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Week 2 CS 361: Advanced Data Structures and Algorithms
Introduction to Algorithms
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Class Overview Start thinking about analyzing a program or algorithm.
Understand algorithm efficiency and running-time complexity. Analysis of an algorithm using Big-O notation. Which Cost More to Feed?
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Algorithm Efficiency There are often many approaches (algorithms) to solve a problem. How do we choose between them? There are two (sometimes conflicting) goals at the heart of computer program design. To design an algorithm that: is easy to understand, code, debug. makes efficient use of the resources. Goal (1) is the concern of Software Engineering. Goal (2) is the concern of data structures and algorithm analysis. When goal (2) is important, how do we measure an algorithm’s cost?
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Estimation Techniques
Known as “back of the envelope” or “back of the napkin” calculation Determine the major parameters that effect the problem. Derive an equation that relates the parameters to the problem. Select values for the parameters, and apply the equation to yield and estimated solution. Look over Chapter 2, read as needed depending on your familiarity with this material. A set has no duplicates, a sequence may have duplicates. Logarithms: We almost always use log to base 2. That is our default base. Essentially, you need to understand the problem
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Estimation Example How many library bookcases does it take to store books totaling one million pages? Estimate: Pages/inch Shelf/Feet Shelves/bookcase Pages/inch: Guess 500 Feet/shelf: Guess 4, actually 3 Shelves/bookcase: Guess 5, actually 7 Units check: pages/in x ft/shelf x shelf/bookcase pages/bookcase
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Best, Worst, Average Cases
Not all inputs of a given size take the same time to run. Sequential search for K in an array of n integers: Begin at first element in array and look at each element in turn until K is found Best case: Worst case: Average case: Best: Find at first position. Cost is 1 compare. Worst: Find at last position. Cost is n compares. Average: (n+1)/2 compares IF we assume the element with value K is equally likely to be in any position in the array.
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Provides upper and lower bounds of running time.
Time Analysis Provides upper and lower bounds of running time. Different types of analysis: - Worst case - Best case - Average case
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Worst Case Provides an upper bound on running time.
An absolute guarantee that the algorithm would not run longer, no matter what the inputs are.
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Best Case Provides a lower bound on running time.
Input is the one for which the algorithm runs the fastest.
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Average Case Provides an estimate of “average” running time.
Assumes that the input is random. Useful when best/worst cases do not happen very often i.e., few input cases lead to best/worst cases.
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Which Analysis to Use? While average time appears to be the fairest measure, It may be difficult to determine. For example, algorithms that are designed to operate on strings of text. Why is the worst case time important? In some situations it may be necessary to use a pessimistic analysis in order to guarantee safety. Recall the “bookcase” problem. Average time analysis requires knowledge of distributions. For example, the assumption of distribution used for average case in the last example. Worst-case time is important for real-time algorithms.
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How to Measure Efficiency?
Critical resources: Time, memory, programmer effort, user effort Factors affecting running time: For most algorithms, running time depends on “size” of the input. Running time is expressed as T(n) for some function T on input size n. Empirical comparison is difficult to do “fairly” and is time consuming. Critical resources: Time. Space (disk, RAM). Programmers effort. Ease of use (user’s effort). Factors affecting running time: Machine load. OS. Compiler. Problem size. Specific input values for given problem size. 12
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How do we analyze an algorithm?
Need to define objective measures. (1) Compare execution times? Not good: times are specific to a particular machine. (2) Count the number of statements? number of statements varies with programming language and style.
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How do we analyze an algorithm? (cont.)
(3) Express running time T as a function of problem size n (i.e., T=f(n) ) Asymptotic Algorithm Analysis Given two algorithms having running times f(n) and g(n), find which functions grows faster? Compare “rates of growth” of f(n) and g(n). Such an analysis is independent of machine time, programming style, etc.
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Understanding Rate of Growth
Consider the example of feeding elephants and goldfish: Total Cost: (cost_of_feeding_elephants) + (cost_of_feeding_goldfish) Approximation: Total Cost ~ cost_of_feeding_elephants
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Understanding Rate of Growth (cont’d)
The low order terms of a function are relatively insignificant for large n n n2 + 10n + 50 Approximation: n4 Highest order term determines rate of growth!
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Visualizing Orders of Growth
On a graph, as you go to the right, a faster growing function eventually becomes larger...
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Growth Rate Graph
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Common orders of magnitude
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Orders of Magnitude
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Rate of Growth ≡ Asymptotic Analysis
Using rate of growth as a measure to compare different functions implies comparing them asymptotically i.e., as n If f(x) is growing faster than g(x), then f(x) always eventually becomes larger than g(x) in the limit i.e., for large enough values of x Because we prefer the worst-case analysis !
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Complexity Let us assume two algorithms A and B that solve the same class of problems. The time complexity of A is 5,000n, T = f(n) = 5000*n the one for B is 2n for an input with n elements, T= g(n) = 2n For n = 10, A requires 5*104 steps, but B only 1024, so B seems to be superior to A. For n = 1000, A requires 5*106 steps, while B requires 1.07*10301 steps.
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Asymptotic Notation worst-case analysis
O notation: asymptotic “less than”: f(n) = O(g(n)) implies: f(n) “≤” c*g(n) in the limit, c is a constant In English: “ f(n) grows asymptotically no faster than g(n) ” c is a constant worst-case analysis
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Asymptotic Notation best-case analysis *formal definition in CS477/677
notation: asymptotic “greater than”: f(n) = (g(n)) implies: f(n) “≥” c*g(n) in the limit , c is a constant In English: “ f(n) grows asymptotically faster than g(n) ” c is a constant best-case analysis *formal definition in CS477/677
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(best and worst cases are same)
Asymptotic Notation notation: asymptotic “equality”: f(n)= (g(n)) implies: f(n) “=” c*g(n) in the limit , c is a constant In English: “ f(n) grows asymptotically as fast as g(n) ” c is a constant tight bound analysis (best and worst cases are same) *formal definition in CS477/677
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Common Misunderstanding
Worst case & Upper bound Upper bound refers to a limit for the run-time of that algorithm. Worst case refers to the worst input among the choices for possible inputs of a given size.
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Big O in practice Figure out T=f(n): run-time (number of basic operations) required on an input of size n Remove low-order terms
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More on big-O O(g(n)) can be related to a set of functions f(n)
f(n) = O(g(n)) if “f(n)≤c*g(n)” Big-O notation provides a machine independent means for determining the efficiency of an algorithm.
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Names of Orders of Magnitude
O(1) bounded (by a constant) time O(log2N) logarithmic time O(N) linear time O(N*log2N) N*log2N time O(N2) quadratic time O(N3) cubic time O(2N ) exponential time
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Constant Time Algorithms
An algorithm is O(1) when its running time is independent of the number of data items. The algorithm runs in constant time. The storing of the element involves a simple assignment statement and thus has efficiency O(1).
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Linear Time Algorithms
An algorithm is O(n) when its running time is proportional to the size of the list. When the number of elements doubles, the number of operations doubles.
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Logarithmic Time Algorithms
The logarithm of n, base 2, is commonly used when analyzing computer algorithms. For example, sorting algorithms. Ex. log2(2) = 1 log2(75) = When compared to the functions n and n2, the function log2n grows very slowly.
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How do we calculate T=f(n) for a program/algorithm?
Associate a "cost" with each statement Find total number of times each statement is executed Add up the costs
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Running Time Examples i = 0; while (i<N) { X=X+Y; // O(1)
result = mystery(X); // O(N) i++; // O(1) } The body of the while loop: O(N) Loop is executed: N times Running time of the entire iteration? N x O(N) = O(N2)
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Running Time Examples (cont.’d)
if (i<j) for ( i=0; i<N; i++ ) X = X+i; else X=0; O(N) O(1) Running time of the entire if-else statement? Max (O(N), O(1)) = O(N)
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Complexity Examples What does the following algorithm compute?
returns the maximum difference between any two numbers in the input array # of Comparisons: n-1 + n-2 + n-3 + … + 1 = (n-1)n/2 = 0.5n n Time complexity is O(n2) int who_knows(int a[n]) { int m = 0; for {int i = 0; i<n; i++} for {int j = i+1; j<n; j++} if (abs(a[i] – a[j]) > m ) m = abs(a[i] – a[j]); return m; }
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Time complexity is O(n).
Complexity Examples Another algorithm solving the same problem: # of Comparisons: 2n - 2 Time complexity is O(n). int max_diff(int a[n]) { int min = a[0]; int max = a[0]; for {int i = 1; i<n; i++} if (a[i] < min ) min = a[i]; else if (a[i] > max ) max = a[i]; } return max-min;
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Running time of various statements
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Examples (cont.’d)
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Examples (cont.’d)
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Analyze the complexity of the following code segments
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Homework #2: Algorithm analysis
Already assigned on BB, due on 9/14/2014, 11:59PM
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Next class & Reading Next class: ADTs of Lists, Stacks, and Queues
Book Chapter 3: “Lists, Stacks, and Queues”
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