4.1 Performance. Running Time. The Challenge. Scientific Method

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Running Time 4.1 Performance As soon as an Analytic Engine exists, it will necessarily guide the future course of the science.

Charles Babbage how many times do you have to turn the crank?

Challenge Scientific Method Q. Will my program be able to solve a large practical problem? Scientific method. Observe some feature of the natural world.

1 Running Time 4.1 Performance 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 how many times do you have to turn the crank? Charles Babbage (1864) Introduction to Programming in Java: An Interdisciplinary Approach Robert Sedgewick and Kevin Wayne Copyright Analytic Engine 3/30/11 8:32 PM 2 The Challenge Scientific Method Q. Will my program be able to solve a large practical problem? 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. Hypothesis must be falsifiable. compile debug solve problems in practice Key insight. [Knuth 1970s] Use the scientific method to understand performance. 3 4

Reasons to Analyze Algorithms Algorithmic Successes Predict performance.

Will this change make my program faster? How can I make my program faster?

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

Freidrich Gauss 1805 Basis for inventing new ways to solve problems.

5 6 Algorithmic Successes Three-Sum Problem -body Simulation.

Application: cosmology, semiconductors, fluid dynamics, Brute force: 2 steps.

2 Reasons to Analyze Algorithms Algorithmic Successes Predict performance. Will my program finish? When will my program finish? Compare algorithms. Will this change make my program faster? How can I make my program faster? 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. Freidrich Gauss 1805 Basis for inventing new ways to solve problems. Enables new technology. Enables new research. 5 6 Algorithmic Successes Three-Sum Problem -body Simulation. Simulate gravitational interactions among bodies. Application: cosmology, semiconductors, fluid dynamics, Brute force: 2 steps. Barnes-Hut algorithm: log steps, enables new research. Andrew Appel PU '81 Three-sum problem. Given integers, how many triples sum to 0? Context. Deeply related to problems in computational geometry. % more 8ints.txt % java ThreeSum < 8ints.txt Q. How would you write a program to solve the problem? 7 8

a[j] + a[k] = 0 public static void main(string[] args) { int[] a = StdArrayIO.readInt1D(); StdOut.println(count(a)); 9 Empirical Analysis Stopwatch Empirical analysis.

3 Three-Sum: Brute-Force Solution public class ThreeSum { Empirical Analysis public static int count(int[] a) { int = a.length; int cnt = 0; for (int j = i+1; j < ; j++) for (int k = j+1; k < ; k++) if (a[i] + a[j] + a[k] == 0) cnt++; return cnt; all possible triples i < j < k such that a[i] + a[j] + a[k] = 0 public static void main(string[] args) { int[] a = StdArrayIO.readInt1D(); StdOut.println(count(a)); 9 Empirical Analysis Stopwatch Empirical analysis. Run the program for various input sizes. Q. How to time a program? A. A stopwatch time Running Linux on Sun-Fire-X4100 with 16GB RAM Caveat. If is too small, you will measure mainly noise

4 Stopwatch Stopwatch Q. How to time a program? A. A Stopwatch object. Q. How to time a program? A. A Stopwatch object. public class Stopwatch { private final long start; public Stopwatch() { start = System.currentTimeMillis(); public double elapsedtime() { return (System.currentTimeMillis() - start) / ; public static void main(string[] args) { int[] a = StdArrayIO.readInt1D(); Stopwatch timer = new Stopwatch(); StdOut.println(count(a)); StdOut.println(timer.elapsedTime()); Empirical Analysis Empirical Analysis Data analysis. Plot running time vs. input size. Initial hypothesis. Running time approximately obeys a power law T () = a b. Data analysis. Plot running time vs. input size on a log-log scale. slope = 3 Consequence. Power law yields straight line. slope = b Refined hypothesis. Running time grows as cube of input size: a 3. slope Q. How fast does running time grow as a function of input size? 15 16

5 Doubling Hypothesis Performance Challenge 1 Doubling hypothesis. Quick way to estimate b in a power law hypothesis. Let T() be running time of main() as a function of input size. Run program, doubling the size of the input? time ratio seems to converge to a constant c = 8 public static void main(string[] args) { int = Integer.parseInt(args[0]); Scenario 1. T(2) / T() converges to about 4. Q. What is order of growth of the running time? Hypothesis. Running time is about a b with b = lg c Performance Challenge 2 Prediction and Validation Let T() be running time of main() as a function of input. Hypothesis. Running time is about a 3 for input of size. public static void main(string[] args) { int = Integer.parseInt(args[0]); Q. How to estimate a? A. Run the program! time = a a = Scenario 2. T(2) / T() converges to about 2. Refined hypothesis. Running time is about seconds. Q. What is order of growth of the running time? Prediction. 1,100 seconds for = 16,384. Observation. time validates hypothesis 19 20

Mathematical Analysis Mathematical Analysis Running time. Count up frequency of execution of each instruction and weight by its execution time.

(no zeros) and 2 (all zeros) increment 2 22 Mathematical Analysis Tilde otation Running time. Count up frequency of execution of each instruction and weight by its execution time.

6 Mathematical Analysis Mathematical Analysis Running time. Count up frequency of execution of each instruction and weight by its execution time. int count = 0; if (a[i] == 0) count++; Donald Knuth Turing award '74 operation variable declaration variable assignment frequency 2 2 less than comparison equal to comparison array access + 1 between (no zeros) and 2 (all zeros) increment 2 22 Mathematical Analysis Tilde otation Running time. Count up frequency of execution of each instruction and weight by its execution time. int count = 0; for (int j = i+1; j < ; j++) if (a[i] + a[j] == 0) count++; Tilde notation. Estimate running time as a function of input size. Ignore lower order terms. when is large, terms are negligible when is small, we don't care Ex ~ 6 3 Ex / ~ 6 3 Ex log ~ 6 3 operation frequency variable declaration ( 1) = 1/2 ( 1) discard lower-order terms (e.g., = 1000: 6 trillion vs. 169 million) variable assignment + 2 less than comparison equal to comparison 1/2 ( + 1) ( + 2) 1/2 ( 1) becoming very tedious to count Technical definition. f() ~ g() means lim f () g() = 1 array access ( 1) increment

7 Mathematical Analysis Constants in Power Law Running time. Count up frequency of execution of each instruction and weight by its execution time. Power law. Running time of a typical program is ~ a b. Exponent b depends on: algorithm. Leading constant a depends on: Algorithm. Input data. Caching. Machine. Compiler. Garbage collection. Just-in-time compilation. CPU use by other applications. system independent effects system dependent effects Inner loop. Focus on instructions in "inner loop." Our approach. Use doubling hypothesis (or mathematical analysis) to estimate exponent b, run experiments to estimate a Analysis: Empirical vs. Mathematical Order of Growth Classifications Empirical analysis. Measure running times, plot, and fit curve. Easy to perform experiments. Model useful for predicting, but not for explaining. Mathematical analysis. Analyze algorithm to estimate # ops as a function of input size. May require advanced mathematics. Model useful for predicting and explaining. Observation. A small subset of mathematical functions suffice to describe running time of many fundamental algorithms. lg = log 2 while ( > 1) { = / 2; lg public static void g(int ) { if ( == 0) return; g(/2); g(/2); lg Critical difference. Mathematical analysis is independent of a particular machine or compiler; applies to machines not yet built. for (int j = 0; j < ; j++) 2 public static void f(int ) { if ( == 0) return; f(-1); f(-1);

8 Order of Growth Classifications Order of Growth: Consequences Sequential Search vs. Binary Search Binary Search Sequential search in an unordered array. Examine each entry until finding a match (or reaching the end). Takes time proportional to length of array in worst case Binary search in an ordered array. Examine the middle entry. If equal, return index. If too large, search in left half (recursively). If too small, search in right half (recursively)

Binary Search: Java Implementation Binary Search: Mathematical Analysis Invariant. If key appears in the array, then a[lo] key a[hi]. // precondition: array a[] is sorted!

9 Binary Search: Java Implementation Binary Search: Mathematical Analysis Invariant. If key appears in the array, then a[lo] key a[hi]. // precondition: array a[] is sorted! public static int search(int key, int[] a) {! int lo = 0;! int 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; // not found! Java library implementation. Arrays.binarySearch(). 33 Proposition. Binary search in an ordered array of size takes at most 1 + log 2 3-way compares. Pf. After each 3-way compare, problem size decreases by a factor of 2. / 2 / 4 / 8 1 Q. How many times can you divide by 2 until you reach 1? A. About log Typical Memory Requirements for Primitive Types Memory Bit. 0 or 1. Byte. 8 bits. Megabyte (MB). 1 million bytes ~ 2 10 bytes. Gigabyte (GB). 1 billion bytes ~ 2 20 bytes. Q. How much memory (in bytes) does your computer have? 36

10 Typical Memory Requirements for Reference Types Typical Memory Requirements for Array Types Memory of an object. Memory for each instance variable, plus Object overhead = 8 bytes on a 32-bit machine. Memory of an array. Memory for each array entry. Array overhead = 16 bytes on a 32-bit machine. 16 bytes on a 64-bit machine 24 bytes on a 64-bit machine Memory of a reference. 4 byte pointer on a 32-bit machine. 8 bytes on a 64-bit machine Q. What's the biggest double[] array you can store on your computer? Summary Q. How can I evaluate the performance of my program? A. Computational experiments, mathematical analysis, scientific method. Q. What if it's not fast enough? ot enough memory? Understand why. Buy a faster computer or more memory. Learn a better algorithm. see COS 226 Discover a new algorithm. see COS 423 attribute cost applicability better machine $$$ or more makes "everything" run faster better algorithm $ or less does not apply to some problems improvement quantitative improvements dramatic qualitative improvements possible 40

Running Time. Analytic Engine. Charles Babbage (1864) how many times do you have to turn the crank?

Running Time. Analytic Engine. Charles Babbage (1864) how many times do you have to turn the crank? 4.1 Performance Introduction to Programming in Java: An Interdisciplinary Approach Robert Sedgewick and Kevin Wayne Copyright 2002 2010 3/30/11 8:32 PM Running Time As soon as an Analytic Engine exists,