1.4 Analysis of Algorithms

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1.4 Analysis of Algorithms Cast of characters Programmer needs to develop a working solution. Client wants to solve problem efficiently. Student might play any or all of these roles someday.

[this lecture] Algorithms, 4 th Edition Robert Sedgewick and Kevin Wayne Copyright 2002 2010 February 2, 2011 9:34:36 AM 2 Running time Reasons to analyze algorithms As soon as an Analytic Engine

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?

1 1.4 Analysis of Algorithms Cast of characters Programmer needs to develop a working solution. Client wants to solve problem efficiently. Student might play any or all of these roles someday. observations mathematical models order-of-growth classifications dependencies on inputs memory Theoretician wants to understand. Basic blocking and tackling is sometimes necessary. [this lecture] Algorithms, 4 th Edition Robert Sedgewick and Kevin Wayne Copyright February 2, :34:36 AM 2 Running time Reasons to analyze 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 (1864) Predict performance. Compare algorithms. Provide guarantees. this course (COS 226) Understand theoretical basis. theory of algorithms (COS 423) how many times do you have to turn the crank? Primary practical reason: avoid performance bugs. client gets poor performance because programmer did not understand performance characteristics Analytic Engine 3 4

Some algorithmic successes Some algorithmic successes Discrete Fourier transform.

Applications: DVD, JPEG, MRI, astrophysics,. Brute force: 2 steps.

Simulate gravitational interactions among bodies. Brute force: 2 steps.

Andrew Appel PU '81 time 64T quadratic time 64T quadratic 32T 32T 16T linearithmic 16T

Scientific method applied to analysis of algorithms Q.

A framework for predicting performance and comparing algorithms. Why is my program so slow?

Hypothesize a model that is consistent with the observations.

2 Some algorithmic successes Some algorithmic successes Discrete Fourier transform. Break down waveform of samples into periodic components. Applications: DVD, JPEG, MRI, astrophysics,. Brute force: 2 steps. FFT algorithm: log steps, enables new technology. Friedrich Gauss body simulation. Simulate gravitational interactions among bodies. Brute force: 2 steps. Barnes-Hut algorithm: log steps, enables new research. Andrew Appel PU '81 time 64T quadratic time 64T quadratic 32T 32T 16T linearithmic 16T linearithmic 8T linear 8T linear size 1K 2K 4K 8K size 1K 2K 4K 8K 5 6 The challenge Scientific method applied to analysis of algorithms Q. Will my program be able to solve a large practical input? A framework for predicting performance and comparing algorithms. Why is my program so slow? Why does it run out of memory? 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. Key insight. [Knuth 1970s] Use scientific method to understand performance. Feature of the natural world = computer itself. 7 8

Example: 3-sum 3-sum. Given distinct integers, how many triples sum to exactly zero? % more 8ints.

memory % java ThreeSum < 8ints.txt 4 3-40 40 0 0 4-10 0 10 0 Context. Deeply related to problems in computational geometry.

txt public class ThreeSum public static int count(int[] a) int = a.

3 Example: 3-sum 3-sum. Given distinct integers, how many triples sum to exactly zero? % more 8ints.txt a[i] a[j] a[k] sum observations mathematical models order-of-growth classifications dependencies on inputs memory % java ThreeSum < 8ints.txt Context. Deeply related to problems in computational geometry sum: brute-force algorithm Measuring the running time Q. How to time a program? % java ThreeSum < 1Kints.txt 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+1; j < ; j++) for (int k = j+1; k < ; k++) if (a[i] + a[j] + a[k] == 0) count++; return count; check each triple we ignore any integer overflow A. Manual. tick tick tick 70 % java ThreeSum < 2Kints.txt 528 % java ThreeSum < 4Kints.txt public static void main(string[] args) int[] a = StdArrayIO.readInt1D(); StdOut.println(count(a));

4 Measuring the running time Measuring the running time Q. How to time a program? Q. How to time a program? A. Automatic. A. Automatic. public class Stopwatch public class Stopwatch Stopwatch() create a new stopwatch Stopwatch() create a new stopwatch double elapsedtime() time since creation (in seconds) double elapsedtime() time since creation (in seconds) public static void main(string[] args) int[] a = StdArrayIO.readInt1D(); Stopwatch stopwatch = new Stopwatch(); StdOut.println(ThreeSum.count(a)); double time = stopwatch.elapsedtime(); client code public class Stopwatch private final long start = System.currentTimeMillis(); public double elapsedtime() long now = System.currentTimeMillis(); return (now - start) / ; implementation (part of stdlib.jar) Empirical analysis Data analysis Run the program for various input sizes and measure running time. Standard plot. Plot running time T () vs. input size. time (seconds) standard plot , , , , running time T() ,000? 1K 2K 4K 8K problem size 15 16

5 Data analysis Prediction and validation (experimental) Log-log plot. Plot running time vs. input size using log-log scale. Hypothesis. The running time is about 1.006! 10 10! seconds. log-log plot lg(t()) straight line of slope 3 lg(t ()) = b lg + c b = c = T () = a b, where a = 2 c Predictions seconds for = 8, seconds for = 16,000. Observations. time (seconds) order of growth of running time is about 3.4 8, Regression. Fit straight line through data points: a b. 1K 2K 4K 8K lg power law slope Hypothesis. The running time is about 1.006! 10 10! seconds. 8, , , validates hypothesis! Experimental algorithmics System independent effects. Algorithm. Input data. determines exponent b in power law System dependent effects. Hardware: CPU, memory, cache, Software: compiler, interpreter, garbage collector, System: operating system, network, other applications, Bad news. Difficult to get precise measurements. Good news. Much easier and cheaper than other sciences. helps determines constant a in power law observations mathematical models order-of-growth classifications dependencies on inputs memory e.g., can run huge number of experiments 19 20

Mathematical models for running time Cost of basic operations Total running time: sum of cost! frequency for all operations.

operation example nanoseconds integer add a + b 2.1 integer multiply a * b 2.4 integer divide a / b 5.4 floating-point add a + b 4.

3 Donald Knuth 1974 Turing Award arctangent Math.atan2(y, x) 129.0......... Running OS X on Macbook Pro 2.

21 22 Cost of basic operations Example: 1-sum operation example nanoseconds variable declaration int a c1 assignment statement a = b c2

int count = 0; for (int i = 0; i < ; i++) if (a[i] == 0) count++; array element access a[i] c4 array length a.

6 Mathematical models for running time Cost of basic operations 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. operation example nanoseconds integer add a + b 2.1 integer multiply a * b 2.4 integer divide a / b 5.4 floating-point add a + b 4.6 floating-point multiply a * b 4.2 floating-point divide a / b 13.5 sine Math.sin(theta) 91.3 Donald Knuth 1974 Turing Award arctangent Math.atan2(y, x) Running OS X on Macbook Pro 2.2GHz with 2GB RAM In principle, accurate mathematical models are available Cost of basic operations Example: 1-sum operation example nanoseconds variable declaration int a c1 assignment statement a = b c2 integer compare a < b c3 Q. How many instructions as a function of input size? int count = 0; for (int i = 0; i < ; i++) if (a[i] == 0) count++; array element access a[i] c4 array length a.length c5 1D array allocation new int[] c6 operation frequency 2D array allocation new int[][] c7 2 string length s.length() c8 substring extraction s.substring(/2, ) c9 variable declaration 2 assignment statement 2 less than compare + 1 string concatenation s + t c10 ovice mistake. Abusive string concatenation. equal to compare array access increment to

Example: 2-sum Simplification 1: cost model Q. How many instructions as a function of input size? Cost model. Use some basic operation as a proxy for running time.

7 Example: 2-sum Simplification 1: cost model Q. How many instructions as a function of input size? 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+1; j < ; j++) if (a[i] + a[j] == 0) count++; int count = 0; for (int i = 0; i < ; i++) for (int j = i+1; j < ; j++) if (a[i] + a[j] == 0) count++; operation frequency ( 1) = 1 ( 1) 2 = 2 operation frequency ( 1) = 1 ( 1) 2 = 2 variable declaration + 2 variable declaration + 2 assignment statement + 2 assignment statement + 2 less than compare ½ ( + 1) ( + 2) less than compare ½ ( + 1) ( + 2) equal to compare ½ ( 1) array access ( 1) tedious to count exactly equal to compare ½ ( 1) array access ( 1) cost model = array accesses increment to 2 increment to Simplification 2: tilde notation Simplification 2: 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 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 1. ⅙ ! ~ ⅙ 3 Ex 2. ⅙ /3 + 56! ~ ⅙ 3 166,666,667 3 /6 2 /2 + /3 Ex 3. ⅙ 3 - " 2 + ⅓! ~ ⅙ 3 166,167,000 discard lower-order terms (e.g., = 1000: 500 thousand vs. 166 million) 1,000 Leading-term approximation operation frequency tilde notation variable declaration + 2 ~ assignment statement + 2 ~ less than compare ½ ( + 1) ( + 2) ~ ½ 2 Technical definition. f() ~ g() means f () lim " # g() = 1 equal to compare ½ ( 1) ~ ½ 2 array access ( 1) ~ 2 increment to 2 ~ to ~

8 Example: 2-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? int count = 0; for (int i = 0; i < ; i++) for (int j = i+1; j < ; j++) if (a[i] + a[j] == 0) count++; "inner loop" int count = 0; for (int i = 0; i < ; i++) for (int j = i+1; j < ; j++) for (int k = j+1; k < ; k++) if (a[i] + a[j] + a[k] == 0) count++; "inner loop" A. ~ 2 array accesses ( 1) = 1 ( 1) 2 = 2 A. ~ " 3 array accesses. 3 = ( 1)( 2) 3! Bottom line. Use cost model and tilde notation to simplify frequency counts. Bottom line. Use cost model and tilde notation to simplify frequency counts Estimating a discrete sum Mathematical models for running time Q. How to estimate a discrete sum? A1. Take COS 340. A2. Replace the sum with an integral, and use calculus! Ex i i=1 x=1 xdx 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) Ex /2 + 1/ /. Ex 3. 3-sum triple loop. i=1 1 i 1 i=1 j=i k=j 1 dx = ln x=1 x x=1 y=x z=y dz dy dx T = c1 A + c2 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

A reasonable model The running time of your program is ~ a b (lg ) c Specific models of this form are known for many algorithms (stay tuned). General laws of this form are known in many circumstances.

9 A reasonable model The running time of your program is ~ a b (lg ) c Specific models of this form are known for many algorithms (stay tuned). General laws of this form are known in many circumstances. (Interested? Take courses in combinatorics and complex analysis) Computing the constants (the hard way) Knuth showed that it is possible in principle to precisely predict running time develop a mathematical model for the frequency of execution of each instruction in the program determine the time required to execute each instruction multiply and sum otes The existence of the constant a is more significant than its value. We often drop the constant and refer to the order of growth. The small set of functions 1, log,, log, 2, and 3 suffices to describe order of growth of running time of typical algorithms. Some algorithms take exponential (~ d ) time (we consider such algorithms in the last few lectures) cycle time instruction set... Hypothesis: T() ~ a c... cache structure code machine-dependent part of the constants (harder to determine now than in the 1970s) GFs model asymptotics analysis algorithm and model-dependent part of the constants (easier to determine now than in the 1970s) Computing the constants (easy way) Predicting performance (the easy way) Run the program! ote: log factor is more difficult Don t bother computing the constants! Hypothesis: T() ~ a b time ratio lg ratio Hypothesis: T() ~ a b time ratio 1. Implement the program Implement the program Compute T(0) and T(20) by running it 3. Calculate b as follows: T(20) a(20) ~ b T(0) a0 b = 2 b lg(t(2 0)/T( 0) b as 0 grows 4. Calculate a as follows: T( 0)/ 0 b a as 0 grows , , , , b 3 a 51.1/ Run it for 0, 20, 40, 80, , Ratio of running times approaches 2 b 2, T(20) a(20) ~ b T(0) a0 b 4,000 8, = 2 b 16, Multiply by ratio 2 b to predict next value predicted value = 51.1 * 8.0 predicted order of growth 3 since lg 8 = 3 Plenty of caveats, but provides a basis for predicting program performance

War story (from COS 126) Q. How long does this program take as a function of? String s = StdIn.readString(); int = s.length();... for (int i = 0; i < ; i++) for (int j = 0; j < ; j++) distance[i][j] =.

10 War story (from COS 126) Q. How long does this program take as a function of? String s = StdIn.readString(); int = s.length();... for (int i = 0; i < ; i++) for (int j = 0; j < ; j++) distance[i][j] = time 1, , time observations mathematical models order-of-growth classifications dependencies on inputs memory 4, , , , Jenny ~ c1 2 seconds Kenny ~ c2 seconds Common order-of-growth classifications Common order-of-growth classifications Good news. the small set of functions 1, log,, log, 2, 3, and 2 suffices to describe order of growth of the running time of typical algorithms. growth rate name typical code framework description example T(2) / T() 1 constant a = b + c; statement add two numbers 1 log-log plot 512T exponential cubic quadratic linearithmic linear log logarithmic linear while ( > 1) = / 2;... for (int i = 0; i < ; i++)... divide in half binary search ~ 1 find the loop maximum 2 64T time log linearithmic [see mergesort lecture] divide and conquer mergesort ~ 2 8T 4T 2 quadratic for (int i = 0; i < ; i++) for (int j = 0; j < ; j++)... double loop check all pairs 4 2T T logarithmic constant 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 1K 2K 4K 8K 512K size Typical orders of growth 2 exponential [see combinatorial search lecture] exhaustive search check all subsets T() 39 40

Practical implications of order-of-growth Binary search growth rate problem size solvable in minutes 1970s 1980s 1990s 2000s Goal. Given a sorted array and a key, find index of the key in the array?

11 Practical implications of order-of-growth Binary search growth rate problem size solvable in minutes 1970s 1980s 1990s 2000s Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for any any any any log any any any any log millions hundreds of thousands tens of millions millions hundreds of millions millions 2 hundreds thousand thousands billions hundreds of millions tens of thousands lo mid hi 3 hundred hundreds thousand thousands s 20s 30 Bottom line. eed linear or linearithmic alg to keep pace with Moore's law Binary search Binary search Goal. Given a sorted array and a key, find index of the key in the array? Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for 33. Successful search. Binary search for lo mid hi lo mid hi 43 44

12 Binary search Binary search: Java implementation Goal. Given a sorted array and a key, find index of the key in the array? Successful search. Binary search for 33. Trivial to implement? First binary search published in 1946; first bug-free one published in Java bug in Arrays.binarySearch() not fixed until lo = hi mid return 4 public static int binarysearch(int[] a, int key) int lo = 0, hi = a.length-1; while (lo <= hi) int mid = lo + (hi - lo) / 2; if (key < a[mid]) hi = mid - 1; else if (key > a[mid]) lo = mid + 1; else return mid; return -1; one 3-way compare Invariant. If key appears in the array a[], then a[lo] " key " a[hi] Binary search: mathematical analysis An 2 log algorithm for 3-sum Proposition. Binary search uses at most 1 + lg compares to search in a sorted array of size. Def. T () # # compares to binary search in a sorted subarray of size. Binary search recurrence. T ()! T ( / 2) + 1 for > 1, with T (1) = 1. left or right half Pf sketch. T ()! T ( / 2) + 1! T ( / 4) ! T ( / 8) ! T ( / ) = 1 + lg given apply recurrence to first term apply recurrence to first term stop applying, T(1) = 1 Step 1. Sort the numbers. Step 2. For each pair of numbers a[i] and a[j], binary search for -(a[i] + a[j]). Analysis. Order of growth is 2 log. Step 1: 2 with insertion sort. Step 2: 2 log with binary search. input sort binary search (-40, -20) 60 (-40, -10) 30 (-40, 0) 40 (-40, 5) 35 (-40, 10) 30 (-40, 40) 0 (-10, 0) 10 (-20, 10) 10 ( 10, 30) -40 ( 10, 40) -50 ( 30, 40) -70 only count if a[i] < a[j] < a[k] to avoid double counting 47 48

13 Comparing programs Hypothesis. The 2 log three-sum algorithm is significantly faster in practice than the brute-force 3 one. time (seconds) time (seconds) 1, , , , ThreeSum.java 1, , , , , , , observations mathematical models order-of-growth classifications dependencies on inputs memory ThreeSumDeluxe.java Bottom line. Typically, better order of growth $ faster in practice Types of analyses 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. Average case. Expected cost. Actual data might not match input model? eed to understand input to effectively process it. Approach 1: design for the worst case. Approach 2: randomize, depend on probabilistic guarantee. Ex 1. Array accesses for brute-force 3 sum. Best: ~ " 3 Average: ~ " 3 Worst: ~ " 3 Ex 2. Compares for binary search. Best: ~ 1 Average: ~ lg Worst: ~ lg 51 52

14 Commonly-used notations Tilde notation vs. big-oh notation notation provides example shorthand for used to Tilde leading term ~ log provide approximate model We use tilde notation whenever possible. Big-Oh notation suppresses leading constant. Big-Oh notation only provides upper bound (not lower bound). Big Theta asymptotic growth rate Θ( 2 ) Big Oh Θ( 2 ) and smaller O( 2 ) ½ log log + 3 classify algorithms develop upper bounds time/memory values represented by O(f()) f() time/memory values represented by ~ c f() c f() Big Omega Θ( 2 ) and larger Ω( 2 ) ½ log + 3 develop lower bounds Common mistake. Interpreting big-oh as an approximate model. input size input size O-notation considered harmful How to predict performance (and to compare algorithms)? ot the scientific method: O-notation Theorem: Running time is O( c )! not at all useful for predicting performance Scientific method calls for tilde-notation. Hypothesis: Running time is ~a c an effective path to predicting performance (stay tuned) observations mathematical models order-of-growth classifications dependencies on inputs memory O-notation is useful for many reasons, BUT Common error: Thinking that O-notation is useful for predicting performance. 56

15 Typical memory requirements for primitive types in Java Typical memory requirements for arrays in Java Bit. 0 or 1. Array overhead. 16 bytes. Byte. 8 bits. Megabyte (MB). 1 million bytes. Gigabyte (GB). 1 billion bytes. type bytes type bytes char[] char[][] ~ 2 M type bytes int[] int[][] ~ 4 M boolean 1 byte 1 char 2 double[] for one-dimensional arrays double[][] ~ 8 M for two-dimensional arrays int 4 float 4 long 8 double 8 Ex. An -by- array of doubles consumes ~ 8 2 bytes of memory. for primitive types Typical memory requirements for objects in Java Typical memory requirements for objects in Java Object overhead. 8 bytes. Reference. 4 bytes. Object overhead. 8 bytes. Reference. 4 bytes. Ex 1. A Complex object consumes 24 bytes of memory. Ex 2. A virgin String of length consumes ~ 2 bytes of memory. public class Complex private double re; private double im; bytes 8 bytes (object overhead) 8 bytes (double) 8 bytes (double) 24 bytes public class String private int offset; private int count; private int hash; private char[] value;... 8 bytes (object overhead) 4 bytes (int) 4 bytes (int) 4 bytes (int) 4 bytes (reference to array) bytes (char[] array) bytes object overhead real imag double value object overhead value offset count hash reference int values 59 60

16 Example Turning the crank: summary Q. How much memory does WeightedQuickUnionUF use as a function of? public class WeightedQuickUnionUF private int[] id; private int[] sz; 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] = 1; public boolean find(int p, int q)... public void union(int p, int q)... 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

1.4 Analysis of Algorithms

1.4 Analysis of Algorithms observations mathematical models order-of-growth classifications dependencies on inputs memory Algorithms, 4 th Edition Robert Sedgewick and Kevin Wayne Copyright 2002 2010 September