Introduction to Data Structures and Algorithms

1. Chapter 1Introduction Instructors: C. Y. Tang and J. S. Roger Jang All the material are integrated from the textbook "Fundamentals of Data Structures in C" and some supplement from the slides of Prof. Hsin-Hsi Chen (NTU).

2. How to create programs • Requirements • Analysis: bottom-up vs. top-down • Design: data objects and operations • Refinement and Coding • Verification • Program Proving • Testing • Debugging

3. Data Structure: How data are organized and hence operated • You’ve been very familiar with arrays in your programming assignments. They are basic (yet powerful!) data structures. • They can hold data (objects)—e.g., integers. • They are structured—structured in a way that the data held inside can be operated. • Each element in an array has an index. With that, you can store or retrieve an element.

4. Learning Data Structures, and Algorithms • You want your tasks to be performed efficiently. You need good methods (algorithms). • Data must be structured in some manner to be operated. • Good structures can be operated efficiently.

5. Example • For example, suppose that you are building a database storing the data of all (past, present, and future) students of NTHU, which are growing in size. • You’ll need to find anybody’s data in the database (to search/retrieve). • To enter new entries into it (to insert). • Etc.

6. Example • You can use an array to implement the database. • To insert a new entry, simply add it to the first empty array cell. • If the current array is full, allocate a new array whose size is the double of the current. Then copy all the original entries from the old array to the new one. • To search for an entry, simply looks at all entries in the array, one by one.

7. Example • Your programming experience, however, tells you that this is not a good method. • In Chapter 10, you’ll see sophisticated data structures that can be operated (searched, inserted/deleted, etc.) efficiently. • Then, you’ll just feel how data can be cleverly structured to serve as the basis of fast algorithms.

8. Example 2 • This time you want to work with polynomials, i.e., functions of the form f(x) = anxn + an-1xn-1 + … + a0 • You’ll need to store them, as well as to multiply or add them. • You may allocate an array to store the coefficients of a polynomial. • E.g., A[i] stores ai.

9. Example 2 • Alternatively, you can use a linked list to store it. • Each node needs to store the coefficient and the exponent. E.g., to store 3x2 + 5, a linked list like [3|2] -> [5|0] is constructed. • To represent polynomials, both data structures have their relative advantages and drawbacks in time and space considerations, etc (see later chapters).

10. Searching in Arrays • You are able to write, in minutes, a program to search sequentially in an array. • However, when the numbers in an array are sorted, you can do much faster… • …using binary search, which you probably are also familiar with. • You may say that, a sorted array is not the same structure as a general array.

11. Binary Search • Let A[0..n - 1] be an array of n integers. • We want to ask: is some integer x stored in A? • Suppose that A is sorted (say, in ascending order), i.e., (for simplicity assume that the numbers are distinct) A[0] < A[1] < … < A[n - 1]. • To be more concrete, let n be 5.

12. Binary Search • Observation: If x > A[2], then x must fall in A[3..4], (or x is not in A). If x < A[2], then x must fall in A[0..1]. • Ex: Let x = 11. • A = 3 5 7 11 13 • x (= 11) > A[2] (= 7), so we need not consider A[0..1].

13. Binary Search • Example: Let x = 17. Let A contain the following 8 integers. Initially, let left = 0 and right = 7 (shown in red). • left and right define the range to be searched. • 2 3 5 7 11 13 17 19 • mid = (0 + 7) / 2 = 3 (rounded). • A[mid] = 7 < x, so let left = mid + 1 = 4, and continue the search.

14. 2 3 5 7 11 13 17 19 • mid = (4 + 7) / 2 = 5, A[mid] = 13, x = 17. • A[mid] < x, so let left = mid + 1 = 6, and continue the search. • 2 3 5 7 11 13 1719 • mid = (6 + 7) / 2 = 6, A[mid] = 17, x = 17. • A[mid] = x, so return mid = 6, the desired position, and we’re done!

15. Binary Search • Binsearch(A, x, left, right) /* finds x in A[left..right] */ while left <= rightdo mid = (left + right) / 2; ifx < A[mid] thenright = mid– 1; else ifx == A[mid] then returnmid; elseleft = mid + 1; return -1; /* not found */

16. Recursive Functions • A function that invokes (or is defined in terms of) itself directly or indirectly is a recursive function. • Fibonacci sequence: Fn = Fn-1 + Fn-2 • Summation: sum(n) = sum(n - 1) + A[n] • Where sum(i) is the sum of the first i items in A • Binomial coefficient (n choose k): C(n, k) = C(n– 1, k) + C(n – 1, k– 1) • The combinatorial interpretation of this is profound—try it out if you’ve not learned it yet!

17. Recursive Binary Search • Binsearch(A, x, left, right) ifleft > rightthenreturn -1; mid = (left + right) / 2; ifx < A[mid] thenreturn Binsearch(A, x, left, right– 1); elseifx == A[mid] then returnmid; else return Binsearch(A, x, left + 1, right);

18. Data Abstraction • Before implementation, we need first to know the specification of the objects to be stored as well as the operations that should be supported, before we can implement it. • In our previous polynomial example, first we have the demand to store polynomials (the objects), supporting multiplications and other operations. The specification is independent of how it is implemented (e.g. using arrays or linked lists).

19. Abstract Data Type (ADT) • An abstract data type (ADT) is a data type that is organized in such a way that the specification of the objects and the specification of the operations on the objects is separated from the representation of the objects and the implementation of the operations. • No fixed syntax to describe them. The specifications need only be clear.

20. ADT for Natural Numbers (an example) • Structure Natural_Number: • Objects: an ordered subrange of the integers starting at 0 and ending at the maximum integer (INT_MAX) on the computer. • Functions: • Nat_Num Zero() ::= 0 • Nat_Num Add(x, y) ::= if (x+y) <= INT_MAX then return x + y, else return INT_MAX. • And so on. This example is actually too simple so that you may feel that the implementation of the functions have been stated. But this is generally not the case.

21. *Structure 1.1:Abstract data type Natural_Number (p.17)structure Natural_Number isobjects: an ordered subrange of the integers starting at zero and ending at the maximum integer (INT_MAX) on the computerfunctions: for all x, y  Nat_Number; TRUE, FALSE  Boolean and where +, -, <, and == are the usual integer operations.Nat_No Zero ( ) ::= 0Boolean Is_Zero(x) ::= if (x) returnFALSEelse returnTRUENat_No Add(x, y) ::= if ((x+y) <= INT_MAX) return x+y else returnINT_MAXBoolean Equal(x,y) ::= if (x== y) returnTRUEelse returnFALSENat_No Successor(x) ::= if (x == INT_MAX) return xelse return x+1Nat_No Subtract(x,y) ::= if (x<y) return 0else return x-yend Natural_Number ::= is defined as

22. Performance Analysis • We’re most interested in the time and space requirements of an algorithm. • The space complexity of a program is the amount of memory that it needs to run to completion. The time complexity is the amount of computer time that it needs to run to completion.

23. 什麼是 Algorithm ? • A number of rules, which are to be followed in a prescribed order, for solving a specific type of problems. • Computer Algorithm的額外考慮： • Finiteness（有限個steps） • Definiteness（每一個step做的事確定） • Effectiveness（不只確定還要足夠的短能做完） • Input/Output（O.S.永不terminate只能稱是computational procedure）

24. Algorithm is everywhere ! • Operating Systems • System Programming • Numerical Applications • Non-numerical Applications ： • 任何一field都要用Algorithm去解決該field所發生的問題。 • Algorithm Implement的方式： • Software • Hardware • Firmware

25. 為什麼要學Algorithm ? • 1. 如果不學，你可能用一個很花錢（Time, Space）的Algorithm去解決問題 • Life-time Job: • 永遠知道解某一問題最佳的Algorithm是什麼 • 方法：對於自己的領域隨時要查paper，update最佳的algorithms或至少知道可以問誰。

26. 為什麼要學Algorithm ? • 2. 如果不學，你可能去對NP-Complete的問題去找efficient的解 • Life-time Job: • 去知道那些問題是NP-Complete的 • Real Application • Average Performance好的 • 找Approximating的解 • TSP： • n = 20 771世紀 • N3log n in average (B & B) • Planar Graph Coloring (Maximum # = 4 已被證明)

27. Measurements • Criteria • Is it correct? • Is it readable? • … • Performance Analysis (machine independent) • space complexity: storage requirement • time complexity: computing time • Performance Measurement (machine dependent)

28. Space Complexity • The space needed includes two parts: • (1) Fixed space requirement: Not dependent on the number and size of the program’s inputs and outputs. • Instruction space, simple variables (e.g., int), fixed-size structure variables (such as struct), and constants.

29. Space Complexity • (2) Variable space requirement S(I): Depends on the instance I involved. • In recursive calls, many copies of simple variables (e.g. many int’s) may exist. Such space requirement is included in S(I). • S(I) may depend on some characteristics of I. The characteristic we’ll most often encounter is n, the size of the instance. • In this case we denote S(I) as S(n).

30. Space Complexity float abc(float a, float b, float c) { return a+b+b*c + (a+b-c) / (a+b) + 4; } • Sabc(I) = 0. • Only has fixed space requirement.

31. Space Complexity float sum(float list[], int n) {/* adds up list[] */ float tempsum = 0; int i; for (i=0; i < n; i++) tempsum += list[i]; return tempsum; } • In C, the array is passed using the address. So Ssum(n) = 0. • In Pascal, the array may be passed by copying values. If this is the case, then Ssum(n) = n.

32. Space Complexity float rsum(float list[], int n) { if (n > 0) return rsum(list, n-1) + list[n-1]; return 0; } • For each recursive call, the OS must save: parameters, local variables, and the return address. • In this example, two parameters (list[] and n) and the return address (internally) are saved for each recursive call.

33. To add a list of n numbers, there are n recursive calls in total. • So Srsum(n) = (c1 + c2 + c3) * n, where c1, c2 and c3 are the number of bytes (or other unit of interest) needed for each of these types (list[], n, and return address).

34. Time Complexity • We’re interested in the number of steps taken by a program. But what is a step? • A program step is a syntactically or semantically meaningful program segment whose execution time is independent of the instance characteristics. • For a program, everybody can have his/her own steps defined. The important thing is the independency of the instance size, etc.

35. s/e: #steps per execution.

36. Time Complexity • For some programs, for a fixed n, the time taken by different instances may still be different. • For example, the binary search algorithm depends on the position of x in array A. • Let A = <3, 5, 7, 11>. • If x = 5, then we need less steps than what if x = 7. • But in both cases, n = 4.

37. Time Complexity • So we may consider the worst case, average case, or best case time complexity of an algorithm. • Worst case: the maximum number of steps needed for any possible instance of size n. • Most commonly used. The concept of guarantee. • Average case: under some assumption of instance distribution, the expected number of steps needed. • Useful. But usually most complicated to analyze. • Best case: the opposite of worst case. • Rarely seen.

38. Asymptotic Notations • To make exact step counts is often not necessary. • The concept of “step” is even inexact itself. • It is more impressive to obtain a “functional” improvement in the time complexity than an improvement by a constant multiple. • It is good to improve 2n2 to n2. • It is even better to improve 2n2 to 1000n. • For nlarge enough, you know that n2 is much larger than 1000n.

39. *Figure 1.7:Function values (p.38)

41. *Figure 1.9:Times on a 1 billion instruction per second computer(p.40)

42. Asymptotic Notations • Therefore, in many cases we do not worry about the coefficients in the time complexity. • Not in “all” cases since, when we cannot have functional improvement, we’ll still want improvements in the coefficients. • We regard 2n2 as equivalent to n2. They belong to the same class in this sense. • Similarly, 2n2 + 100n and n2 belong to the same class, since the quadratic term is dominant. • n2 and n belong to different classes.

43. Asymptotic Notation (O) • Definitionf(n) = O(g(n)) iff there exist positive constants c and n0 such that f(n)  cg(n) for all n, n  n0. • Examples • 3n+2=O(n) /* 3n+24n for n2 */ • 3n+3=O(n) /* 3n+34n for n3 */ • 100n+6=O(n) /* 100n+6101n for n10 */ • 10n2+4n+2=O(n2) /* 10n2+4n+211n2 for n5 */ • 6*2n+n2=O(2n) /* 6*2n+n2 7*2n for n4 */

44. Example • Complexity of c1n2+c2n and c3n • for sufficiently large of value, c3n is faster than c1n2+c2n • for small values of n, either could be faster • c1=1, c2=2, c3=100 --> c1n2+c2n  c3n for n  98 • c1=1, c2=2, c3=1000 --> c1n2+c2n  c3n for n  998 • break even point • no matter what the values of c1, c2, and c3, the n beyond which c3n is always faster than c1n2+c2n

45. O(1): constant • O(n): linear • O(n2): quadratic • O(n3): cubic • O(2n): exponential • O(logn) • O(nlogn)

46. Asymptotic Notations • To make these concepts more precise, asymptotic notations are introduced. • f(n) = O(g(n)) if there exist positive constants c and n0 such that f(n) cg(n) for all n > n0. • 1000n = O(2n2). This is to say that 1000n is no larger than () 2n2 in this sense. • You can choose c = 500 and n0 = 1. • Then 1000n 500 * 2n2 = 1000n2 for all n > 1.

47. Asymptotic Notations • 1000n = O(n) since you can choose c = 1000 and n0 = 1. • 1000 = O(1). • For constant functions, the choice of n0 is arbitrary. • n2 O(n) since for any c > 0 and any n0 > 0, there always exists some n > n0 such that n2 > cn.

48. Asymptotic Notations • 2n2 + 100n = O(n2), since 2n2 + 100n < 200n2 = O(n2) • This last equation may be validated by choosing c = 200 and n0 = 1. • So you can feel that in asymptotic notations we only care about the most dominant term. Simply throw out the minor terms. Throw out the constants, too. • log2n = O(n). May choose c = 1 and n0 = 2. • n log2n = O(n2).

49. Asymptotic Notations • n100 = O(2n). You may choose c = 1 and n0 = 1000. • O(1): constant, O(n): linear, O(n2): quadratic, O(n3): cubic, O(log n): logarithmic, O(n log n), O(2n): exponential. • These are the most commonly encountered time complexities. • The base of the log is not relevant asymptotically, since logba = (1/log2b) log2a, different only by a constant multiple 1/log2b.

50. Asymptotic Notations • f(n) = O(g(n)) is just a notation. f(n) and O(g(n)) are not the same thing. • So you can’t write O(g(n)) = f(n). • It is also common to view O(g(n)) as a set of functions, and f(n) = O(g(n)) actually means f(n)  O(g(n)). • O(g(n)) = {f(n): for some c > 0 and n0 > 0 such that f(n) < cg(n) for all n > n0}