Algorithm Design and Analysis Liao Minghong School of Computer Science and Technology of HIT July, 2003

Course Description Course Name: Algorithm Design and Analysis Course Type: Selected Undergraduate Course Teaching hours: 30 Teaching materials: Algorithms Design Techniques and Analysis, M. H. Alsuwaiyel, Publishing House of Electronics Industry, 2003

Reference Materials Thomas H.Cormen, et al., Introduction to Algorithms, Higher Education Press, The MIT Press, 2002 王晓东，算法设计与分析，清华大学出 版社，

Course Contents Part 1: Basic Concepts and Introduction to Algorithms Part 2: Techniques Based on Recursion Part 3: First-Cut Techniques Part 4: Complexity of Problems Part 5: Coping with Hardness

Course Contents_Part 1 Chapter 1:Basic Concepts in Algorithmic Analysis Chapter 2: Mathematical Preliminaries Chapter 3: Data Structure Chapter 4: Heaps and the Disjoint Sets Structures

Course Contents_Part 2 Chapter 5: Introduction Chapter 6: Divide and Conquer Chapter 7: Dynamic Programming

Course Contents_Part 3 Chapter 8: The Greedy Approach Chapter 9: Graph Traversal

Course Contents_Part 4 Chapter 10: NP-Complete Problems Chapter 11: Introduction to computational complexity Chapter 12: Lower Bounds

Course Contents_Part 5 Chapter 13: Backtracking Chapter 14: Randomized Algorithms Chapter 15: Approximation Algorithms

Chapter 1: Basic Concepts in Algorithmic Analysis Basic Concepts of Algorithms Time Complexity How to Estimate the running Time of an Algorithm Worst case and average case analysis Amortized analysis

1.1 Basic Concepts of Algorithms Architecture of Computational Science

Process of solving problem problemcomputabilitycomplexityAlgorithmsprogrammingSoftware system Architecture Computability theory Computable & non computable Computational model Equivalence of model Complexity theory Worst case Best case Average case P =? NP Algorithms design & analysis Design algorithms Analysis technique Optimal algorithms C language Java language …

1.1 Basic Concepts of Algorithms Algorithm: is a procedure that consists of a finite set of instructions which, given an input from some set of possible inputs,enables us to obtain an output if such an output exists or else obtain nothing at all if there is no output for that particular input through a systematic execution of the instructions.

1.1 Basic Concepts of Algorithms Characteristics of Algorithms –Corrective –Concrete steps –Determinative –Limited –Halted

1.1 Basic Concepts of Algorithms_some algorithms Linear Search Binary Search Selection sort Insertion sort Merge Sort Quick sort Etc.

1.2 Time Complexity Order of growth The O-notation The Ω-notation The Θ-notation The o-notation

1.2 Time Complexity _Order of growth Definition1.1 elementary operation We denote by an “elementary operation” any computational step whose cost is always upperbounded by a constant amount of time regardless of the input data or the algorithm used.

1.2 Time Complexity _Order of growth Fig. 1.5 growth of some typical functions that represent running times

Definition 1.2 The O-notation Let f(n) and g(n) be the functions from the set of natural numbers to the set of nonnegative real numbers. f(n) is said to be O(g(n)) if there exists a natural number n 0 and a constant c>0 such that n>=n 0, f(n)<=cg(n). Consequently, if lim f(n)/g(n) exists, then lim, implies f(n)=O(g(n))

Definition 1.3 The Ω-notation Let f(n) and g(n) be two functions from the set of natural numbers to the set of nonnegative real numbers. f(n) is said to be Ω(g(n)) if there exists a natural number n 0 and a constant c>0 such that n>=n 0, f(n)>=cg(n). Consequently, if lim f(n)/g(n) exists, then lim, implies f(n)= Ω(g(n))

Definition 1.4 The Θ-notation Let f(n) and g(n) be two functions from the set of natural numbers to the set of nonnegative real numbers. f(n) is said to be Θ(g(n)) if there exists a natural number n 0 and two positive constants c1 and c2 such that n>=n 0, c1g(n)<=f(n)<=c2g(n). Consequently, if lim f(n)/g(n) exists, then Lim, implies f(n)= Θ(g(n)) Where c is a constant strictly greater than 0.

Definition 1.5 The o-notation Let f(n) and g(n) be the functions from the set of natural numbers to the set of nonnegative real numbers. f(n) is said to be o(g(n)) if for every constant c>0 there exists a positive integer n 0 such that f(n) =n 0. Consequently, if lim f(n)/g(n) exists, then Lim, implies f(n)=o(g(n))

Definition 1.6 Complexity Classes Let R be the relation on the set of complexity functions defined by f R g if and only if f(n)= Θ(g(n)). It is easy to see that R is reflexive, symmetric and transitive, i.e., an equivalence relation. The equivalence classes induced by this relation are called complexity classes. e.g. all polynomials of degree 2 belong to the same complexity class n 2.

Definition 1.7 Space Complexity We define the space used by an algorithm to be the number of memory cells needed to carry out the computational steps required to solve an instance of the problem excluding the space allocated to hold the input. All definition of order of growth and asymptotic bounds pertaining to time complexity carry over to space complexity. Let T(n) and S(n) denote, respectively, the time and space complexity of an algorithm, then S(n)=O(T(n))

1.3 Worst case analysis In worst case analysis of time complexity we select the maximum cost among all possible inputs of size n. Under the worst case assumption, the notions of upper and lower bounds in many algorithms coincide and, consequently, we may say that an algorithm runs in time Θ (f(n)) in the worst case. But the upper and lower bounds are not always coincide.

1.4 Average case analysis The running time is taken to be the average time over all inputs of size n. It is necessary to know the probabilities of all input occurrences. The analysis is in many cases complex and lengthy

1.5 Amortized analysis Unable to express the time complexity in terms of the Θ-notation,we will be content with the O- notation, but the algorithm may be much faster than our estimate even in the worst case. If the operation takes a large amount of time occasionally and runs much faster most of the time, then this is an indication that amortized analysis should be employed.

1.5 Amortized analysis In amortized analysis, we average out the time taken by the operation throughout the execution of the algorithm, and refer to this average as the amortized running time of that operation. Different with average case analysis: –The average is taken over all instances of the same size. –The average need to assume that the probability distribution of the input.

1.5 Amortized analysis Example 1.32 We have a doubly linked list that initially consists of one node which contains the integer 0. We have as input an array A[1..n] of n positive integers that are to be processed in the following way. If the current integer x is odd, then append x to the list. If it is even, then first append x and then remove all odd elements before x in the list. Let us analyze the running time of this algorithm.

The algorithm: for j = 1 to n x=a[j] append x to the list if x is even then while pred(x) is odd delete pred(x) end while end if end for

1.5 Amortized analysis Time complexity analysis: General analysis: O(n 2 ) Amortized analysis: Θ(n) (number of insertions: n, and the number of deletions: 0~n-1, so the total operations: n~2n-1)