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ANALYSIS OF ALGORITHMS

ANALYSIS OF ALGORITHMS. Data Structures. Algorithms. Definition A step by step procedure for performing some task in a finite amount of time Analysis Methodology for analyzing a “good” algorithm (i.e. Archimedes) Look at Running Time. How are algorithms described ? PSEUDO-CODE.

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ANALYSIS OF ALGORITHMS

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  1. ANALYSIS OF ALGORITHMS Data Structures

  2. Algorithms • DefinitionA step by step procedure for performing some task in a finite amount of time • Analysis Methodology for analyzing a “good” algorithm (i.e. Archimedes) • Look at Running Time

  3. How are algorithms described ? PSEUDO-CODE • representation of an algorithm • combines • natural language • familiar programming language structures • facilitates the high-level analysis of an algorithm

  4. Pseudo-Code Conventions • Expressions • Algorithm Structures • Control Structures

  5. Expressions • Standard math symbols + - * / ( ) • Relational operators • Boolean operators and or not • Assignment operator , = • Array indexing A[i]

  6. Algorithm Structure • Algorithm heading Algorithm name(param1, param2,...): Input : input elements Output : output elements • Statements call object.method(arguments) return statement return value control structures

  7. Control Structures • if ... then ... [else ...] • while ... do • repeat ... until ... • for ... do • decision structures • while loops • repeat loops • for loop

  8. General Rules • communicate high level ideas and not implementation details (programming language specifics) • clear and informative

  9. Example - Problem • Problem : Write pseudo-code for an algorithm that will determine the maximum element from a list (array)

  10. Example - Algorithm Algorithm arrayMax(A,n): Input: An array A storing n integers. Output: The maximum element in A. currentMax A[0] for i 1 to n - 1 do if currentMax < A[i] then currentMax A[i] return currentMax

  11. Analysis of Algorithms • Running time • depends on input size n • number of primitive operations performed • Primitive operation • unit of operation that can be identified in the pseudo-code

  12. Primitive Operations-Types • Assigning a value to a variable • Calling a method/procedure • Performing an arithmetic operation • Comparing two numbers • Returning from a method/procedure

  13. Primitive Operations - Some General Rules • for Loops • Nested for Loops • Conditional Statements

  14. for Loops The running time of a for loop is at most the running time of the statements inside the for loop (including tests) times the number of iterations. Example for i 0 to n - 1 do A[i] 0

  15. Nested for Loops Analyze these inside out. The total running time of a statement inside a group of nested for loops is the running time of the statement multiplied by the product of the sizes of all the for loops. Example for i 0 to n - 1 do for j 0 to n - 1 do k++

  16. Conditional Statements if (cond) then S1 else S2 The running time of an if-else statement is never more than the running time of the test plus the larger of the running times of S1 and S2.

  17. Primitive Operations - Example • Count the number of primitive operations in this program : i = 0 a = 0 for i = 1 to n do print i a=a+i return i

  18. Primitive Operations - Example • Estimate : 3n + 3 • Linear Time- (i.e. 6n + 2, 3n+5, n+1,etc.)

  19. Primitive Operations - Example 2 • Count the number of primitive operations in this program : i = 0 a = 0 for i = 1 to n do print i for a = 1 to n do print a return i

  20. Primitive Operations - Example2 • Estimate : n2 + n + 3 • Polynomial (Quadratic) Time

  21. Example 3 - Class Algorithm sum(int n): Input: Integer n Output: Sum of the cubes of 1 to n partial_sum 0 for i 1 to n do partial_sum partial_sum + i*i*i return (partial_sum)

  22. Primitive Operations -Considerations • Arbitrariness of measurement • Worst-case versus average-case (difficult to be exact : range) • Implicit operations • loop control actions • array access

  23. Analysis of Algorithms • take-out the effect of constant factors (i.e. 2n vs 3n : same level) • this can be attributed to differences in hardware and software environments • compare algorithms across types (linear vs polynomial vs exponential) • look at growth rates

  24. Asymptotic Notation • Goal: • To simplify analysis by getting rid of irrelevant information • The “Big-Oh” Notation • Given functions f(n) and g(n), f(n) is O (g(n)) if f(n) <= c g(n) for n>= n0 where c and n0 are constants

  25. Big-Oh Notation • Rule : Drop lower order terms and constant factors 5n + 2 is O (n) 4n3log n + 6n3 + 1 is O (n3logn)

  26. Big-Oh Notation - Relatives • Big Omega (reverse) • Big Theta (both ways, same level) • General Rule : Use the most descriptive notation to describe the algorithm

  27. Practical Significance of Big-Oh Notation • Example for i 0 to n-1 do A[i] 0 • Running time is 2n, if we count assignments & array accesses 3n, if we include implicit loop assignments 4n, if we include implicit loop comparisons • Regardless, running time is O (n)

  28. Algorithms - Common Classifications • Typical functions that classify algorithms constant O (1) logarithmic O (log n) linear O (n) quadratic O (n2) exponential O (an), n > 1

  29. Linear Search vs Binary Search • Linear Search O (n) • Binary Search O (log n) • Example : to search for a number from 1024 entries using the binary search algorithm will only take 10 steps

  30. O(log n) Algorithm

  31. O(log n) Algorithm Binary Search Algorithm bin_search(int a[], int x, int n) Input: Array a, search target x, array size n Output: Position of x in the array (if found)

  32. ` { low 0; high n - 1 while (low <= high) do mid (low + high)/2 if(a[mid] < x) then low mid + 1 else if (a[mid] > x) then high mid - 1 else return (mid) return (NOT_FOUND) }

  33. Prefix Average Problem • Given an array X storing n numbers, we want to compute an array A such that A[1] is the average of elements X[0] … X[I] for I = 0 … n-1. • algorithm A = O (n2) • algorithm B = O (n)

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