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Algorithms

Algorithms. Al-Khwarizmi , arab mathematician, 8 th century Wrote a book: al-kitab… from which the word Algebra comes Oldest algorithm : Euclidian algorithm for finding the largest common divisor of two numbers. Ten Big Ideas in Algorithms. Orders of Magnitude and Big Oh notation

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Algorithms

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  1. Algorithms Al-Khwarizmi, arab mathematician, 8th century Wrote a book: al-kitab… from which the word Algebra comes Oldest algorithm: Euclidian algorithm for finding the largest common divisor of two numbers

  2. Ten Big Ideas in Algorithms • Orders of Magnitude and Big Oh notation • Complexity: Polynomial versus Exponential • Iteration versus Recursion • Recurrence Relations • Divide-and-Conquer • Graph Traversal: Depth-First, Breadth-First • Greedy Algorithm • Dynamic Programming • Lower Bounds • NP-completeness

  3. 1. Orders of Magnitude • How to compare algorithm A with algorithm B in an implementation-independent manner? • Asymptotic analysis: compare them when n goes to infinity (becomes very large) • [An aside: massive data sets; the Internet and search engines; Bioinformatics] • a(n)=running time of alg A, b(n)=of alg B • A is better than B if lim a(n)/b(n) = 0 (b “grows faster” than b) • Examples in Mathematica

  4. Big Oh Notation • Too many functions, not all components relevant • Big Oh helps keep the big picture. • Example: a(n) = 3 n2 + 1 and b(n) = 5 n2 + 3 n + 100 grow roughly at the same rate • lim a(n) / b(n) = constant • a(n) = O(n2), b(n) = O(n2)

  5. Complexity: Polynomial versus Exponential • Polynomial: order growth of O(n), O(n2), O(n3), etc. is tractable. • Exponential: order growth of O(2n), O(nn) or larger is intractable and leads to impractical algorithms

  6. 3. Iteration versus Recursion • Find Min of n elements: iterate over the array, at each step do a comparison. Time = n • Sort n elements by Selection Sort: iterate over array, at each iteration Find Min, put in right position Time = n + (n-1) + (n-2) + … + 3 + 2 + 1 • Find closed form for summation: Time = n(n+1)/2= O(n2)

  7. Recursion • Binary Search: split the list in two, search in the half where it should be. • Merge Sort: divide the array in two equal parts; sort each part recursively; merge them. • Quick Sort: use first element to partition array in two, first being smaller than second; sort each part recursively.

  8. 4. Recurrence Relations • Time for Merge Sort: F(n) = 2 F(n/2) + n • Solve recurrence relation: F(n) = n log n

  9. 5. Divide and Conquer • Paradigm for algorithm design: split in two or more parts, solve recursively, combine solutions • Binary Search, Merge Sort and Quick Sort illustrate it. • Leads to improved algorithms for many problems: • Median finding • Matrix Multiplication • Computational Geometry algorithms (convex hulls, etc.) • Leads to divide and conquer recurrence relations.

  10. 6.Graph Traversal: Depth-First and Breadth-First • Techniques for designing very efficient (linear time) algorithms • Apply to a variety of graph problems: • Connectivity • Finding cycles in graphs • Strongly connected components in directed graphs • Biconnected components in graphs • Planarity testing

  11. 7. Greedy Algorithm • Proceed iteratively. At each step, choose an element which maximizes some “profit” or minimizes some “cost”. • Minimum spanning tree in a graph (min cost connecting network): at each step, pick up the least expensive edge that does not create a cycle in the graph. • Shortest paths in a graph from given vertex a: at each step, choose a new vertex which is closest to an already constructed set of vertices.

  12. 8. Dynamic Programming • A scheme for combining several results for smaller values to obtain the current result. • Often leads to polynomial time algorithms • All pairs shortest paths: compute the shortest path between all pairs of nodes in a graph.

  13. 9. Lower Bounds • When do you stop improving an algorithm? How to prove that what you found is the best possible algorithm? • Lower bounds: techniques for proving an algorithm is the best possible. • Work under certain assumptions about the model of computation • Classical bounds: finding the minimum, sorting. • Information Theoretical Lower Bound • Adversary arguments

  14. 10.NP-Completeness • Despite all efforts, hundreds of very practical problems have no efficient (polynomial time) known algorithm (doesn’t mean that one may not be found) • Nobody has been able to prove that no polynomial time algorithm exists for any of these problems. • But one can show that if a polynomial time algorithm exists for one of these problems, then it exists for all of them. • A problem is NP-complete if it is as hard (NP-hard) as any of the problems in this class, and if a solution to the problem can be verified in polynomial time (it is in NP)

  15. Some NP-complete Problems • Traveling Salesman Problem • Hamiltonian Cycle in a graph • Maximum Clique in a graph • Satisfiability for boolean formulas and circuits • The game of Push-Push 

  16. Overview of the course • We cover the 10 big ideas • Lots of examples of algorithms • Start with some algebra and calculus • Need data structures and recursion • Use Leda software for demos (all algorithms are implemented and visualized) • Abstract, high level class • Need to be comfortable with abstract thinking • A little bit of programming – mostly theory

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