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  1. Discrete Math and Its Application to Computer Science UBİ 501 Lecture - 3 İlker Kocabaş E.Ü Uluslararası Bilgisayar Enstitüsü Bornova - İzmir

  2. Flow ALGORITHMS Introduction Algorithmic Complexity Growing Functions NUMBER THEORY Modular Arithmetic Primary Numbers Greatest Common Divisor (gcd) & Least Common Multipier (lcd) Ecludian Algorithm for gcd Number Systems: Decimal, Binary, Octal, ….

  3. AlgorithmsIntroduction (1) • Algorithm: • A finite set of precise instructions for performing a computation or for solving a problem. • Synonyms for a algorithm are: program, recipe, procedure, and many others. • Pseudocode (T: Sözde Kod) • Describing an algorithm by using a specific computure language: Complex instructions and difficult to understand. • Intemadiate step between Natural Language & Programming Language

  4. Algorithms (1)Pseudocode Example • Algorithm-1: Finding the maximum element in a finite sequence INPUT procedure max(a1,a2,a3….an: integers) max := a1 fori:=0 ton ifmax < aithenmax:= ai outputmax DIFINITENESS OUTPUT

  5. AlgorithmsBasic Problems in CS • Searching (T: Arama) Algorithms • Finding element ai equals to x • Linear Search, Binary Search, … • Algorithm 2: Linear Search Algorithm procedure max(x: integer, a1,a2,a3….an: distinct integers) i:=1 while (i ≤ n and x ≠ ai) i := i + 1 if i ≤ n then location := i else location:= 0 outputlocation

  6. Algorithms (1)Basic Problems in CS • Linear Search Example • Find x = 5 while (i ≤ n and x ≠ ai) i := i + 1 if i ≤ n then location := i else location:= 0

  7. Algorithms (1)Basic Problems in CS • Sorting (Sıralama) Algorithms • Sort a sequence A for a given order criteria • Buble Sort, Insertion Sort, …..

  8. Algorithms (1)Basic Problems in CS • Merging (T: Birleştirme) Algorithms • Merge ordered sequences A & B

  9. AlgorithmsAlgorithmic Complexity (2) • How can the efficiency of an algorithm be analyzed? • Time: “Time used by a computer” to solve a problem • Space: “The amount of Computer memory” required to implement algorithm

  10. Algorithms (2)Running Time • Running time: • Measure the actual time spent by implementation of algorithm. • Deficiencies: • Actual running time changes paltform to platform (1Ghz ≠ 2 Ghz) • There is no information wrt varying n (input size) and input order. • Count the basic operations or steps processed by algorithm

  11. Algorithms (2)Running Time • Running time: • Count the basic operations or steps executed by algorithm • Comparision (T: karşılaştırma) [ Eg. X < Y ] • Assignment (T: Atama) [ Eg. X = 5 ] • Increment/Decriment [ Eg. X = X  1 ] • Function Output [ Eg. return/output X ] • Addition/Substruction/Multiplication/Division • ………..

  12. Algorithms (2)Running Time • Count the basic operations or steps processed by algorithm • Best Case Analysis: Minimum number of operations executed wrt input behaviour of a given size. • Average Case Analysis: Average number of operations used to solve the problem over all inputs of a given size. • Worst Case Analysis: Maximum number of operationsnumbers of steps executed wrt input behaviour of a given size.

  13. Algorithms (2)Algorithm 3: Surjectivity procedureisOnto( f [(1, 2,…, n)  (1, 2,…, m)]: function) • if( m > n ) 1 step comp. • return false 1 step End if exec. • soFarIsOnto:= true 1 step ass. • forj:= 1 to mm loops: 1 step comp. +1 step increment • soFarIsOnto:= false1 step ass. • fori := 1 to n n loops: 2 steps comp. + inc. • if ( f(i) = j ) 1 step comp. • soFarIsOnto:= true1 step ass. • if( !soFarIsOnto ) 1 step negation • return false 1 step End • return true;1 step End

  14. Algorithms (2)Algorithm 3: Surjectivity • Best Case Analysis: 1 operation • if( m > n ) 1 step comp. • return false 1 step End if exec. • Worst Case Analysis:2+ m(5n+3) = 5mn +3m+2 • if( m > n ) 1 • return false 1 step End if exec. • soFarIsOnto:= true1 • forj:= 1 to m n : [ 1 +1 • soFarIsOnto:= false 1 • fori := 1 to n n: (1 + 1 • if ( f(i) = j ) 1 • soFarIsOnto:= true1 • if( !soFarIsOnto ) 1 )] • return false 1 step End • return true;1 step End

  15. Algorithm (2)Comparing Running Times • At most 5mn+3m+2 for first algorithm • At most 5m+2n+2 for second algorithm Worst case when m n so replace m by n: 5n 2+3n+2 vs. 8n+2 To tell which is better, look at dominant term: 5n2+3n+2 vs. 8n+2 So second algorithm is better.

  16. Running Times IssuesBig-O Response Asymptotic notation (Big-O, Big- , Big-) gives partial resolution to problems: • For large n the largest term dominates so 5n 2+3n+2 is modeled by just n 2.

  17. Running Times IssuesBig-O Response Asymptotic notation (Big-O, Big- , Big-) gives partial resolution to problems: • Different lengths of basic steps, just change 5n 2 to Cn 2 for some constant, so doesn’t change largest term

  18. Running Times IssuesBig-O Response Asymptotic notation (Big-O, Big- , Big-) gives partial resolution to problems: • Basic operations on different (but well-designed) platforms will differ by a constant factor. Again, changes 5n 2 to Cn2 for some constant.

  19. Running Times IssuesBig-O Response Asymptotic notation (Big-O, Big- , Big-) gives partial resolution to problems: • Even if overestimated by assuming iterations of while-loops that never occurred, may still be able to show that overestimate only represents different constant multiple of largest term.

  20. Big-O, Big-, Big- • Useful for computing algorithmic complexity, i.e. the amount of time that it takes for computer program to run.

  21. Notational Issues Big-O notation is a way of comparing functions. Notation unconventional: EG: 3x 3 + 5x 2 – 9 = O (x 3) Doesn’t mean “3x 3 + 5x 2 – 9 equals the function O (x 3)” Which actually means “3x 3+5x 2 –9 is dominated by x 3” Read as: “3x 3+5x 2 –9 is big-Oh of x 3”

  22. Intuitive Notion of Big-O Asymptotic notation captures behavior of functions for large values of x. EG: Dominant term of 3x 3+5x 2 –9 is x 3. As x becomes larger and larger, other terms become insignificant and only x 3 remains in the picture:

  23. Intuitive Notion of Big-Odomain – [0,2] y = 3x 3+5x 2 –9 y = x 3 y = x 2 y = x

  24. Intuitive Notion of Big-Odomain – [0,5] y = 3x 3+5x 2 –9 y = x 3 y = x 2 y = x

  25. Intuitive Notion of Big-Odomain – [0,10] y = 3x 3+5x 2 –9 y = x 3 y = x 2 y = x

  26. Intuitive Notion of Big-Odomain – [0,100] y = 3x 3+5x 2 –9 y = x 3 y = x 2 y = x

  27. Intuitive Notion of Big-O In fact, 3x 3+5x 2 –9 is smaller than 5x 3 for large enough values of x: y = 5x 3 y = 3x 3+5x 2 –9 y = x 2 y = x

  28. Big-O. Formal Definition f (x ) is asymptotically dominated by g (x ) if there’s a constant multiple of g (x ) bigger than f (x ) as x goes to infinity: DEF: Let f , g be functions with domain R0 or N and codomain R. If there are constants C and k such  x > k,|f (x )|  C  |g (x )| then we write: f (x ) = O ( g (x ) )

  29. Common Misunderstanding It’s true that 3x 3 + 5x 2 – 9 = O (x 3) as we’ll prove shortly. However, also true are: • 3x 3 + 5x 2 – 9 = O (x 4) • x 3 = O (3x 3 + 5x 2 – 9) • sin(x)= O (x 4) NOTE: C.S. usage of big-O typically involves mentioning only the most dominant term. “The running time is O (x 2.5)” Mathematically big-O is more subtle.

  30. Big-O. Example EG: Show that 3x 3 + 5x 2 – 9 = O (x 3). Previous graphs show C = 5 good guess. Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k

  31. EG: Show that3x 3 + 5x 2 – 9 = O (x 3). Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k • Collect terms: 5x 2 ≤ 2x 3 + 9

  32. EG: Show that3x 3 + 5x 2 – 9 = O (x 3). Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k • Collect terms: 5x 2 ≤ 2x 3 + 9 • What k will make 5x 2 ≤ x 3 for x > k ?

  33. EG: Show that3x 3 + 5x 2 – 9 = O (x 3). Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k • Collect terms: 5x 2 ≤ 2x 3 + 9 • What k will make 5x 2 ≤ x 3 for x > k ? • k = 5 !

  34. EG: Show that3x 3 + 5x 2 – 9 = O (x 3). Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k • Collect terms: 5x 2 ≤ 2x 3 + 9 • What k will make 5x 2 ≤ x 3 for x > k ? • k = 5 ! • So for x > 5, 5x 2 ≤ x 3 ≤ 2x 3 + 9

  35. EG: Show that3x 3 + 5x 2 – 9 = O (x 3). Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k • Collect terms: 5x 2 ≤ 2x 3 + 9 • What k will make 5x 2 ≤ x 3 for x > k ? • k = 5 ! • So for x > 5, 5x 2 ≤ x 3 ≤ 2x 3 + 9 • Solution: C = 5, k = 5 (not unique!)

  36. EG: Show that3x 3 + 5x 2 – 9 = O (x 3). Find k so that 3x 3 + 5x 2 – 9  5x 3 for x > k • Collect terms: 5x 2 ≤ 2x 3 + 9 • What k will make 5x 2 ≤ x 3 for x > k ? • k = 5 ! • So for x > 5, 5x 2 ≤ x 3 ≤ 2x 3 + 9 • Solution: C = 5, k = 5 (not unique!)

  37. Big-O. Negative Example x 4O (3x 3 + 5x 2 – 9) : No pair C, k exist for which x > k implies C (3x3 + 5x 2 – 9)x 4 Argue using limits: x 4 always catches up regardless of C. 

  38. Big-O and limits LEMMA: If the limit as x   of the quotient |f (x) / g (x)| exists then f (x ) = O ( g (x ) ). EG: 3x 3 + 5x 2 – 9 = O (x 3 ). Compute: …so big-O relationship proved.

  39. Little-o and limits DEF: If the limit as x   of the quotient |f (x) / g (x)| = 0 then f (x ) = o (g (x ) ). EG: 3x 3 + 5x 2 – 9 = o (x 3.1 ). Compute:

  40. Big- and Big- Big-: reverse of big-O. I.e. f (x ) = (g (x ))  g (x ) = O (f (x )) so f (x ) asymptotically dominatesg (x ). Big-: domination in both directions. I.e. f (x ) = (g (x ))  f (x ) = O (g (x ))  f (x ) = (g (x )) Synonym for f = (g): “f is of orderg ”

  41. Useful facts • Any polynomial is big- of its largest term • EG: x 4/100000 + 3x 3 + 5x 2 – 9 =(x 4) • The sum of two functions is big-O of the biggest • EG: x 4 ln(x ) + x 5 = O (x 5) • Non-zero constants are irrelevant: • EG: 17x 4 ln(x ) = O (x 4 ln(x ))

  42. Big-O, Big-, Big-. Examples Q: Order the following from smallest to largest asymptotically. Group together all functions which are big- of each other:

  43. Big-O, Big-, Big-. Examples A: 1. 2. 3. , (change of base formula) 4. 5. 6. 7. 8. 9. 10.

  44. Incomparable Functions Given two functions f (x ) and g (x ) it is not always the case that one dominates the other so that f and g are asymptotically incomparable. E.G: f (x) = |x 2 sin(x)| vs. g (x) = 5x 1.5

  45. Incomparable Functions y = x2 y = |x 2 sin(x)| y = 5x1.5

  46. Incomparable Functions y = x2 y = 5x1.5 y = |x 2 sin(x)|

  47. Big-OA Grain of Salt Big-O notation gives a good first guess for deciding which algorithms are faster. In practice, the guess isn’t always correct. Consider time functions n 6 vs. 1000n 5.9. Asymptotically, the second is better. Often catch such examples of purported advances in theoretical computer science publications. The following graph shows the relative performance of the two algorithms:

  48. Big-OA Grain of Salt Running-time In days Assuming each operation takes a nano-second, so computer runs at 1 GHz T(n) = 1000n 5.9 T(n) = n 6 Input size n

  49. Big-OA Grain of Salt In fact, 1000n 5.9 only catches up to n 6 when 1000n 5.9 = n 6, i.e.: 1000= n 0.1,i.e.: n = 100010 = 1030 operations = 1030/109 = 1021 seconds  1021/(3x107)  3x1013 years  3x1013/(2x1010)  1500 universe lifetimes!

  50. AlgorithmsExtra-1 • The world of computation can be subdivided into three classes: • Tractable Problems • Polynomial worst-case complexity (P Class) • (nc), c ≥ 1 constant [Eg.Bubble Sort Algorithm is (n2)] • Intractable Problems (NP Class) • Exponential worst-case complexity (E Class) • (cn), c ≥ 1 constant [Eg.Satisfiability Algorithm is (2n)] • Factorial worst-case complexity (F Class) • (n!), [Eg.Traveling Salesman Algorithm is (n!)] • Unsolvable Problems • No algorithms exists for solving them • Halting Problem