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Complexity analysis - The Big O Notation
Complexity analysis determines how resource requirements like time and memory scale with problem size. Computation time depends on hardware, while complexity analyzes algorithm scaling. Big O notation describes asymptotic function growth. Common complexities are O(1) constant, O(log n) logarithmic, O(n) linear, O(n^2) quadratic. Statements are O(1), if/else max branch, loops run n times, nested loops run n*m times, functions match calling structure, and when statements have undefined time.
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Introduction to complexity analysis in object-oriented programming and data structures.
Differentiation between computation time and complexity, focusing on their definitions in computer science.
Examples provided for mathematical expressions showing complexity calculations with variables K and L.
Explanation of Big O notation and its role in denoting the growth rates of functions in complexity theory.
Various complexity functions represented in Big O notation, detailing constants to exponential complexity.
Discussion on the efficiency of algorithms covering CPU usage, memory, disk, and network resources.
Contrast between performance metrics during execution and the scaling requirements of algorithms based on complexity.
List of basic operations that impact the time required by functions or procedures in computation.
Overview of varying complexities for algorithms based on problem size expressed using Big O notation.
Identifying six cases to consider for calculating time complexity in programming constructs.
Explains calculation of total time for a series of statements executed in a constant time approach.
Analyzes the worst-case time complexity for an if-else statement based on block execution times.
Calculating time complexity for loops based on the number of executions relative to input size.
Total execution time for nested loops determined by the product of outer and inner loop iterations.
Behavior of function calls in terms of complexity based on how they are invoked in programming.
Examination of complexities associated with uncertain statements in algorithms, particularly in AI training.
Conclusion of the presentation with a wrap-up of topics discussed.
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Complexity analysis - The Big O Notation
- 1. Complexity Analysis OOP ANDDATA STRUCTURES ENGR. JAWAD ALI http://web.mit.edu/16.070/www/lecture/big_o.pdfDocument available on:
- 2. Difference between complexityand computation time Computation time: The interval of solving a problem based on the embedded system architecture. (in the field of computer sciences) Complexity: The art of handling a problem based on the algorithm designed to solve a case. OR The difficulty faced by the processor in solving a deployed case on it.
- 3. Example ex = 1+ x + x2/2 + x3/3... x is Real K=2*3 L=2^3 f(x) = 2 +3x for x = 5 K=2+2+2 L=2*2*2
- 4. Big O Notation Big O notation (with a capital letter O, not a zero), also called Landau's symbol, is a symbolism used in complexity theory, computer science, and mathematics to describe the asymptotic behavior of functions. Basically, it tells you how fast a function grows or declines
- 5. Functions defined inbig o notation O(1) constant(slowest) O(log(n)) logarithmic O((log(n))c) polylogarithmic (same as O(log(n)) ) O(n) linear O(n2) quadratic O(nc) polynomial O(cn) exponential(fastest)
- 6. Understanding big o Efficiency covers lots of resources, including: 1. CPU (time) usage (The most important) 2. Memory usage 3. Disk usage 4. Network usage
- 7. Performance vs complexity 1. Performance: how much time/memory/disk/... is actually used when a program is run. This depends on the machine, compiler, etc. as well as the code. 2. Complexity: how do the resource requirements of a program or algorithm scale, i.e., what happens as the size of the problem being solved gets larger?
- 8. More about performance The time required by a function/procedure is proportional to the number of "basic operations" that it performs, like; 1. one arithmetic operation (e.g., +, *). 2. one assignment (e.g. x := 0) 3. one test (e.g., x = 0) 4. one read (of a primitive type: integer, float, character, Boolean) 5. one write (of a primitive type: integer, float, character, Boolean)
- 9. Regarding computing We expresscomplexity using big-O notation. For a problem of size N: A constant-time algorithm is "order 1": O(1) A linear-time algorithm is "order N": O(N) A quadratic-time algorithm is "order N squared": O(N2) Infinite Time algorithm is “Order infinity”: O(inf)
- 10. Finding complexity Generally, wehave 6 cases 1. Statements 2. If else 3. Loop 4. Nested loop 5. Function call 6. When
- 11. Statement statement 1; statement 2; ... statementk; The total time is found by adding the times for all statements: total time = time(statement 1) + time(statement 2) + ... + time(statement k) If each statement is "simple" (only involves basic operations) then the time for each statement is constant and the total time is also constant: O(1).
- 12. If Else if (cond)then block 1 (statements) else block 2 (statements) end if; Here, either block 1 will execute, or block 2 will execute. Therefore, the worst-case time is the slower of the two possibilities: max(time(block 1), time(block 2)) If block 1 takes O(1) and block 2 takes O(N), the if-then- else statement would be O(N)
- 13. LOOP for I in1 .. N loop sequence of statements end loop The loop executes N times, so the sequence of statements also executes N times. If we assume the statements are O(1), the total time for the for loop is N * O(1), which is O(N) overall.
- 14. Nested LOOP for Iin 1 .. N loop for J in 1 .. M loop sequence of statements end loop; end loop; The statements in the inner loop execute a total of N * M times. Thus, the complexity is O(N * M).
- 15. Function Calls The behaviorof function is same as statement if called once Its behavior is statement in loop if it is called in loop Its behavior is more like nested loop if it is called inside loop and it has an characteristic loop inside as well
- 16. When The behaviorof such statement is not defined by time or cycles of processing It may occur the very next moment It might not occur even after the device is expired Such algorithms are limited by some thresholds or bounds, becomes O(N) Used in training and testing of Artificial Neural Networks and such
- 17.