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Chapter 3: The Fundamentals: Algorithms, the Integers, and Matrices

Chapter 3: The Fundamentals: Algorithms, the Integers, and Matrices. § 3.1: Algorithms. § 3.1 – Algorithms. Algorithms. The foundation of computer programming. Most generally, an algorithm just means a definite procedure for performing some sort of task .

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Chapter 3: The Fundamentals: Algorithms, the Integers, and Matrices

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  1. Chapter 3:The Fundamentals: Algorithms, the Integers, and Matrices (c)2001-2002, Michael P. Frank

  2. §3.1: Algorithms § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  3. Algorithms • The foundation of computer programming. • Most generally, an algorithm just means a definite procedure for performing some sort of task. • A computer program is simply a description of an algorithm in a language precise enough for a computer to understand, requiring only operations the computer already knows how to do. • We say that a program implements (or “is an implementation of”) its algorithm. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  4. Programming Languages • Some common programming languages: • Newer:Java, C, C++, Visual Basic, JavaScript, Perl, Tcl, Pascal • Older: Fortran, Cobol, Lisp, Basic • Assembly languages, for low-level coding. • In this class we will use an informal, Pascal-like “pseudo-code” language. • You should know at least 1 real language! § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  5. Algorithm Example (English) • Task: Given a sequence {ai}=a1,…,an, aiN, say what its largest element is. • Set the value of a temporary variablev (largest element seen so far) to a1’s value. • Look at the next element ai in the sequence. • If ai>v, then re-assign v to the number ai. • Repeat previous 2 steps until there are no more elements in the sequence, & return v. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  6. Executing an Algorithm • When you start up a piece of software, we say the program or its algorithm are being run or executed by the computer. • Given a description of an algorithm, you can also execute it by hand, by working through all of its steps on paper. • Before ~WWII, “computer” meant a person whose job was to run algorithms! § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  7. Executing the Max algorithm • Let {ai}=7,12,3,15,8. Find its maximum… • Set v = a1 = 7. • Look at next element: a2 = 12. • Is a2>v? Yes, so change v to 12. • Look at next element: a2 = 3. • Is 3>12? No, leave v alone…. • Is 15>12? Yes, v=15… § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  8. Algorithm Characteristics Some important features of algorithms: • Input. Information or data that comes in. • Output. Information or data that goes out. • Definiteness. Precisely defined. • Correctness. Outputs correctly relate to inputs. • Finiteness. Won’t take forever to describe or run. • Effectiveness. Individual steps are all do-able. • Generality. Works for many possible inputs. • Efficiency. Takes little time & memory to run. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  9. procedurename(argument: type) variable:=expression informal statement beginstatementsend {comment} ifcondition then statement [else statement] for variable:=initial value to final valuestatement whileconditionstatement procname(arguments) Not defined in book: returnexpression Declaration STATEMENTS Our Pseudocode Language: §A2 § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  10. procedure procname(arg: type) • Declares that the following text defines a procedure named procname that takes inputs (arguments) named arg which are data objects of the type type. • Example:proceduremaximum(L: list of integers) [statements defining maximum…] § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  11. variable:=expression • An assignment statement evaluates the expression expression, then reassigns the variable variable to the value that results. • Example:v:= 3x+7 (If x is 2, changes v to 13.) • In pseudocode (but not real code), the expression might be informal: • x:= the largest integer in the list L § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  12. Informal statement • Sometimes we may write a statement as an informal English imperative, if the meaning is still clear and precise: “swap x and y” • Keep in mind that real programming languages never allow this. • When we ask for an algorithm to do so-and-so, writing “Do so-and-so” isn’t enough! • Break down algorithm into detailed steps. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  13. Groups a sequence of statements together:beginstatement 1statement 2 …statement nend Allows sequence to be used like a single statement. Might be used: After a procedure declaration. In an if statement after then or else. In the body of a for or while loop. beginstatementsend § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  14. {comment} • Not executed (does nothing). • Natural-language text explaining some aspect of the procedure to human readers. • Also called a remark in some real programming languages. • Example: • {Note that v is the largest integer seen so far.} § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  15. ifconditionthenstatement • Evaluate the propositional expression condition. • If the resulting truth value is true, then execute the statement statement; otherwise, just skip on ahead to the next statement. • Variant: ifcondthenstmt1elsestmt2Like before, but iff truth value is false, executes stmt2. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  16. whileconditionstatement • Evaluate the propositional expression condition. • If the resulting value is true, then execute statement. • Continue repeating the above two actions over and over until finally the condition evaluates to false; then go on to the next statement. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  17. whileconditionstatement • Also equivalent to infinite nested ifs, like so: if conditionbeginstatement if conditionbeginstatement …(continue infinite nested if’s)endend § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  18. forvar:=initial to finalstmt • Initial is an integer expression. • Final is another integer expression. • Repeatedly execute stmt, first with variable var :=initial, then with var :=initial+1, then with var :=initial+2, etc., then finally with var :=final. • What happens if stmt changes the value that initial or final evaluates to? § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  19. forvar:=initial to finalstmt • For can be exactly defined in terms of while, like so: beginvar:=initialwhilevar finalbeginstmtvar:=var + 1endend § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  20. procedure(argument) • A procedure call statement invokes the named procedure, giving it as its input the value of the argument expression. • Various real programming languages refer to procedures as functions (since the procedure call notation works similarly to function application f(x)), or as subroutines, subprograms, or methods. § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  21. Max procedure in pseudocode proceduremax(a1, a2, …, an: integers) v:=a1 {largest element so far} fori:= 2 ton {go thru rest of elems} ifai > vthen v:=ai {found bigger?} {at this point v’s value is the same as the largest integer in the list} returnv § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  22. Another example task • Problem of searching an ordered list. • Given a list L of n elements that are sorted into a definite order (e.g., numeric, alphabetical), • And given a particular element x, • Determine whether x appears in the list, • and if so, return its index (position) in the list. • Problem occurs often in many contexts. • Let’s find an efficient algorithm! § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  23. Search alg. #1: Linear Search procedurelinear search(x: integer, a1, a2, …, an: distinct integers)i:= 1while (i n  x  ai)i:=i + 1ifi n then location:=ielselocation:= 0return location {index or 0 if not found} § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  24. Search alg. #2: Binary Search • Basic idea: On each step, look at the middle element of the remaining list to eliminate half of it, and quickly zero in on the desired element. <x <x <x >x § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  25. Search alg. #2: Binary Search procedurebinary search(x:integer, a1, a2, …, an: distinct integers)i:= 1 {left endpoint of search interval}j:=n {right endpoint of search interval}while i<j begin {while interval has >1 item}m:=(i+j)/2 {midpoint}ifx>amtheni := m+1 else j := mendifx = aithenlocation:=ielselocation:= 0returnlocation § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  26. Sorting alg. : Bubble Sort procedurebubblesort(a1, a2, …, an)for i:= 1 ton-1for j:= 1 to n-i ifaj>aj+1then interchange aj and aj+1 {a1, a2, …, an is in increasing order} § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  27. Sorting alg. : Insertion Sort procedureinsertionsort(a1, a2, …, an) for j:= 2 ton begini:= 1 whileaj > ai i:=i+1 m:=aj for k:= 0 to j-i-1 aj-k:=aj-k-1 ai:=m end{a1, a2, …, an are sorted} § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  28. Greedy Change-Making Alg. procedurechange(c1, c2, …, cr : c1> c2> …> cr；n) for i:= 1 tor whilen  ci begin add a coin with value cito the change n:=n- ci end § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  29. Practice exercises • 3.1.3: Devise an algorithm that finds the sum of all the integers in a list. [2 min] • proceduresum(a1, a2, …, an: integers)s:= 0 {sum of elems so far}fori:= 1 ton {go thru all elems}s:=s + ai {add current item}{at this point s is the sum of all items}returns § 3.1 – Algorithms (c)2001-2002, Michael P. Frank

  30. §3.2: The Growth of Functions § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  31. Orders of Growth • For functions over numbers, we often need to know a rough measure of how fast a function grows. • If f(x) is faster growing than g(x), then f(x) always eventually becomes larger than g(x) in the limit (for large enough values of x). • Useful in engineering for showing that one design scales better or worse than another. § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  32. Orders of Growth - Motivation • Suppose you are designing a web site to process user data (e.g., financial records). • Suppose database program A takes fA(n)=30n+8 microseconds to process any n records, while program B takes fB(n)=n2+1 microseconds to process the n records. • Which program do you choose, knowing you’ll want to support millions of users? § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  33. Visualizing Orders of Growth • On a graph, asyou go to theright, a fastergrowingfunctioneventuallybecomeslarger... fA(n)=30n+8 Value of function  fB(n)=n2+1 Increasing n  § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  34. Concept of order of growth • We say fA(n)=30n+8is order n, or O(n). It is, at most, roughly proportional to n. • fB(n)=n2+1 is order n2, or O(n2). It is roughly proportional to n2. • Any O(n2) function is faster-growing than any O(n) function. • For large numbers of user records, the O(n2) function will always take more time. § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  35. Definition: O(g), at mostorder g Let g be any function RR. • Define “at most order g”, written O(g), to be: {f:RR | c,k: x>k: f(x)  cg(x)}. • “Beyond some point k, function f is at most a constant c times g (i.e., proportional to g).” • “f is at most order g”, or “f is O(g)”, or “f=O(g)” all just mean that fO(g). • Sometimes the phrase “at most” is omitted. § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  36. Points about the definition • Note that f is O(g) so long as any values of c and k exist that satisfy the definition. • But: The particular c, k, values that make the statement true are not unique: Any larger value of c and/or k will also work. • You are not required to find the smallest c and k values that work. (Indeed, in some cases, there may be no smallest values!) However, you should prove that the values you choose do work. § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  37. “Big-O” Proof Examples • Show that 30n+8 is O(n). • Show c, k: n>k:30n+8  cn. • Show that n2+1 is O(n2). • Show c, k: n>k: n2+1  cn2. § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  38. cn =31n n>k=8  Big-O example, graphically • Note 30n+8 isn’tless than nanywhere (n>0). • It isn’t evenless than 31neverywhere. • But it is less than31neverywhere tothe right of n=8. 30n+8 30n+8O(n) Value of function  n Increasing n  § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  39. Useful Facts about Big O • Big O, as a relation, is transitive: fO(g)  gO(h)  fO(h) • O with constant multiples, roots, and logs... f (in (1)) & constantsa,bR, with b0,af, f 1-b, and (logbf)aare all O(f). • Sums of functions:If gO(f) and hO(f), then g+hO(f). § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  40. More Big-O facts • c>0, O(cf)=O(f+c)=O(fc)=O(f) • f1O(g1)  f2O(g2)  • f1 f2 O(g1g2) • f1+f2 O(g1+g2) = O(max(g1,g2)) = O(g1) if g2O(g1) (Very useful!) § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  41. Orders of Growth (§3.2) - So Far • For any g:RR, “at most order g”,O(g)  {f:RR | c,k x>k |f(x)|  |cg(x)|}. • Often, one deals only with positive functions and can ignore absolute value symbols. • “fO(g)” often written “f is O(g)”or “f=O(g)”. • The latter form is an instance of a more general convention... § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  42. Order-of-Growth Expressions • “O(f)” when used as a term in an arithmetic expression means: “some function f such that fO(f)”. • E.g.: “x2+O(x)” means “x2 plus some function that is O(x)”. • Formally, you can think of any such expression as denoting a set of functions: “x2+O(x)” : {g | fO(x): g(x)= x2+f(x)} § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  43. Order of Growth Equations • Suppose E1 and E2 are order-of-growth expressions corresponding to the sets of functions S and T, respectively. • Then the “equation” E1=E2 really means fS, gT : f=gor simply ST. • Example: x2 + O(x) = O(x2) means fO(x): gO(x2): x2+f(x)=g(x) § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  44. Useful Facts about Big O •  f,g & constantsa,bR, with b0, • af = O(f); (e.g. 3x2 = O(x2)) • f+O(f) = O(f); (e.g. x2+x = O(x2)) • Also, if f=(1) (at least order 1), then: • |f|1-b = O(f); (e.g.x1 = O(x)) • (logb |f|)a= O(f). (e.g. log x = O(x)) • g=O(fg) (e.g.x = O(x log x)) • fg O(g) (e.g.x log x  O(x)) • a=O(f) (e.g. 3 = O(x)) § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  45. Definition: (g), exactly order g • If fO(g) and gO(f) then we say “g and f are of the same order” or “f is (exactly) order g” and write f(g). • Another equivalent definition:(g)  {f:RR | c1c2k x>k: |c1g(x)||f(x)||c2g(x)| } • “Everywhere beyond some point k, f(x)lies in between two multiples of g(x).” § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  46. Rules for  • Mostly like rules for O( ), except: •  f,g>0 & constantsa,bR, with b>0,af (f), but  Same as with O.f  (fg) unless g=(1)  Unlike O.|f| 1-b (f), and  Unlike with O.(logb |f|)c (f).  Unlike with O. • The functions in the latter two cases we say are strictly of lower order than (f). § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  47.  example • Determine whether: • Solution: § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  48. Other Order-of-Growth Relations • (g) = {f | gO(f)}“The functions that are at least orderg.” • o(g) = {f | c>0 k x>k : |f(x)| < |cg(x)|}“The functions that are strictly lower order than g.” o(g)  O(g)  (g). • (g) = {f | c>0 k x>k : |cg(x)| < |f(x)|}“The functions that are strictly higher order than g.” (g)  (g)  (g). § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  49. Relations Between the Relations • Subset relations between order-of-growth sets. RR ( f ) O( f ) • f o( f ) ( f ) ( f ) § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

  50. Why o(f )O(x)(x) • A function that is O(x), but neither o(x) nor (x): § 3.2 – The Growth of Functions (c)2001-2002, Michael P. Frank

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