Design and Analysis of Algorithms & Computational Complexity

Design and Analysis of Algorithms & Computational Complexity CS490 Koji Tajii

Designing a program means… • 1. Specify what you exactly want to do. • What kind of input and output? • What is the Precondition and Post condition? • 2. Design the Algorithm • Pseudocode • What kind of Data structures are needed? • 3. Translate the program into a programming language • C++ JAVA, etc… • 4. Test and Debug • Does it actually work?

Algorithmic Design Techniques • Divide and Conquer • Decomposing tasks into smaller subtasks • Dynamic Programming • Break down problems into stages • Keep track of what’s going on at each stage as states • Determine the best solution by taking into consideration the states of the previous stage • Greedy • Always choosing the best local solution at the time to eventually get to the final solution.

Example - Traversing a Weighted Graph

Time and Space Complexity • Efficiency of program is determined by its speed and amount of memory space it takes. • Faster and smaller is better but…

Big-O Notation • Order of an algorithm • Rough approximation of number of operations a program will execute to determine its speed 2

Worst, Average, Best Case Scenario • Worst Case • Maximum Number of Operations • Average Case • Average Number of Operations • Best Case • Fewest Number of Operations • Probability of occurrence? • Unless the best-case behavior occurs with high probability, it is generally not used • The worst case occurs often

Example 1 void search(const int a[], int first, int size, int target, bool& found, int& location) { int middle; if(size == 0) found = false; else { middle = first + size/2; if (target == a[middle]) { location = middle; found = true; } else if (target < a[middle]) search(a, first, size/2, target, found, location); else search(a, middle+1, (size-1)/2, target, found, location); } }

Example 2 Void Sort(int A[], int N) { bool sorted = false’ for(int Pass = 1; (Pass < N) && !sorted; ++Pass) { Sorted = true; for(int Index = 0; Index < N-Pass; ++Index) { int NextIndex = Index + 1; if(A[Index] > A[NextIndex]) { swap(A[Index], A[NextIndex]); Sorted = false; } } } }

References • Main Michael & Savitch Walter. Data Structures and Other Objects using C++. Addison-Wesley. 1997. • Walls & Millors. Data Abstraction and Problem Solving with C++. 2nd ed. Addison-Wesley. 1998 • http://mat.gsia.cmu.edu/classes/dynamic/dynamic.html

Design and Analysis of Algorithms & Computational Complexity