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CS 3610/5610N data structures Lecture: complexity analysis

CS 3610/5610N data structures Lecture: complexity analysis. Time matters. To understand a data structure. FIGURE 1-1 Gift shop and each dot representing a house. Algorithm Analysis. Example 50 packages delivered to 50 different houses 50 houses one mile apart, in the same area.

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CS 3610/5610N data structures Lecture: complexity analysis

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  1. CS 3610/5610N data structuresLecture: complexity analysis Data Structures Using C++ 2E

  2. Time matters Data Structures Using C++ 2E

  3. To understand a data structure Data Structures Using C++ 2E

  4. FIGURE 1-1 Gift shop and each dot representing a house Algorithm Analysis • Example • 50 packages delivered to 50 different houses • 50 houses one mile apart, in the same area Data Structures Using C++ 2E

  5. FIGURE 1-2 Package delivering scheme Algorithm Analysis (cont’d.) • Example (cont’d.) • Driver picks up all 50 packages • Drives one mile to first house, delivers first package • Drives another mile, delivers second package • Drives another mile, delivers third package, and so on • Distance driven to deliver packages • 1+1+1+… +1 = 50 miles • Total distance traveled: 50 + 50 = 100 miles Data Structures Using C++ 2E

  6. FIGURE 1-3 Another package delivery scheme Algorithm Analysis (cont’d.) • Example (cont’d.) • Similar route to deliver another set of 50 packages • Driver picks up first package, drives one mile to the first house, delivers package, returns to the shop • Driver picks up second package, drives two miles, delivers second package, returns to the shop • Total distance traveled • 2 * (1+2+3+…+50) = 2550 miles Data Structures Using C++ 2E

  7. Algorithm Analysis (cont’d.) • Example (cont’d.) • n packages to deliver to n houses, each one mile apart • First scheme: total distance traveled • 1+1+1+… +n = 2n miles • Function of n • Second scheme: total distance traveled • 2 * (1+2+3+…+n) = 2*(n(n+1) / 2) = n2+n • Function of n2 Data Structures Using C++ 2E

  8. TABLE 1-1 Various values of n, 2n, n2, and n2 + n Algorithm Analysis (cont’d.) • Analyzing an algorithm • Count number of operations performed • Not affected by computer speed Data Structures Using C++ 2E

  9. Algorithm Analysis (cont’d.) • Example 1-1 • Illustrates fixed number of executed operations Data Structures Using C++ 2E

  10. Analysis of Example 1-1 • Line 1 has one operation, <<; • Line 2 has two operations; • Line 3 has one operation, >=; • Line 4 has one operation, =; • Line 6 has one operation; and • Line 7 has three operations. • Either Line 4 or Line 6 executes. • Therefore, the total number of operations executed in the preceding code is 1 + 2 + 1 + 1 + 3 = 8. In this algorithm, the number of operations executed is fixed. Data Structures Using C++ 2E

  11. Algorithm Analysis (cont’d.) • Example 1-2 • Illustrates dominant operations Data Structures Using C++ 2E

  12. Analysis of Example 1-2 • Before the while loop: 5 operations (lines 1 – 4) • After the while loop: 9 or 8 operations (depends on whether line 11 or line 13 executes) • The while loop: • 1 operation (line 5) • 4 operations (lines 6-8) • Note: the loop head (line 5) will be executed one more time than the body to terminate the loop. • Therefore, if the loop runs k times, the total number of operations is 5 + (k*5 + 1) + 9 or 5 + (k*5 + 1) + 8.

  13. Algorithm Analysis (cont’d.) • Search algorithm • n: represents list size • f(n): count function • Number of comparisons in search algorithm • c: units of computer time to execute one operation • cf(n): computer time to execute f(n) operations • Constant c depends computer speed (varies) • f(n): number of basic operations (constant) • Determine algorithm efficiency • Knowing how function f(n) grows as problem size grows Data Structures Using C++ 2E

  14. Algorithm Analysis (cont’d.) TABLE 1-2 Growth rates of various functions Data Structures Using C++ 2E

  15. TABLE 1-3 Time for f(n) instructions on a computer that executes 1 billion instructions per second Figure 1-4 Growth rate of functions in Table 1-3 Algorithm Analysis (cont’d.) Data Structures Using C++ 2E

  16. The Big-O Notation • Notation useful in describing algorithm behavior • Shows how a function f(n) grows as n increases without bound • Asymptotic • Study of the function f as n becomes larger and larger without bound • Examples of functions • g(n)=n2 (no linear term) • f(n)=n2 + 4n + 20 Data Structures Using C++ 2E

  17. TABLE 1-4 Growth rate of n2 and n2 + 4n + 20n Algorithm Analysis: The Big-O Notation (cont’d.) • As n becomes larger and larger • Term 4n + 20 in f(n) becomes insignificant • Term n2 becomes dominant term Data Structures Using C++ 2E

  18. Algorithm Analysis: The Big-O Notation (cont’d.) • Algorithm analysis • If function complexity can be described by complexity of a quadratic function without the linear term • We say the function is of O(n2)orBig-O of n2 • Let f and g be real-valued functions • Assume f and g nonnegative • For all real numbers n, f(n) >= 0and g(n) >= 0 • f(n) is Big-O of g(n):written f(n) = O(g(n)) • If there exists positive constants c and n0 such that f(n) <= cg(n) for all n >= n0 Data Structures Using C++ 2E

  19. TABLE 1-5 Some Big-O functions that appear in algorithm analysis Algorithm Analysis: The Big-O Notation (cont’d.) Data Structures Using C++ 2E

  20. Three time complexities Data Structures Using C++ 2E

  21. Running time, cont’d Data Structures Using C++ 2E

  22. Running time of iterative functions • The routine “Example 1-2” is an iterative example • The number of operations can usually be calculated through the summation of the operations within individual steps/stages. Data Structures Using C++ 2E

  23. Recursive example: merge sort

  24. Merge() Data Structures Using C++ 2E

  25. Time complexity of Merge Sort Data Structures Using C++ 2E

  26. Time complexity

  27. Big-O notation: revisit Data Structures Using C++ 2E

  28. Asymptotic notation Data Structures Using C++ 2E

  29. Big-O, Big-Omega, Big-Theta Data Structures Using C++ 2E

  30. Example Data Structures Using C++ 2E

  31. Example, cont’d Data Structures Using C++ 2E

  32. Logarithms Data Structures Using C++ 2E

  33. Some rules about Big-O notations Data Structures Using C++ 2E

  34. Some rules about Big-O notations, cont’d Data Structures Using C++ 2E

  35. Some rules about Big-O notations, cont’d Data Structures Using C++ 2E

  36. Important notation Data Structures Using C++ 2E

  37. Big Omega notation Data Structures Using C++ 2E

  38. Big Theta notation

  39. Big-O, Big-Omega, Big-Theta Data Structures Using C++ 2E

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