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Theory of Algorithms: Fundamentals of the Analysis of Algorithm Efficiency

Theory of Algorithms: Fundamentals of the Analysis of Algorithm Efficiency. James Gain and Edwin Blake {jgain | edwin} @cs.uct.ac.za Department of Computer Science University of Cape Town August - October 2004. Objectives. To outline a general Analytic Framework

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Theory of Algorithms: Fundamentals of the Analysis of Algorithm Efficiency

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  1. Theory of Algorithms:Fundamentals of the Analysis of Algorithm Efficiency James Gain and Edwin Blake {jgain | edwin} @cs.uct.ac.za Department of Computer Science University of Cape Town August - October 2004

  2. Objectives • To outline a general Analytic Framework • To introduce the conceptual tools of: • Asymptotic Notations • Base Efficiency Classes • To cover Techniques for the Analysis of Algorithms: • Mathematical (non-recursive and recursive) • Empirical • Visualisation

  3. Analysis of Algorithms • Issues: • Correctness (Is it guaranteed to produce a correct correspondence between inputs and outputs?) • Time efficiency (How fast does it run?) • Space efficiency (How much extra space does it require?) • Optimality (Is this provably the best possible solution?) • Approaches: • Theoretical analysis • Empirical analysis • Visualisation

  4. input size running time Number of times basic operation is executed execution time for basic operation Theoretical analysis of time efficiency • Time efficiency is analyzed by determining the number of repetitions of the basic operation as a function of input size • Basic operation: the operation that contributes most towards the running time of the algorithm. • T(n) ≈copC(n)

  5. Input size and basic operations

  6. Best-, Average- and Worst-cases • For some algorithms efficiency depends on type of input: • Worst case: W(n) – max over inputs of size n • Best case: B(n) – min over inputs of size n • Average case: A(n) – “avg” over inputs of size n • Number of times the basic operation will be executed on typical input • NOT the average of worst and best case • Expected number of basic operations repetitions considered as a random variable under some assumption about the probability distribution of all possible inputs of size n

  7. Amortized Efficiency • From a data structures perspective the total time of a sequence of operations is important • Real-time apps are an exception • Amortized efficiency: time of an operation averaged over a worst-case sequence of operations • Desire “self-organizing” data structures that adapt to the context of use, e.g. red-black trees

  8. Exercise: Sequential search • Problem: Given a list of n elements and a search key K, find an element equal to K, if any. • Algorithm: Scan the list and compare its successive elements with K until either a matching element is found (successful search) of the list is exhausted (unsuccessful search) • Calculate the: • Worst case? • Best case? • Average case?

  9. Types of Formulas for Operation Counts • Exact formula • e.g., C(n) = n(n-1)/2 • Formula indicating order of growth with specific multiplicative constant • e.g., C(n) ≈ 0.5 n2 • Formula indicating order of growth with unknown multiplicative constant • e.g., C(n) ≈cn2

  10. Order of Growth • Most important: Order of growth within a constant multiple as n∞ • Example: • How much faster will algorithm run on computer that is twice as fast? • How much longer does it take to solve problem of double input size?

  11. Asymptotic Notations • A way of comparing functions that ignores constant factors and small input sizes • O(g(n)): • class of functions f(n) that grow no faster than g(n) • f(n) ≤c g(n) for all n ≥n0 • Θ(g(n)): • class of functions f(n) that grow at same rate as g(n) • c2 g(n) ≤ f(n) ≤ c1 g(n) for all n ≥ n0 • Ω(g(n)): • class of functions f(n) that grow at least as fast as g(n) • f(n) ≥ c g(n) for all n ≥ n0

  12. Big-Oh: t(n)  O(g(n))

  13. Big-Omega: t(n)  Ω(g(n))

  14. Big-Theta: t(n)  Θ(g(n))

  15. Establishing Rate of Growth: using Limits • limn→∞ t(n)/g(n) = • 0 implies order of t(n) < order of g(n) • c implies order of t(n) = order of g(n) • ∞ implies order of t(n) > order of g(n) Example: • 10n vs. 2n2 • limn→∞ 10n/2n2 = 5 limn→∞ 1/n = 0 • Exercises: • n(n+1)/2 vs. n2 • logb n vs. logc n

  16. L’Hôpital’s rule • If limn→∞ f(n) = limn →∞ g(n) = ∞ and the derivatives f ', g'exist, Then limn→∞ f(n) / g(n) = limn →∞ f '(n) / g'(n) • Example: • log n vs. n • limn→∞ log n / n = limn→∞ 1/n log e = log e limn→∞ 1/n = 0 • So, order log n < order n

  17. Establishing Rate of Growth: using Definitions • f(n) is O(g(n)) if order of growth of f(n) ≤ order of growth of g(n) (within constant multiple) • There exist positive constant c and non-negative integer n0 such that • f(n) ≤ c g(n) for every n ≥ n0 • This needs to be mathematically provable • Examples: • 10n is O(2n2) because 10n < 2n2for n > 5 • 5n+20 is O(10n) because 5n+20 < 10n for n > 4

  18. Basic Asymptotic Efficiency Classes

  19. Time Efficiency of Non-recursive Algorithms • Steps in mathematical analysis of nonrecursive algorithms: • Decide on parameter n indicating input size • Identify algorithm’s basic operation • Determine worst, average, and best case for input of size n • Set up summation for C(n) reflecting algorithm’s loop structure • Simplify summation using standard formulas (see Appendix A of textbook)

  20. Examples: Analysing non-recursive algorithms • Matrix multiplication • Multiply two square matrices of order n using dot product of rows and columns • Selection sort • Find smallest element in remainder of list and swap with current element • Insertion sort • Assume sub-list is sorted an insert current element • Mystery Algorithm • Exercise

  21. Matrix Multiplication • n = matrix order • Basic op = multiplication or addition • Best case = worst case = average case

  22. Selection Sort • n = number of list elements • basic op = comparison • best case = worst case = average case

  23. Insertion Sort • n = number of list elements • basic op = comparision • best case != worst case != average case • best case: A[j] > v executed only once on each iteration

  24. Exercise: Mystery Algorithm • What does this algorithm compute? • What is its basic operation? • What is its efficiency class? • Suggest an improvement and analyse your improvement // Input: A matrix A[0..n-1, 0..n-1] of real numbers for i  0 to n - 1 do • for j i to n-1 do if A[i,j]  A[j,i] return false return true

  25. Factorial: a Recursive Function • Recursive Evaluation of n! • Definition: • n ! = 1*2*…*(n-1)*n • Recursive form for n!: • F(n) = F(n-1) * n, F(0) = 1 • Algorithm: • IF n=0 THEN F(n)  1 ELSE F(n) F(n-1) * n • RETURN F(n) • Need to find the number of multiplications M(n) to compute n!

  26. Recurrence Relations • Definition: • An equation or inequality that describes a function in terms of its value on smaller inputs • Recurrence: • The recursive step • E.g., M(n) = M(n-1) + 1 for n > 0 • Initial Condition: • The terminating step • E.g., M(0) = 0 [call stops when n = 0, no mults when n = 0] • Must differentiate the recursive function (factorial calculation) from the recurrence relation (number of basic operations)

  27. Recurrence Solution for n! • Need an analytic solution for M(n) • Solved by Method of Backward Substitution: • M(n) = M(n-1) + 1 • Substitute M(n-1) = M(n-2) + 1 • = [M(n-2) + 1] +1 = M(n-2) + 2 • Substitute M(n-2) = M(n-3) + 1 • = [M(n-3) + 1] + 2 = M(n-3) + 3 • Pattern: M(n) = M(n-i) + i • Ultimately: M(n) = M(n-n)+n = M(0) + n = n

  28. Plan for Analysis of Recursive Algorithms • Steps in mathematical analysis of recursive algorithms: • Decide on parameter n indicating input size • Identify algorithm’s basic operation • Determine worst, average, and best case for input of size n • (a) Set up a recurrence relation and initial condition(s) for C(n)-the number of times the basic operation will be executed for an input of size n OR (b) Alternatively, count recursive calls • (a) Solve the recurrence to obtain a closed form OR (b) Estimate the order of magnitude of the solution (see Appendix B)

  29. Iterative Methods for Solving Recurrences • Method of Forward Substitution: • Starting from the initial condition generate the first few terms • Look for a pattern expressible as a closed formula • Check validity by direct substitution or induction • Limited because pattern is hard to spot • Method of Backward Substitution: • Express x(n-1) successively as a function of x(n-2), x(n-3), … • Derive x(n) as a function of x(n-i) • Substitute n-i = base condition • Surprisingly successful

  30. Templates for Solving Recurrences • Decrease by One Recurrence • One (constant) operation reduces problem size by one • Examples: Factorial • T(n) = T(n-1) + c T(1) = d • Solution: T(n) = (n-1)c + d • Linear Order • A pass through input reduces problem size by one • Examples: Insertion Sort • T(n) = T(n-1) + cn T(1) = d • Solution: T(n) = [n(n+1)/2 – 1] c + d • Quadratic Order

  31. More Templates for Solving Recurrences • Decrease by a Constant Factor • One (constant) operation reduces problem size by half • Example: binary search • T(n) = T(n/2) + c T(1) = d • Solution: T(n) = c lg n + d • Logarithmic Order • A pass through input reduces problem size by half • Example: Merge Sort • T(n) = 2T(n/2) + cn T(1) = d • Solution: T(n) = cn lg n + d n • n log n order

  32. Master Method for Solving Recurrences • Divide and Conquer Style Recurrence • T(n) = aT(n/b) + f (n)where • f (n).(nk)is the time spent in dividing and merging • a >= 1, b >= 2 • a < bk T(n) .(nk) • a = bk T(n) .(nk lg n ) • a > bk T(n) .(nlog b a) • Note: the same results hold with O instead of .

  33. Example: Fibonacci Numbers • The Fibonacci sequence: • 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … • Describes the growth pattern of Rabbits. Just short of exponential, ask Australia! • Fibonacci recurrence: • F(n) = F(n-1) + F(n-2) • F(0) = 0 • F(1) = 1 • Another example: • A(n) = 3A(n-1) - 2A(n-2) A(0) = 1 A(1) = 3 • 2ndorder linear homogeneous recurrence relation • with constant coefficients

  34. n F(n-1) F(n) 0 1 = F(n) F(n+1) 1 1 Computing Fibonnaci Numbers Algorithm Alternatives: • Definition based recursive algorithm • Nonrecursive brute-force algorithm • Explicit formula algorithm F(n) = 1/√5 n where is the golden ratio (1 + √5) / 2 • Logarithmic algorithm based on formula: • for n≥1, assuming an efficient way of computing matrix powers.

  35. Empirical Analysis of Time Efficiency • Sometimes a mathematical analysis is difficult for even simple algorithms (limited applicability) • Alternative is to measure algorithm execution • PLAN: • Understand the experiment’s purpose • Are we checking the accuracy of a theoretical result, comparing efficiency of different algorithms or implementations, hypothesising the efficiency class? • Decide on an efficiency measure • Use physical units of time (e.g., milliseconds) • OR • Count actual number of basic operations

  36. Plan for Empirical Analysis • Decide on Characteristics of the Input Sample • Can choose Pseudorandom inputs, Patterned inputs or a Combination • Certain problems already have benchmark inputs • Implement the Algorithm • Generate a Sample Set of Inputs • Run the Algorithm and Record the Empirical Results • Analyze the Data • Regression Analysis is often used to fit a function to the scatterplot of results

  37. Algorithm Visualisation • Definition: • Algorithm Visualisation seeks to convey useful algorithm information through the use of images • Flavours: • Static algorithm visualisation • Algorithm animation • Some new insights: e.g., odd and even disks in Towers of Hanoi • Attributes of Information Visualisation apply - zoom and filter, detail on demand, etc. • Websource: • The Complete Collection of Algorithm Animation • http://www.cs.hope.edu/~alganim/ccaa/

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