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Discrete Mathematics CS 2610

Discrete Mathematics CS 2610. September 24, 2008. Greedy Algorithms. Greedy algorithms make the “best” choice at each step of the algorithm. Greedy algorithms do not “backtrack,” or reconsider previous decisions. Example: Making change in the fewest number of coins (US currency).

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Discrete Mathematics CS 2610

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  1. Discrete Mathematics CS 2610 September 24, 2008

  2. Greedy Algorithms • Greedy algorithms make the “best” choice at each step of the algorithm. • Greedy algorithms do not “backtrack,” or reconsider previous decisions. • Example: Making change in the fewest number of coins (US currency). • Greedy algorithm: Starting with largest coins, use as many as possible before moving to next largest denomination.

  3. Greedy Algorithms • Problem: Assign meeting to conference rooms • Policy: In decreasing order of room capacity, assign a meeting to the next largest available room Meeting 1: 70 Meeting 2: 46 Meeting 3: 125 Meeting 4: 110 Meeting 5: 30 Meeting 6: 87 Room A 200 Room B 150 Room C 150 Room D 100 Room E 75 Room F 50 M1, OK M2, OK M3, OK X M5, OK X This is a suboptimal algorithm!

  4. Greedy Algorithms • Problem: Assign meeting to conference rooms • Policy: In ascending order of room capacity, assign the largest remaining meeting that can be held in the room. Meeting 1: 70 Meeting 2: 46 Meeting 3: 125 Meeting 4: 110 Meeting 5: 30 Meeting 6: 87 Room F 50 Room E 75 Room D 100 Room B 150 Room C 150 Room A 200 M2, OK M1, OK M6, OK M3, OK M4, OK M5, OK Just one of many algorithms.

  5. Order of Growth Terminology O(1) Constant O(log n) Logarithmic (c  v) O(logc n) Polylogarithmic (c  Z+) O(n) Linear O(nc) Polynomial (c  Z+) O(cn) Exponential (c  Z+) O(n!) Factorial Best Worst

  6. Complexity of Problems • Tractable: A problem that can be solved with a deterministic polynomial (or better) worst-case time complexity. • Also denoted as P • ExampleS: • Search Problem • Sorting problem • Find the maximum

  7. Complexity of Problems • Intractable: Problems that are not tractable. • Example: Traveling salesperson problem 􀂄 • Wide use of greedy algorithms to get an approximate solution. • For example under certain circumstances, you can get an approximation that is at most double the optimal solution.

  8. P vs. NP • NP: Solvable problems whose solution can be checkedin polynomial time. • P  NP • The most famous unproven conjecture in computer science is that this inclusion is proper. • 􀂄 P  NP rather than P = NP

  9. Complexity of Problems • Not Solvable: Proven to have no algorithm that computes it • Example: Halting problem (Alan Turing) Determine whether an arbitrary given algorithm, will eventually halt for any given finite input. • Corollary: The question of whether or not a program halts for a given input is unsolvable.

  10. Big-O Notation • Big-O notation is used to express the time complexity of an algorithm. • It is a statement about an algorithm, not a machine! 􀂄 • We can assume that any operation requires the same amount of time. • The time complexity of an algorithm can be described independently of the software and hardware used to implement the algorithm.

  11. Big-O Notation • Def.: Let f , g be functions with domain R≥0 or N and codomain R. f(x) is O(g(x)) if there are constants C and k s.t. x > k, |f (x)| ≤ C ⋅ |g (x)| • We say f (x) is asymptotically dominated by g (x) • C|g(x)| is an upper bound of f(x). • C and k are called witnesses to the relationship between f & g.

  12. Big-O Notation Note that |C ⋅ g(x)| does not need to be greater than |f(x)| for all x, only for all x > k for some k.

  13. Big-O Notation • To prove that a function f(x) is O(g(x)) • Find values for k and C- not necessarily the smallest one, larger values also work!! • It is sufficient to find a certain k and C that works • In many cases, for all x ≥ 0, if f(x) ≥ 0 then |f(x)| = f(x) Example: f(x) = x2 + 2x + 1 is O(x2) for C = 4 and k = 1

  14. Big-O Notation Show that f(x) = x2 + 2x + 1 is O(x2). When x > 1 we know that x ≤ x2 and 1 ≤ x2 then 0 ≤ x2 + 2x + 1 ≤ x2 + 2 x2 + x2 = 4 x2 so, let C = 4 and k = 1 as witnesses, i.e., f(x) = x2 + 2x + 1 < 4 x2 when x > 1 Could try x > 2. Then we have 2x ≤ x2 & 1 ≤ x2 then 0 ≤ x2 + 2x + 1 ≤ x2 + x2 + x2 = 3 x2 so, C = 3 and k = 2 are also witnesses to f(x) being O(x2). Note that f(x) is also O(x3), etc.

  15. Big-O Notation Show that f(x) = 7 x2 is O(x3). When x > 7 we know that 7 x2 < x3 (multiply x > 7 by x2) so, let C = 1 and k = 7 as witnesses. Could try x > 1. Then we have 7 x2 < 7 x3 so, C = 7 and k = 1 are also witnesses to f(x) being O(x3). Note that f(x) is also O(x4), etc.

  16. Big-O Notation Show that f(n) = n2 is not O(n). Show that no pair of C and k exists such that n2 ≤ Cn whenever n > k. When n > 0, divide both sides of n2 ≤ Cn by n to get n ≤ C. No matter what C and k are, n ≤ C will not hold for all n with n > k! (There is no bound on the input.)

  17. Big-O Notation • Observe that g(x) = x2 is O(x2 + 2x + 1) • Def: Two functions f(x) and g(x) have the same order iff g(x) is O(f(x)) and f(x) is O(g(x))

  18. Big-O Notation • Also, the function f(x) = 3 x2 + 2x + 3 is O(x3) • What about O(x4) ? • In fact, the function Cg(x) is an upper bound for f(x), but not necessarily the tightest bound. • When Big-O notation is used, g(x) is usually chosen to be as small as possible. This way it carries the most information possible.

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