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1 TCSS 342, Winter 2005 Lecture Notes Course Overview, Review of Math Concepts, Algorithm Analysis and Big-Oh Notation Weiss book, Chapter 5, pp. 147-181

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2 Course objectives (broad) prepare you to be a good software engineer (specific) learn basic data structures and algorithms data structures – how data is organized algorithms – unambiguous sequence of steps to compute something

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3 Software design goals What are some goals one should have for good software?

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4 Course content data structures algorithms data structures + algorithms = programs algorithm analysis – determining how long an algorithm will take to solve a problem Who cares? Aren't computers fast enough and getting faster?

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5 Given an array of 1,000,000 integers, find the maximum integer in the array. Now suppose we are asked to find the kth largest element (The Selection Problem) … 012999,999 An example

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6 candidate solution 1 sort the entire array (from small to large), using Java's Arrays.sort() pick out the (1,000,000 – k)th element candidate solution 2 sort the first k elements for each of the remaining 1,000,000 – k elements, keep the k largest in an array pick out the smallest of the k survivors Candidate solutions

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7 Is either solution good? Is there a better solution? What makes a solution "better" than another? Is it entirely based on runtime? How would you go about determining which solution is better? could code them, test them could somehow make predictions and analysis of each solution, without coding

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8 Why algorithm analysis? as computers get faster and problem sizes get bigger, analysis will become more important The difference between good and bad algorithms will get bigger being able to analyze algorithms will help us identify good ones without having to program them and test them first

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9 Why data structures? when programming, you are an engineer engineers have a bag of tools and tricks – and the knowledge of which tool is the right one for a given problem Examples: arrays, lists, stacks, queues, trees, hash tables, heaps, graphs

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10 Development practices modular (flexible) code appropriate commenting of code each method needs a comment explaining its parameters and its behavior writing code to match a rigid specification being able to choose the right data structures to solve a variety of programming problems using an integrated development environment (IDE) incremental development

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11 Math background: exponents Exponents X Y = "X to the Y th power"; X multiplied by itself Y times Some useful identities X A X B = X A+B X A / X B = X A-B (X A ) B = X AB X N +X N = 2X N 2 N +2 N = 2 N+1

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12 Logarithms definition: X A = B if and only if log X B = A intuition: log X B means "the power X must be raised to, to get B" a logarithm with no base implies base 2 log B means log 2 B Examples log 2 16 = 4(because 2 4 = 16) log 10 1000 = 3(because 10 3 = 1000)

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13 log AB = log A + log B Proof: (let's write it together!) log A/B = log A – log B log (A B ) = B log A example: log 4 32 = (log 2 32) / (log 2 4) = 5 / 2 Logarithms, continued

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14 Series for some expression Expr (possibly containing i ), means the sum of all values of Expr with each value of i between j and k inclusive Example: = (2(0) + 1) + (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + (2(4) + 1) = 1 + 3 + 5 + 7 + 9 = 25 Arithmetic series

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15 Series identities sum from 1 through N inclusive is there an intuition for this identity? sum of all numbers from 1 to N 1 + 2 + 3 +... + (N-2) + (N-1) + N how many terms are in this sum? Can we rearrange them?

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16 Series sum of powers of 2 0 + 1 + 2 + 4 + 8 + 16 + 32 = 64 - 1 think about binary representation of numbers... sum of powers of any number a ("Geometric progression" for a>0, a ≠ 1)

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17 Algorithm performance How to determine how much time an algorithm A uses to solve problem X ? Depends on input; use input size "N" as parameter Determine function f(N) representing cost empirical analysis: code it and use timer running on many inputs algorithm analysis: Analyze steps of algorithm, estimating amount of work each step takes

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18 Typically use a simple model for basic operation costs RAM (Random Access Machine) model RAM model has all the basic operations: +, -, *, /, =, comparisons fixed sized integers (e.g., 32-bit) infinite memory All basic operations take exactly one time unit (one CPU instruction) to execute RAM model

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19 Critique of the model Strengths: simple easier to prove things about the model than the real machine can estimate algorithm behavior on any hardware/software Weaknesses: not all operations take the same amount of time in a real machine does not account for page faults, disk accesses, limited memory, floating point math, etc

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20 modelreal world Idea: useful statements using the model translate into useful statements about real computers Why use models?

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21 Relative rates of growth most algorithms' runtime can be expressed as a function of the input size N rate of growth: measure of how quickly the graph of a function rises goal: distinguish between fast- and slow- growing functions we only care about very large input sizes (for small sizes, most any algorithm is fast enough) this helps us discover which algorithms will run more quickly or slowly, for large input sizes

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22 Growth rate example Consider these graphs of functions. Perhaps each one represents an algorithm: n 3 + 2n 2 100n 2 + 1000 Which grows faster?

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23 Growth rate example How about now?

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24 Defn: T(N) = O(f(N)) if there exist positive constants c, n 0 such that: T(N) c · f(N) for all N n 0 idea: We are concerned with how the function grows when N is large. We are not picky about constant factors: coarse distinctions among functions Lingo: "T(N) grows no faster than f(N)." Big-Oh notation

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25 Examples n = O(2n)? 2n = O(n)? n = O(n 2 )? n 2 = O(n)? n = O(1)? 100 = O(n)? 10 log n = O(n)? 214n + 34 = O(2n 2 + 8n)?

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26 pick tightest bound. If f(N) = 5N, then: f(N) = O(N 5 ) f(N) = O(N 3 ) f(N) = O(N) preferred f(N) = O(N log N) ignore constant factors and low order terms T(N) = O(N), not T(N) = O(5N) T(N) = O(N 3 ), not T(N) = O(N 3 + N 2 + N log N) Bad style: f(N) O(g(N)) Wrong: f(N) O(g(N)) Preferred big-Oh usage

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27 Big-Oh of selected functions

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28 Ten-fold processor speedup

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29 Defn: T(N) = (g(N)) if there are positive constants c and n 0 such that T(N) c g(N) for all N n 0 Lingo: "T(N) grows no slower than g(N)." Defn: T(N) = (h(N)) if and only if T(N) = O(h(N)) and T(N) = (h(N)). Big-Oh, Omega, and Theta establish a relative order among all functions of N Big omega, theta

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30 Defn: T(N) = o(p(N)) if T(N) = O(p(N)) and T(N) (p(N)) notationintuition O (Big-Oh) (Big-Omega) (Theta) = o (little-Oh) < Intuition, little-Oh

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31 Fact: If f(N) = O(g(N)), then g(N) = (f(N)). Proof: Suppose f(N) = O(g(N)). Then there exist constants c and n 0 such that f(N) c g(N) for all N n 0 Then g(N) (1/c) f(N) for all N n 0, and so g(N) = (f(N)) More about asymptotics

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32 T(N) = O(f(N)) f(N) is an upper bound on T(N) T(N) grows no faster than f(N) T(N) = (g(N)) g(N) is a lower bound on T(N) T(N) grows at least as fast as g(N) T(N) = o(h(N)) T(N) grows strictly slower than h(N) More terminology

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33 Facts about big-Oh If T 1 (N) = O(f(N)) and T 2 (N) = O(g(N)), then T 1 (N) + T 2 (N) = O(f(N) + g(N)) T 1 (N) * T 2 (N) = O(f(N) * g(N)) If T(N) is a polynomial of degree k, then: T(N) = (N k ) example: 17n 3 + 2n 2 + 4n + 1 = (n 3 ) log k N = O(N), for any constant k

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34 Algebra ex. f(N) = N / log N g(N) = log N same as asking which grows faster, N or log 2 N Evaluate: limit isBig-Oh relation 0f(N) = o(g(N)) c 0f(N) = (g(N)) g(N) = o(f(N)) no limitno relation Techniques

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35 L'Hôpital's rule: If and, then example: f(N) = N, g(N) = log N Use L'Hôpital's rule f'(N) = 1, g'(N) = 1/N g(N) = o(f(N)) Techniques, cont'd

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36 for (int i = 0; i < n; i += c) // O(n) statement(s); Adding to the loop counter means that the loop runtime grows linearly when compared to its maximum value n. for (int i = 0; i < n; i *= c) // O(log n) statement(s); Multiplying the loop counter means that the maximum value n must grow exponentially to linearly increase the loop runtime; therefore, it is logarithmic. for (int i = 0; i < n * n; i += c) // O(n 2 ) statement(s); The loop maximum is n 2, so the runtime is quadratic. Program loop runtimes

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37 for (int i = 0; i < n; i += c) // O(n 2 ) for (int j = 0; j < n; i += c) statement; Nesting loops multiplies their runtimes. for (int i = 0; i < n; i += c) statement; for (int i = 0; i < n; i += c) // O(n log n) for (int j = 0; j < n; i *= c) statement; Loops in sequence add together their runtimes, which means the loop set with the larger runtime dominates. More loop runtimes

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38 Maximum subsequence sum The maximum contiguous subsequence sum problem: Given a sequence of integers A 0, A 1,..., A n - 1, find the maximum value of for any integers 0 (i, j) < n. (This sum is zero if all numbers in the sequence are negative.)

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39 First algorithm (brute force) try all possible combinations of subsequences // implement together function maxSubsequence(array[]): max sum = 0 for each starting index i, for each ending index j, add up the sum from A i to A j if this sum is bigger than max, max sum = this sum return max sum What is the runtime (Big-Oh) of this algorithm? How could it be improved?

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40 still try all possible combinations, but don't redundantly add the sums key observation: in other words, we don't need to throw away partial sums can we use this information to remove one of the loops from our algorithm? // implement together function maxSubsequence2(array[]): What is the runtime (Big-Oh) of this new algorithm? Can it still be improved further? Second algorithm (improved)

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41 Third algorithm (improved!) must avoid trying all possible combinations; to do this, we must find a way to broadly eliminate many potential combinations from consideration Claim #1: A subsequence with a negative sum cannot be the start of the maximum-sum subsequence.

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42 Claim #1, more formally: If A i, j is a subsequence such that, then there is no q such that A i,q is the maximum-sum subsequence. Proof: (do it together in class) Can this help us produce a better algorithm? Third algorithm, continued

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43 Claim #2: when examining subsequences left - to - right, for some starting index i, if A i,j becomes the first subsequence starting with i, such that Then no part of A i,j can be part of the maximum-sum subsequence. (Why is this a stronger claim than Claim #1?) Proof: (do it together in class) Third algorithm, continued

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44 Third algorithm, continued These figures show the possible contents of A i,j

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45 Can we eliminate another loop from our algorithm? // implement together function maxSubsequence3(array[]): What is its runtime (Big-Oh)? Is there an even better algorithm than this third algorithm? Can you make a strong argument about why or why not? Third algorithm, continued

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46 Express the running time as f(N), where N is the size of the input worst case: your enemy gets to pick the input average case: need to assume a probability distribution on the inputs amortized: your enemy gets to pick the inputs/operations, but you only have to guarantee speed over a large number of operations Kinds of runtime analysis

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47 References Weiss book, Chapter 5, pp. 147-181

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