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Algorithms

Explore key algorithms like Interval Scheduling, Bipartite Matching, Independent Set, and Polygon Area with discussions on choosing the best algorithm based on running time. Learn Big-Oh, Big-Omega, and Theta notations for performance analysis. Dive into common running times across linear, quadratic, cubic, and exponential complexities in algorithmic problem solving.

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Algorithms

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  1. Algorithms Algorithm: what is it ?

  2. Algorithms • Algorithm: what is it ? • Some representative problems : • Interval Scheduling

  3. Algorithms • Algorithm: what is it ? • Some representative problems : • Interval Scheduling • Bipartite Matching

  4. Algorithms • Algorithm: what is it ? • Some representative problems : • Interval Scheduling • Bipartite Matching • Independent Set

  5. Algorithms • Algorithm: what is it ? • Some representative problems : • Interval Scheduling • Bipartite Matching • Independent Set • Area of a Polygon

  6. Algorithms How to decide which algorithm is better ? Search problem Input : a sequence of n numbers (in an array A) and a number x Output : YES, if A contains x, NO otherwise

  7. Algorithms How to decide which algorithm is better ? Search problem Input : a sequence of n numbers (in an array A) and a number x Output : YES, if A contains x, NO otherwise What if A is already sorted ?

  8. Running Time O(n) – running time of the linear search O(log n) – running time of the binary search Def : Big-Oh (asymptotic upper bound) f(n) = O(g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) · c g(n) Examples : n, n3, log n, 2n, 7n2 + n3/3, 1, 1 + log n, n log n, n + log n

  9. Running Time Def : Big-Oh (asymptotic upper bound) f(n) = O(g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) · c g(n) Example: Prove that n = O(n3)

  10. Running Time Def : Big-Oh (asymptotic upper bound) f(n) = O(g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) · c g(n) Example: Prove that n3 = O(7n2+n3/3)

  11. Running Time Def : Big-Oh (asymptotic upper bound) f(n) = O(g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) · c g(n) Example: Prove that n3 = O(n3/3-7n2)

  12. Running Time Def : Big-Oh (asymptotic upper bound) f(n) = O(g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) · c g(n) Example: Prove that log10 n = O(log n) And that log n = O(log10 n)

  13. Running Time Def : Big-Oh (asymptotic upper bound) f(n) = O(g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) · c g(n) Example: what about 3n and 2n

  14. Running Time O(n) – running time of the linear search O(log n) – running time of the binary search Def : Big-Omega (asymptotic lower bound) f(n) = (g(n)) if there exists a constant c > 0 and a constant n0 such that for every n ¸ n0 we have f(n) ¸ c g(n) Examples : n, n3, log n, 2n, 7n2 + n3/3, 1, 1 + log n, n log n, n + log n

  15. Running Time O(n) – running time of the linear search O(log n) – running time of the binary search Def : Theta (asymptotically tight bound) f(n) = (g(n)) if there exists constants c1, c2 > 0 and a constant n0 such that for every n ¸ n0 we have c1 g(n) · f(n) · c2 g(n) Examples : n, n3, log n, 2n, 7n2 + n3/3, 1, 1 + log n, n log n, n + log n

  16. A survey of common running times Linear • for i=1 to n do • something Also linear : • for i=1 to n do • something • for i=1 to n do • something else

  17. A survey of common running times Example (linear time): Given is a point A=(ax, ay) and n points (x1,y1), (x2,y2), …, (xn,yn) specifying a polygon. Decide if A lies inside or outside the polygon.

  18. A survey of common running times Example (linear time): Given are n points (x1,y1), (x2,y2), …, (xn,yn) specifying a polygon. Compute the area of the polygon.

  19. A survey of common running times O(n log n) Or: • for i=1 to n do • for j=1 to log(n) do • something • for i=1 to n do • j=n • while j>1 do • something • j = j/2

  20. A survey of common running times Quadratic • for i=1 to n do • for j=1 to n do • something

  21. A survey of common running times Example (CONVEX HULL): Given are n points (x1,y1), (x2,y2), …, (xn,yn) in the plane. Find their convex hull, i.e. smallest convex polygon containing all points.

  22. A survey of common running times Example (CONVEX HULL): Given are n points (x1,y1), (x2,y2), …, (xn,yn) in the plane. Find their convex hull, i.e. smallest convex polygon containing all points.

  23. A survey of common running times Cubic

  24. A survey of common running times O(nk) – polynomial (if k is a constant)

  25. A survey of common running times Exponential, e.g., O(2k)

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