Algorithms. Algorithms. Algorithms 1.4 ANALYSIS OF ALGORITHMS

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1 Announcements Algorithms ROBERT SEDGEWICK KEVI WAYE First programming assignment. Due Tuesday at pm. Try electronic submission system today. "Check All Submitted Files. will perform checks on your code You may use this up to 0 times. Can still submit after you use up your checks. Should not be your primary testing technique! Registration. Register for Piazza. Official course announcements are posted there. Register for Coursera (see Piazza post). ROBERT Register for Poll Everywhere (see Piazza post). Algorithms F O U R T H E D I T I O SEDGEWICK KEVI WAYE AALYSIS OF ALGORITHMS introduction observations mathematical models order-of-growth classifications theory of algorithms memory Running time.4 AALYSIS OF ALGORITHMS As soon as an Analytic Engine exists, it will necessarily guide the future course of the science. Whenever any result is sought by its aid, the question will arise By what course of calculation can these results be arrived at by the machine in the shortest time? Charles Babbage (864) Algorithms ROBERT SEDGEWICK KEVI WAYE introduction observations mathematical models order-of-growth classifications theory of algorithms memory how many times do you have to turn the crank? Analytic Engine 4

2 The Life of the Philosopher Motivation. The iron folding-doors of the small-room or oven were opened. Captain Kater and myself entered, and they were closed upon us... The thermometer marked, if I recollect rightly, 65 degrees. The pulse was quickened, and I ought to have to have counted but did not count the number of inspirations per minute. Perspiration commenced immediately and was very copious. We remained, I believe, about five or six minutes without very great discomfort, and I experienced no subsequent inconvenience from the result of the experiment Charles Babbage, From the Life of the Philosopher Our analysis last time was ad hoc. Formalize our analysis. Justify our formalization. algorithm initialize union connected quick-find quick-union weighted QU lg lg includes cost of finding roots 65 Fahrenheit / 30 Celsius public QuickFindUF(int ) id = new int[]; for (int i = 0; i < ; i++) id[i] = i; set id of each object to itself ( array accesses) 5 6 Reasons to analyze algorithms Some algorithmic successes Predict performance. Compare algorithms. Provide guarantees. this course Discrete Fourier transform. Break down waveform of samples into periodic components. Applications: DVD, JPEG, MRI, astrophysics,. Brute force: steps. Friedrich FFT algorithm: log steps, enables new technology. Gauss 805 Understand theoretical basis. theory of algorithms time 64T quadratic... Primary practical reason: avoid performance bugs (e.g. the ////// exploit). 3T 9 client gets poor performance because programmer did not understand performance characteristics 6T size 8T linearithmic linear K K 4K 8K 7 8

3 Some algorithmic successes The challenge -body simulation. Simulate gravitational interactions among bodies. Brute force: steps. Barnes-Hut algorithm: log steps, enables new research. Andrew Appel PU '8 Q. Will my program be able to solve a large practical input? Why is my program so slow? Why does it run out of memory? time 64T quadratic 3T 6T size 8T linearithmic linear K K 4K 8K Insight. [Knuth 970s] Use scientific method to understand performance. 9 0 Scientific method applied to analysis of algorithms A framework for predicting performance and comparing algorithms. Scientific method. Observe some feature of the natural world. Hypothesize a model that is consistent with the observations. Predict events using the hypothesis. Verify the predictions by making further observations. Validate by repeating until the hypothesis and observations agree. Principles. Experiments must be reproducible. Hypotheses must be falsifiable. Algorithms ROBERT SEDGEWICK KEVI WAYE AALYSIS OF ALGORITHMS introduction observations mathematical models order-of-growth classifications theory of algorithms memory Feature of the natural world. Computer itself.

4 Example: 3-SUM 3-SUM: brute-force algorithm 3-SUM. Given distinct integers, how many triples sum to exactly zero? % more 8ints.txt % java ThreeSum 8ints.txt a[i] a[j] a[k] sum public class ThreeSum public static int count(int[] a) int = a.length; int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) for (int k = j+; k < ; k++) if (a[i] + a[j] + a[k] == 0) count++; return count; check each triple for simplicity, ignore integer overflow Context. Deeply related to problems in computational geometry. public static void main(string[] args) int[] a = In.readInts(args[0]); StdOut.println(count(a)); 3 4 Measuring the running time Measuring the running time Q. How to time a program? A. Manual. % java ThreeSum Kints.txt tick tick tick Q. How to time a program? A. Automatic. 70 % java ThreeSum Kints.txt 58 % java ThreeSum 4Kints.txt public class Stopwatch Stopwatch() (part of stdlib.jar ) create a new stopwatch double elapsedtime() time since creation (in seconds) public static void main(string[] args) int[] a = In.readInts(args[0]); Stopwatch stopwatch = new Stopwatch(); StdOut.println(ThreeSum.count(a)); double time = stopwatch.elapsedtime(); client code 6

5 Empirical analysis Empirical analysis Run the program for various input sizes and measure running time. Run the program for various input sizes and measure running time. time (seconds) ,000 0., , , ,000? 7 8 Data analysis Data analysis Standard plot. Plot running time T () vs. input size. Hard to form a useful hypothesis. standard plot running time T() Log-log plot. Plot running time T () vs. input size using log-log scale. log-log plot 3 orders of magnitude lg(t()) straight line of slope 3 T() = a b power law lg(t ()) = b lg + lg a lg(t ()) = b lg + c b =.999 c = a = K K 4K 8K lg K K 4K 8K problem size Regression. Fit straight line through data points: lg(t()) y = b x + = cb lg() + c Interpretation. T() = a b, where a = c Hypothesis. The running time is about seconds. slope 9 0

6 Prediction and validation Doubling hypothesis Hypothesis. The running time is about seconds. Predictions. 5.0 seconds for = 8, seconds for = 6,000. "order of growth" of running time is about 3 [stay tuned] Doubling hypothesis. Another way to build models of the form T() = a b Run program, doubling the size of the input. time (seconds) ratio lg ratio Observations. time (seconds) 8, , ,000 5., , , , lg (5. / 6.40) = 3.0 6, validates hypothesis! Hypothesis. Running time is about a b with b = lg ratio. seems to converge to a constant b 3 Doubling hypothesis Experimental algorithmics Doubling hypothesis. Quick way to estimate b in a power-law relationship. Q. How to estimate a (assuming we know b)? A. Run the program (for a sufficient large value of ) and solve for a. time (seconds) 8, = a , a = System independent effects. Algorithm. Input data. System dependent effects. Hardware: CPU, memory, cache, Software: compiler, interpreter, garbage collector, determines exponent b in power law System: operating system, network, other apps, determines constant a in power law 8, Caveat. In some cases, b can depend on system (e.g. virtualization) Bad news. Difficult to get precise measurements. Hypothesis. Running time is about seconds. Good news. Much easier and cheaper than other sciences. almost identical hypothesis to one obtained via linear regression 3 e.g., can run huge number of experiments 4

7 Mathematical models for running time.4 AALYSIS OF ALGORITHMS Total running time: sum of cost frequency for all operations. eed to analyze program to determine set of operations. Cost depends on machine, compiler. Frequency depends on algorithm, input data. Algorithms ROBERT SEDGEWICK KEVI WAYE introduction observations mathematical models order-of-growth classifications theory of algorithms memory Donald Knuth 974 Turing Award In principle, accurate mathematical models are available. 6 Cost of basic operations Cost of basic operations operation example nanoseconds integer add a + b. operation example nanoseconds integer multiply a * b.4 variable declaration int a c integer divide a / b 5.4 assignment statement a = b c floating-point add a + b 4.6 integer compare a < b c3 floating-point multiply a * b 4. array element access a[i] c4 floating-point divide a / b 3.5 sine Math.sin(theta) 9.3 arctangent Math.atan(y, x) 9.0 array length a.length c5 D array allocation new int[] c6 D array allocation new int[][] c string length s.length() c8 substring extraction s.substring(/, ) c9 Running OS X on Macbook Pro.GHz with GB RAM string concatenation s + t c0 Computer Architecture Caveats (see COS 475). Most computers are more like assembly lines than oracles (pipelining). ovice Register vs. cache vs. RAM vs. hard disk (Java is a high level language) 7 mistake. Abusive string concatenation. 8

8 Example: -SUM Example: -SUM Q. How many instructions as a function of input size? Q. How many instructions as a function of input size? int count = 0; for (int i = 0; i < ; i++) if (a[i] == 0) count++; int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) count++; array accesses operation frequency Frequency, =0000 variable declaration assignment statement less than compare equal to compare 0000 Pf ( ) = ( ) = array access 0000 increment to 0000 to ( ) = half of half of 30 Example: -SUM Simplifying the calculations Q. How many instructions as a function of input size? int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) count++; operation frequency ( ) = ( ) = It is convenient to have a measure of the amount of work involved in a computing process, even though it be a very crude one. We may count up the number of times that various elementary operations are applied in the whole process and then given them various weights. We might, for instance, count the number of additions, subtractions, multiplications, divisions, recording of numbers, and extractions of figures from tables. In the case of computing with matrices most of the work consists of multiplications and writing down numbers, and we shall therefore only attempt to count the number of multiplications and recordings. Alan Turing variable declaration + assignment statement + less than compare ½ ( + ) ( + ) equal to compare ½ ( ) array access ( ) increment ½ ( ) to ( ) tedious to count exactly ROUDIG-OFF ERRORS I MATRIX PROCESSES By A. M. TURIG ational Physical Laboratory, Teddington, Middlesex) [Received 4 ovember 947] SUMMARY A number of methods of solving sets of linear equations and inverting matrices are discussed. The theory of the rounding-off errors involved is investigated for some of the methods. In all cases examined, including the well-known 'Gauss elimination process', it is found that the errors are normally quite moderate: no exponential build-up need occur. 3 3

9 Simplification : cost model Cost model. Use some basic operation as a proxy for running time. int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) count++; operation frequency variable declaration + assignment statement ( ) = ( ) = Simplification : tilde notation Estimate running time (or memory) as a function of input size. Ignore lower order terms. when is large, terms are negligible when is small, we don't care Ex. ⅙ ! ~ ⅙ 3 Ex. ⅙ /3 + 56! ~ ⅙ 3 Ex 3. ⅙ 3 - ½ + ⅓! ~ ⅙ 3 discard lower-order terms (e.g., = 000: million vs million) 66,666,667,000 3 /6 3 /6 / + /3 66,67,000 Leading-term approximation less than compare ½ ( + ) ( + ) equal to compare ½ ( ) array access ( ) increment ½ ( ) to ( ) cost model = array accesses (we assume compiler/jvm do not optimize any array accesses away!) Technical definition. f() ~ g() means f () lim g() = Simplification : tilde notation Estimate running time (or memory) as a function of input size. Ignore lower order terms. when is large, terms are negligible when is small, we don't care Example: -SUM Q. Approximately how many array accesses as a function of input size? int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) count++; "inner loop" operation frequency tilde notation variable declaration + ~ assignment statement + ~ less than compare ½ ( + ) ( + ) ~ ½ equal to compare ½ ( ) ~ ½ array access ( ) ~ A. ~ array accesses. Because (½ ) = ( ) = ( ) = increment ½ ( ) to ( ) ~ ½ to ~ Bottom line. Use cost model and tilde notation to simplify counts

10 Example: 3-SUM Estimating a discrete sum Q. Approximately how many array accesses as a function of input size? int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) for (int k = j+; k < ; k++) if (a[i] + a[j] + a[k] == 0) count++; "inner loop" Q. How to estimate a discrete sum? A. Take discrete mathematics course. A. Replace the sum with an integral, and use calculus! Ex i= i x= xdx A. ~ ½ 3 array accesses. Because (3/6 3 ) = ½ 3 3 = ( )( ) 3! 6 3 Ex. k + k + + k. Ex 3. + / + /3 + + /. i= i= i k i x= x= x k dx dx x = ln k + k+ Bottom line. Use cost model and tilde notation to simplify counts. Ex 4. 3-sum triple loop. i= j=i k=j x= y=x z=y dz dy dx Estimating a discrete sum Estimating a discrete sum Q. How to estimate a discrete sum? A. Take discrete mathematics course. A. Replace the sum with an integral, and use calculus! Q. How to estimate a discrete sum? A3. Use Maple or Wolfram Alpha. Ex 4. + ½ + ¼ + ⅛ + i=0 i = x=0 x dx = ln.447 wolframalpha.com [wayne:nobel.princeton.edu] > maple5 \^/ Maple 5 (X86 64 LIUX)._ \ / _. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 0 \ MAPLE / All rights reserved. Maple is a trademark of < > Waterloo Maple Inc. Type? for help. > factor(sum(sum(sum(, k=j+..), j = i+..), i =..)); Caveat. Integral trick doesn't always work! ( - ) ( - )

11 Mathematical models for running time In principle, accurate mathematical models are available. In practice, Formulas can be complicated. Realities of hardware impact accuracy of formulas. Advanced mathematics might be required. Exact models best left for experts. costs (depend on machine, compiler) T = c A + c B + c3 C + c4 D + c5 E A = array access B = integer add C = integer compare D = increment E = variable assignment frequencies (depend on algorithm, input) Algorithms ROBERT SEDGEWICK KEVI WAYE AALYSIS OF ALGORITHMS introduction observations mathematical models order-of-growth classifications theory of algorithms memory Bottom line. We use approximate models in this course: T() ~ c 3. 4 Order-of-growth Definition. If f() ~ a g(), then the order-of-growth of f() is just g() Example: Runtime of 3SUM: ~ /6 t 3 [see page 8] Order-of-growth of the runtime of 3SUM: 3 We often say order-of-growth of 3SUM as shorthand for the runtime. Common order-of-growth classifications Good news. the small set of functions, log,, log,, 3, and suffices to describe order-of-growth of typical algorithms. log-log plot 5T exponential cubic quadratic order of growth discards leading coefficient linearithmic linear 64T time int count = 0; for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) for (int k = j+; k < ; k++) if (a[i] + a[j] + a[k] == 0) count++; Time to execute: t 8T 4T T T logarithmic constant K K 4K 8K 5K size Typical orders of growth 43 44

12 Common order-of-growth classifications Practical implications of order-of-growth order of growth name typical code framework description example T() / T() constant a = b[0] + b[]; statement log logarithmic linear while ( > ) = / ;... for (int i = 0; i < ; i++)... log linearithmic [see mergesort lecture] quadratic for (int i = 0; i < ; i++) for (int j = 0; j < ; j++)... add two array elements divide in half binary search ~ loop divide and conquer double loop find the maximum mergesort ~ check all pairs 4 problem size solvable in minutes growth rate 970s 980s 990s 000s any any any any log any any any any millions tens of hundreds of millions millions billions log hundreds of hundreds of millions millions thousands millions hundreds thousand thousands tens of thousands 3 cubic for (int i = 0; i < ; i++) for (int j = 0; j < ; j++) for (int k = 0; k < ; k++)... triple loop check all triples 8 3 hundred hundreds thousand thousands 0 0s 0s 30 exponential [see combinatorial search lecture] exhaustive search check all subsets T() Bottom line. eed linear or linearithmic alg to keep pace with Moore's law Binary search demo Binary search: Java implementation Goal. Given a sorted array and a key, find index of the key in the array? Binary search. Compare key against middle entry. Too small, go left. Too big, go right. lo Equal, found. successful search for hi Trivial to implement? First binary search published in 946; first bug-free one in 96. Bug in Java's Arrays.binarySearch() discovered in 006. public static int binarysearch(int[] a, int key) int lo = 0, hi = a.length-; while (lo <= hi) int mid = lo + (hi - lo) / ; if (key < a[mid]) hi = mid - ; else if (key > a[mid]) lo = mid + ; else return mid; return -; one "3-way compare" Invariant. If key appears in the array a[], then a[lo] key a[hi]

13 Binary search: mathematical analysis An log algorithm for 3-SUM Proposition. Binary search uses at most + lg key compares to search in a sorted array of size. Def. T () # key compares to binary search a sorted subarray of size. Sorting-based algorithm. Step : Sort the (distinct) numbers. Step : For each pair of numbers a[i] and a[j], binary search for -(a[i] + a[j]). input sort Binary search recurrence. T () T ( / ) + for >, with T () =. Pf sketch. T () T ( / ) + T ( / 4) + + T ( / 8) T ( / ) = + lg left or right half possible to implement with one -way compare (instead of 3-way) given apply recurrence to first term apply recurrence to first term stop applying, T() = Analysis. Order of growth is log. Step : with insertion sort. Step : log with binary search. binary searches, each log Remark. Can achieve by modifying binary search step. binary search (-40, -0) 60 (-40, -0) 50 (-40, 0) 40 (-40, 5) 35 (-40, 0) 30 (-40, 40) 0 (-0, -0) 30 (-0, 0) 0 ( 0, 30) -40 ( 0, 40) -50 ( 30, 40) -70 only count if a[i] < a[j] < a[k] to avoid double counting Comparing programs Hypothesis. The sorting-based log algorithm for 3-SUM is significantly faster in practice than the brute-force 3 algorithm. time (seconds), time (seconds), AALYSIS OF ALGORITHMS, , , ThreeSum.java, , , , , , Algorithms ROBERT SEDGEWICK KEVI WAYE introduction observations mathematical models order-of-growth classifications theory of algorithms memory ThreeSumDeluxe.java Guiding principle. Typically, better order of growth faster in practice. 5

14 Types of analyses: Performance depends on input Types of analyses Best case. Lower bound on cost. Determined by easiest input. Provides a goal for all inputs. Worst case. Upper bound on cost. Determined by most difficult input. Provides a guarantee for all inputs. Average case. Expected cost for random input. eed a model for random input. Provides a way to predict performance. Best case. Lower bound on cost. Worst case. Upper bound on cost (guarantee). Average case. Expected cost. Primary practical reason: avoid performance bugs (e.g. the ////// exploit). Actual data might not match input model? eed to understand input to effectively process it. Approach : depend on worst case guarantee. Example: Use Mergesort instead of Quicksort Approach : randomize, depend on probabilistic guarantee. Example: Quicksort Ex. Compares for binary search. Best: ~ Average: ~ lg Worst: ~ lg Ex. Array accesses for brute-force 3-SUM. Best: ~ ½ 3 Average: ~ ½ 3 Worst: ~ ½ Theory of algorithms -- probing the worst case Theory of algorithms -- probing the worst case ew Goals. Establish difficulty of a problem, e.g. how hard is 3SUM? Develop optimal algorithm. Approach: Use order-of-growth in worst case Analysis is asymptotic, i.e. for very large Suppress details in analysis: analysis is to within a constant factor. Use order-of-growth instead of tilde notation Eliminate variability in input model by focusing on the worst case Testing optimality of algorithm A for problem P Find worst case order of growth guarantee for algorithm A, g() Find best possible guarantee for any algorithm that solves P, h() If they match, i.e. g() = h(), then: Worst case performance of A is asymptotically optimal Optimal algorithm for P has order of growth g() If they don t, g() at least provides an upper bound Example: The -SUM problem (how many 0s?) Let A be the brute force algorithm where we simply look at each entry and count the zeros Order of growth: g() = Of any algorithm that solves P, order of growth can be no better than Order of growth: h() = g() = h(). A is optimal! 55 56

15 Commonly-used notations in the theory of algorithms Theory of algorithms: example notation provides example shorthand for used to Goals. Establish difficulty of a problem and develop optimal algorithms. Ex. -SUM = Is there a 0 in the array? Big Theta asymptotic order of growth Θ( ) ½ log classify algorithms Upper bound: O(g() ). Ex. Brute-force algorithm for -SUM: Look at every array entry. Running time of the brute-force algorithm is Θ( ). Running time of the unknown optimal algorithm for -SUM is O( ). g() Big Oh Θ( ) and smaller O( ) Big Omega Θ( ) and larger Ω( ) 00 log + 3 ½ log + 3 develop upper bounds develop lower bounds Lower bound Ω(h() ). Proof that no algorithm can do better than Θ(h() ). Ex. Have to examine all entries (any unexamined one might be 0). Running time of the unknown optimal algorithm for -SUM is Ω( ). Optimal algorithm. Lower bound equals upper bound: g() = h() = Ex. Brute-force algorithm for -SUM is optimal: its running time is Θ( ). h() Theory of algorithms: example Theory of algorithms: example Goals. Establish difficulty of a problem and develop optimal algorithms. Ex. 3-SUM. Upper bound. A specific algorithm. Ex. Brute-force algorithm for 3-SUM. Running time of the optimal algorithm for 3-SUM is O( 3 ). Goals. Establish difficulty of a problem and develop optimal algorithms. Ex. 3-SUM. Upper bound. A specific algorithm. Ex. Improved algorithm for 3-SUM. Running time of the optimal algorithm for 3-SUM is O( log ). Lower bound. Proof that no algorithm can do better. Ex. Have to examine all entries to solve 3-SUM. Running time of the optimal algorithm for solving 3-SUM is Ω( ). 59 Open problems. Optimal algorithm for 3-SUM? Subquadratic algorithm for 3-SUM? Quadratic lower bound for 3-SUM? 60

16 Algorithm design approach Commonly-used notations Start. Develop an algorithm. Prove a lower bound. Gap? Lower the upper bound (discover a new algorithm). Raise the lower bound (more difficult). Golden Age of Algorithm Design (970s-). Steadily decreasing upper bounds for many important problems. Many known optimal algorithms. Caveats. Overly pessimistic to focus on worst case? eed better than to within a constant factor to predict performance. Asymptotic performance not always useful (e.g. matrix multiplication) 6 notation provides example shorthand for used to Tilde leading term ~ log 0 Big Theta asymptotic growth rate Θ( ) Big Oh Θ( ) and smaller O( ) Big Omega Θ( ) and larger Ω( ) ½ log log + 3 ½ Common mistake. Interpreting big-oh as an approximate model. This course. Focus on approximate models: use Tilde-notation log + 3 provide approximate model classify algorithms develop upper bounds develop lower bounds 6.4 AALYSIS OF ALGORITHMS Basics Bit. 0 or. Byte. 8 bits. hard drives IST most computer scientists Megabyte (MB). million or 0 bytes. Gigabyte (GB). billion or 30 bytes. Algorithms ROBERT SEDGEWICK KEVI WAYE introduction observations mathematical models order-of-growth classifications theory of algorithms memory 64-bit machine. We assume a 64-bit machine with 8 byte pointers. Can address more memory. Pointers use more space. some JVMs "compress" ordinary object pointers to 4 bytes to avoid this cost 64

17 Typical memory usage for primitive types and arrays Typical memory usage for objects in Java type bytes boolean type bytes char[] + 4 Object overhead. 6 bytes. Reference. 8 bytes. Padding. Each object uses a multiple of 8 bytes. byte char int 4 int[] double[] for one-dimensional arrays Ex. A Date object uses 3 bytes of memory. float 4 long 8 double 8 for primitive types type char[][] int[][] double[][] bytes ~ M ~ 4 M ~ 8 M public class Date private int day; private int month; private int year;... object overhead day month year padding int values 6 bytes (object overhead) 4 bytes (int) 4 bytes (int) 4 bytes (int) 4 bytes (padding) 3 bytes for two-dimensional arrays Typical memory usage summary Example Total memory usage for a data type value: Primitive type: 4 bytes for int, 8 bytes for double, Object reference: 8 bytes. Array: 4 bytes + memory for each array entry. Object: 6 bytes + memory for each instance variable + 8 bytes if inner class (for pointer to enclosing class). Padding: round up to multiple of 8 bytes. Shallow memory usage: Don't count referenced objects. Deep memory usage: If array entry or instance variable is a reference, add memory (recursively) for referenced object. Q. How much memory does WeightedQuickUnionUF use as a function of? Use tilde notation to simplify your answer. public class WeightedQuickUnionUF private int[] id; private int[] sz; private int count; public WeightedQuickUnionUF(int ) id = new int[]; sz = new int[]; for (int i = 0; i < ; i++) id[i] = i; for (int i = 0; i < ; i++) sz[i] = ;... 6 bytes (object overhead) 8 + (4 + 4) each reference + int[] array 4 bytes (int) 4 bytes (padding) A ~ 8 bytes

18 Turning the crank: summary Empirical analysis. Execute program to perform experiments. Assume power law and formulate a hypothesis for running time. Model enables us to make predictions. Mathematical analysis. Analyze algorithm to count frequency of operations. Use tilde notation to simplify analysis. Model enables us to explain behavior. Scientific method. Mathematical model is independent of a particular system; applies to machines not yet built. Empirical analysis is necessary to validate mathematical models and to make predictions. 69