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Analysis-handouts-2

Analysis-handouts-2 - Analysis of Algorithms Input...

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1 Analysis of Algorithms Algorithm Input Output An algorithm is a step-by-step procedure for solving a problem in a finite amount of time. Analysis of Algorithms 2 Running Time (§1.1) Most algorithms transform input objects into output objects. The running time of an algorithm typically grows with the input size. Average case time is often difficult to determine. We focus on the worst case running time. Easier to analyze Crucial to applications such as games, finance and robotics 0 20 40 60 80 100 120 Running Time 1000 2000 3000 4000 Input Size best case average case worst case
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2 Analysis of Algorithms 3 Experimental Studies (§ 1.6) Write a program implementing the algorithm Run the program with inputs of varying size and composition Use a method like System.currentTimeMillis() to get an accurate measure of the actual running time Plot the results 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 50 100 Input Size Time (ms) Analysis of Algorithms 4 Limitations of Experiments It is necessary to implement the algorithm, which may be difficult Results may not be indicative of the running time on other inputs not included in the experiment. In order to compare two algorithms, the same hardware and software environments must be used
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3 Analysis of Algorithms 5 Theoretical Analysis Uses a high-level description of the algorithm instead of an implementation Characterizes running time as a function of the input size, n . Takes into account all possible inputs Allows us to evaluate the speed of an algorithm independent of the hardware/software environment Analysis of Algorithms 6 Pseudocode (§1.1) High-level description of an algorithm More structured than English prose Less detailed than a program Preferred notation for describing algorithms Hides program design issues Algorithm arrayMax ( A , n ) Input array A of n integers Output maximum element of A currentMax A [0] for i 1 to n 1 do if A [ i ] > currentMax then currentMax A [ i ] return currentMax Example: find max element of an array
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4 Analysis of Algorithms 7 Pseudocode Details Control flow if then [ else …] while do repeat until for do Indentation replaces braces Method declaration Algorithm method ( arg [, arg …]) Input Output Method call var.method ( arg [, arg …]) Return value return expression Expressions Assignment (like = in Java) = Equality testing (like == in Java) n 2 Superscripts and other mathematical formatting allowed Analysis of Algorithms 8 The Random Access Machine (RAM) Model A CPU An potentially unbounded bank of memory cells, each of which can hold an arbitrary number or character 0 1 2 Memory cells are numbered and accessing any cell in memory takes unit time.
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