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CS483 Midterm Review:
Analysis of Algorithms
General Motivation: Objective of the course: how to design and analyze algorithms Algorithms: recipes for solving problems , they are realized by programs Programs: sequences of instructions
One alternative: take a program and count the number of instructions (this may be depend on implementation)
We need some measure how to compare them which would be independent of machine, details and it would depend on the METHOD used to solve the problem
In general given a program the number of instructions will depend on input:
- depends of the size of input (number of input elements n)
- depends on particular input
Comparing algorithms/programs
We want to compare algorithms in the same class: (trying to solve the same problems, without worrying about details of speed of computer, implementation)
Running time -> Number of instructions as a function of input size n:
Algorithm 1 takes f(n) steps, algorithm 2 takes g(n) steps, we Want to compare them = > compare functions f, g, for large n
Order of growth of functions:
Definitions of classes of functions Upper bound, lower bound, tight bound, not tight upper (lower) bound
Asymptotic Efficiency
- Measuring input size
- What is elementary operation
- Definition of asymptotic notations
- Basic Efficiency classes
- Asymptotic running time of some
elementary algorithms (sequential search,
binary search, brute force algorithms,
closest pair, convex hull etc)
- (computing summations – see appendix)
Techniques how to compare functions
Facts: polynomials are dominated by higher order terms
To show it from definition choosec andno such that inequalities hold
Facts:Facts:^ exponential functions grow faster then polynomials
Comparing functions using limits (L’Hospital rule):
Worst case running-time
Longest running time on any input Regardless of the input it will takeat mostat most w(n) steps It is the upper bound of the algorithm on any input
Lower bound on the worst-case running time There is some input that the algorithm will perform at leastat least f(n) steps
Worst case result on a particular array Best case result on a particular array If we make statement about the algorithm, it has to be for any input
Design and analysis of algorithms
Types of algorithms: Iterative (multiple loops)
Compute exact running time (bounding summations)
Design and analysis of algorithms
Iterative (multiple loops) Compute exact running time (bounding summations)
Recursive algorithms – partition into smaller sub-problems
To establish the running time, we need to solve the recurrent equations. Techniques for solving recurrences:
- Expansion (expand and compute (or bound summations)
- Master’s Theorem (for divide and conquer problems)
Examples of recursive algorithms: (Merge Sort, Quick Sort, Binary Search, Selection, Tower of Hanoi etc., number of bits in n’s binary represent.)
Brute force algorithms
- Selection Sort – worst case running time
- Bubble Sort – worst case running time
- Sequential Search, String Matching
- Closest Pair, Convex Hull problems
- Exhaustive Search – knapsack, traveling
salesman, assignment problems
Transform and Conquer
- Pre-sorting
- AVL trees – properties of AVLtrees
efficiency of insert operation
- How to do rotations
- Heaps and heapsort
Median and order statistics
MinMin MaxMax^
worst case (at most) and best casen-
Selection problemSelection problem^ expected average running time randomized partition
Selection problemSelection problem^ average case running time quick sort idea
Heapsort
Implicit data structure (correspondence heap – array) Heap properties
Build HeapBuild Heap
Loose bound upper bound
Tighter bound
HeapifyHeapify
Heap SortHeap Sort (^) extract max, replace root, Heapify(A,1)
Heapsort is
Need to know how these work Does it depend on whether array is sorted or not?
