysis of efficiency • Ultimate goal: Determining calculation time for the algorithms design (well, that is if we don’t look at usage of other ressources such as memory, network, other kinds of I/O). • Challenge: Even in this case the calculation time depends on factors such as hardware, OS, set of instructions – and inputs for the algorithm. • In any case we would like to make an estimate of runtimes. • Usual method: Analyse the worst-case complexity using Big-O notation (asymptotic notation). • This is often fine, but be aware that in some situations best-case or average-case analysis may be more informative.

  14. Asymptotic notation • Big O-notation, the usual, fast and often fine method. • Variants: , O,  - just a short explanation (next slide). The book is more formal. • Mini-exercise: How does Insertion sort behave? • In the afternoon we go into more details with different kinds of algorithms, such as divide-and-conquer and merge sort.

  15. Asymptotic notation (blackboard) • Big O-notation: Can be considered an upper limit for calculation time. • Variant: Little o-notation (not asymptotically tight). • Big Theta () – There exists both c1 and c2 which (given n0) makes upper and lower bounds for growth. • Big Omega () – Only lower boundary.

  16. Problems • From the book: • 1.1-1, 1.1-2, 1.1-3, 1.1-4, 1.1-5, 1.2-2, 1.2-3, 3.1-2, 3.1-3. • Project related: • Identify and list the most important algorithms in your project (as well as you can at this stage) • For each of these algorithms, discuss if they are critical with respect to design - and what are the critical factors? (e.g. time, correctness, computational power, memory/storage). • Select one or two of the algorithms, and try to make an estimate of complexity using Big O-notation. • Discuss what data structures could be used..

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