Algorithms. Algorithms. Algorithms 1.4 ANALYSIS OF ALGORITHMS. introduction. introduction observations. observations

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1 Algorithms ROBERT SEDGEWICK KEVI WAYE.4 AALYSIS OF ALGORITHMS.4 AALYSIS OF ALGORITHMS introduction introduction observations observations Algorithms F O U R T H E D I T I O mathematical models order-of-growth classifications Algorithms mathematical models order-of-growth classifications ROBERT SEDGEWICK KEVI WAYE memory ROBERT SEDGEWICK KEVI WAYE memory Cast of characters Running time Programmer needs to develop a working solution. 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) Client wants to solve problem efficiently. Student might play any or all of these roles someday. how many times do you have to turn the crank? Theoretician wants to understand. Analytic Engine 3 4

Reasons to analyze algorithms Some algorithmic successes Predict performance.

this course (COS 6) Discrete Fourier transform.

Applications: DVD, JPEG, MRI, astrophysics,. Brute force: steps.

performance characteristics 6T 8T size linearithmic linear K K 4K 8K 5 6 Some

Simulate gravitational interactions among bodies. Brute force: steps.

2 Reasons to analyze algorithms Some algorithmic successes Predict performance. Compare algorithms. Provide guarantees. this course (COS 6) 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 (COS 43) time 64T quadratic Primary practical reason: avoid performance bugs. 3T client gets poor performance because programmer did not understand performance characteristics 6T 8T size linearithmic linear K K 4K 8K 5 6 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 8T size linearithmic linear K K 4K 8K Insight. [Knuth 970s] Use scientific method to understand performance. 7 8

Scientific method applied to analysis of algorithms A framework for predicting performance and comparing algorithms. Scientific method. Observe some feature of the natural world.

Validate by repeating until the hypothesis and observations agree. Principles. Experiments must be reproducible. Hypotheses must be falsifiable. Algorithms ROBERT SEDGEWICK KEVI WAYE http://algs4.cs.

9 Example: 3-SUM 3-SUM: brute-force algorithm 3-SUM. Given distinct integers, how many triples sum to exactly zero? % more 8ints.txt 8 30-40 -0-0 40 0 0 5 % java ThreeSum 8ints.

3 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 memory Feature of the natural world. Computer itself. 9 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; 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) return count; check each triple for simplicity, ignore integer overflow Context. Deeply related to problems in computational geometry. public static void main(string[] args) In in = new In(args[0]); int[] a = in.readallints(); StdOut.println(count(a));

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.

jar ) create a new stopwatch double elapsedtime() time since creation (in seconds) public static void main(string[] args) In in = new In(args[0]); int[] a = in.

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) In in = new In(args[0]); int[] a = in.readallints(); Stopwatch stopwatch = new Stopwatch(); StdOut.println(ThreeSum.count(a)); client code double time = stopwatch.elapsedtime(); StdOut.println("elapsed time " + time); 4 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? 5 6

5 Data analysis Data analysis Standard plot. Plot running time T () vs. input size. Log-log plot. Plot running time T () vs. input size using log-log scale. standard plot running time T() log-log plot lg(t()) 3 orders of magnitude straight line of slope 3 lg(t ()) = b lg + c b =.999 c = T () = a b, where a = c. 0 K K 4K 8K lg K K 4K 8K problem size Regression. Fit straight line through data points: a b. power law slope Hypothesis. The running time is about seconds. 7 8 Prediction and validation Doubling hypothesis Hypothesis. The running time is about seconds. Doubling hypothesis. Quick way to estimate b in a power-law relationship. Predictions. 5.0 seconds for = 8, seconds for = 6,000. "order of growth" of running time is about 3 [stay tuned] Run program, doubling the size of the input. time (seconds) ratio lg ratio T () T () = a()b a b = b Observations. time (seconds), , , , , , lg (6.4 / 0.8) = 3.0 8, , validates hypothesis! seems to converge to a constant b 3 Hypothesis. Running time is about a b with b = lg ratio. Caveat. Cannot identify logarithmic factors with doubling hypothesis. 9 0

Doubling hypothesis Experimental algorithmics Doubling hypothesis.

Run the program (for a sufficient large value of ) and solve for a. time (seconds) 8,000 5. 5. = a 8000 3 8,000 5.

Hardware: CPU, memory, cache, Software: compiler, interpreter, garbage collector, determines exponent b in power

Running time is about 0.998 0 0 3 seconds. Bad news. Difficult to get precise measurements. Good news.

g., can run huge number of experiments Mathematical models for running time.

eed to analyze program to determine set of operations. Cost depends on machine, compiler.

6 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 Hypothesis. Running time is about seconds. Bad news. Difficult to get precise measurements. Good news. Much easier and cheaper than other sciences. almost identical hypothesis to one obtained via linear regression e.g., can run huge number of experiments 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 memory Donald Knuth 974 Turing Award In principle, accurate mathematical models are available. 4

7 Cost of basic operations Cost of basic operations Challenge. How to estimate constants. Observation. Most primitive operations take constant time. operation example nanoseconds integer add a + b. integer multiply a * b.4 integer divide a / b 5.4 floating-point add a + b 4.6 floating-point multiply a * b 4. floating-point divide a / b 3.5 sine Math.sin(theta) 9.3 operation example nanoseconds variable declaration int a c assignment statement a = b c integer compare a < b c 3 array element access a[i] c 4 array length a.length c 5 D array allocation new int[] c 6 D array allocation new int[][] c 7 arctangent Math.atan(y, x) Running OS X on Macbook Pro.GHz with GB RAM Caveat. on-primitive operations often take more than constant time. 5 novice mistake: abusive string concatenation 6 Example: -SUM Example: -SUM Q. How many instructions as a function of input size? Q. How many instructions as a function of input size? for (int i = 0; i < ; i++) if (a[i] == 0) for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) operation frequency array accesses Pf. [ n even] ( ) = ( ) = variable declaration assignment statement less than compare + equal to compare array access increment to ( ) = half of square half of diagonal 8

String theory infinite sum Example: -SUM ++3+4+...= Q. How many instructions as a function of input size?

..+( ) = ( ) = variable declaration + assignment statement + less than compare ½ ( + ) ( + ) equal to compare ½ ( ) array access ( ) tedious to count exactly http://www.nytimes.

html increment ½ ( ) to ( ) 9 30 Simplifying the calculations Simplification : cost model It is convenient to have a measure of the amount of work involved in a computing process, even though it be a

8 String theory infinite sum Example: -SUM = Q. How many instructions as a function of input size? for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) operation frequency ( ) = ( ) = variable declaration + assignment statement + less than compare ½ ( + ) ( + ) equal to compare ½ ( ) array access ( ) tedious to count exactly increment ½ ( ) to ( ) 9 30 Simplifying the calculations Simplification : cost model 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 Cost model. Use some basic operation as a proxy for running time. for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) operation frequency ( ) = ( ) = 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. variable declaration + assignment statement + 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!) 3 3

9 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 3 /6 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 66,666,667 3 /6 / + /3 Ex 3. ⅙ 3 - ½ + ⅓! ~ ⅙ 3 66,67,000 discard lower-order terms (e.g., = 000: million vs million),000 Leading-term approximation operation frequency tilde notation variable declaration + ~ assignment statement + ~ less than compare ½ ( + ) ( + ) ~ ½ equal to compare ½ ( ) ~ ½ Technical definition. f() ~ g() means lim f () g() = array access ( ) ~ increment ½ ( ) to ( ) ~ ½ to ~ Example: -SUM Example: 3-SUM Q. Approximately how many array accesses as a function of input size? Q. Approximately how many array accesses as a function of input size? for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) if (a[i] + a[j] == 0) "inner loop" 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) "inner loop" A. ~ array accesses ( ) = ( ) = A. ~ ½ 3 array accesses. 3 = ( )( ) 3! 6 3 Bottom line. Use cost model and tilde notation to simplify counts. Bottom line. Use cost model and tilde notation to simplify counts

10 Diversion: estimating a discrete sum Estimating a discrete sum Q. How to estimate a discrete sum? A. Take a discrete mathematics course (COS 340). A. Replace the sum with an integral, and use calculus! Q. How to estimate a discrete sum? A. Take a discrete mathematics course (COS 340). A. Replace the sum with an integral, and use calculus! Ex Ex. k + k + + k. Ex 3. + / + /3 + + /. i= i= i= i i k i x= x= x= xdx x k dx dx = ln x k + k+ Ex 4. + ½ + ¼ + ⅛ + i=0 x=0 i = x dx = ln.447 Ex 4. 3-sum triple loop. i= j=i k=j x= y=x z=y dz dy dx 6 3 Caveat. Integral trick doesn't always work! Estimating a discrete sum Mathematical models for running time Q. How to estimate a discrete sum? A3. Use Maple or Wolfram Alpha. In principle, accurate mathematical models are available. In practice, Formulas can be complicated. Advanced mathematics might be required. Exact models best left for experts. costs (depend on machine, compiler) 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 =..)); 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) ( - ) ( - ) Bottom line. We use approximate models in this course: T() ~ c

11 Analysis of algorithms quiz How many array accesses does the following code fragment make as a function of? A. ~ 3 B. ~ 3/ lg C. ~ 3/ 3 D. ~ 3 3 E. I don't know. for (int i = 0; i < ; i++) for (int j = i+; j < ; j++) for (int k = ; k < ; k = k*) if (a[i] + a[j] >= a[k]) Algorithms ROBERT SEDGEWICK KEVI WAYE AALYSIS OF ALGORITHMS introduction observations mathematical models order-of-growth classifications memory 4 Common order-of-growth classifications Common order-of-growth classifications Definition. If f () ~ c g() for some constant c > 0, then the order of growth of f () is g(). Ignores leading coefficient. Ignores lower-order terms. Ex. The order of growth of the running time of this code is 3. 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) Good news. The set of functions, log,, log,, 3, and suffices to describe the order of growth of most common algorithms. log-log plot time 5T 64T 8T 4T exponential cubic quadratic linearithmic linear Typical usage. With running times. where leading coefficient depends on machine, compiler, JVM, T T K logarithmic constant K 4K 8K 5K size Typical orders of growth 44

Common order-of-growth classifications Binary search order of growth name typical code framework description example T() / T() Goal. Given a sorted array and a key, find index of the key in the array?

12 Common order-of-growth classifications Binary search order of growth name typical code framework description example T() / T() Goal. Given a sorted array and a key, find index of the key in the array? constant a = b + c; statement log logarithmic linear while ( > ) = /;... for (int i = 0; i < ; i++)... add two numbers divide in half binary search ~ loop find the maximum Binary search. Compare key against middle entry. Too small, go left. Too big, go right. Equal, found. log linearithmic [see mergesort lecture] divide and conquer mergesort ~ quadratic for (int i = 0; i < ; i++) for (int j = 0; j < ; j++)... double loop check all pairs cubic for (int i = 0; i < ; i++) for (int j = 0; j < ; j++) for (int k = 0; k < ; k++)... triple loop check all triples exponential [see combinatorial search lecture] exhaustive search check all subsets T() Binary search: implementation Binary search: Java implementation Trivial to implement? First binary search published in 946. First bug-free one in 96. Bug in Java's Arrays.binarySearch() discovered in 006. Invariant. If key appears in array a[], then a[lo] key a[hi]. public static int binarysearch(int[] a, int key) int lo = 0, hi = a.length - ; while (lo <= hi) why not mid = (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"

Binary search: mathematical analysis TECHICAL ITERVIEW QUESTIOS Proposition.

size. Def. T () = # key compares to binary search a sorted subarray of size.

left or right half (floored division) Pf sketch.

of 3-way) T () T ( / ) + [ given ] T ( / 4) + + [ apply recurrence to first

stop applying, T() = ] = + lg lg 49 50 WHY ARE MAHOLE COVERS ROUD?

Given distinct integers, find three such that a + b + c = 0. Version 0.

13 Binary search: mathematical analysis TECHICAL ITERVIEW QUESTIOS 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. Binary search recurrence. T () T ( / ) + for >, with T () =. left or right half (floored division) Pf sketch. [assume is a power of ] possible to implement with one -way compare (instead of 3-way) T () T ( / ) + [ given ] T ( / 4) + + [ apply recurrence to first term ] T ( / 8) [ apply recurrence to first term ] T ( / ) [ stop applying, T() = ] = + lg lg WHY ARE MAHOLE COVERS ROUD? THE 3-SUM PROBLEM 3-SUM. Given distinct integers, find three such that a + b + c = 0. Version 0. 3 time, space. Version. log time, space. Version. time, space. ote. For full credit, running time should be worst case. ew York, ew York Geneva, Switzerland Zermatt, Switzerland 5 5

An log algorithm for 3-SUM Comparing programs Algorithm. Step : Sort the (distinct) numbers. Step : For each pair of numbers a[i] and a[j], binary search for -(a[i] + a[j]).

time (seconds) time (seconds) binary search,000 0.,000 0.4 Analysis. Order of growth is log. Step : with insertion sort. Step : log with binary search.

14 An log algorithm for 3-SUM Comparing programs 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 Hypothesis. The sorting-based log algorithm for 3-SUM is significantly faster in practice than the brute-force 3 algorithm. time (seconds) time (seconds) binary search,000 0., Analysis. Order of growth is log. Step : with insertion sort. Step : log with binary search. (-40, -0) 60 (-40, -0) 50 (-40, 0) 40 (-40, 5) 35 (-40, 0) 30, , , ThreeSum.java, , , , Remark. Can achieve by modifying binary search step. (-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 3, , ThreeSumDeluxe.java Guiding principle. Typically, better order of growth faster in practice Basics.4 AALYSIS OF ALGORITHMS Bit. 0 or. Byte. 8 bits. 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 memory 64-bit machine. We assume a 64-bit machine with 8-byte pointers. Can address more memory. some Pointers use more space. JVMs "compress" ordinary object pointers to 4 bytes to avoid this cost 56

15 Typical memory usage for primitive types and arrays Typical memory usage for objects in Java type bytes boolean byte char int 4 type bytes char[] + 4 int[] double[] one-dimensional arrays Object overhead. 6 bytes. Reference. 8 bytes. Padding. Each object uses a multiple of 8 bytes. Ex. A Date object uses 3 bytes of memory. float 4 long 8 double 8 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 two-dimensional arrays Typical memory usage summary Memory analysis quiz 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. Padding: round up to multiple of 8 bytes. ote. Depending on application, we may want to count memory for any referenced objects (recursively). + 8 extra bytes per inner class object (for reference to enclosing class) How much memory does a WeightedQuickUnionUF use as a function of? A. ~ 4 bytes B. ~ 8 bytes C. ~ 4 bytes D. ~ 8 bytes E. I don't know 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] = ;

16 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. 6

Algorithms. Algorithms. Algorithms 1.4 ANALYSIS OF ALGORITHMS. introduction. introduction observations. observations

Algorithms. Algorithms. Algorithms 1.4 ANALYSIS OF ALGORITHMS. introduction. introduction observations. observations Algorithms ROBERT SEDGEWICK KEVI WAYE.4 AALYSIS OF ALGORITHMS.4 AALYSIS OF ALGORITHMS introduction introduction observations observations Algorithms F O U R T H E D I T I O mathematical models order-of-growth