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CSE 3358 Note Set 2

CSE 3358 Note Set 2. Data Structures and Algorithms. Overview:. What are we measuring and why? Computational complexity introduction Big-O Notation . Problem. For a problem Different ways to solve – differing algorithms

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CSE 3358 Note Set 2

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  1. CSE 3358 Note Set 2 Data Structures and Algorithms

  2. Overview: • What are we measuring and why? • Computational complexity introduction • Big-O Notation

  3. Problem • For a problem • Different ways to solve – differing algorithms • Problem: Searching an element in an arrayPossible searching algorithms?

  4. The Selection Problem • Problem: Find the Kth largest number in a set • Solution Possibilities:

  5. The Selection Problem • If dataset size = 10,000,000 and k = 5,000,000 • Previous algos could take several days • Is one algorithm better than the other? • Are there better algorithms?

  6. Efficiency • Limited amount of resources to use in solving a problem. • Resource Examples: • We can use any metric to compare various algorithms intended to solve the same problem. • Using some metrics in for some problems might not be enlightening…

  7. Computational Complexity • Computational complexity – the amount of effort needed to apply an algorithm or how costly it is. • Two most common metrics (and the ones we’ll use) • Time (most common) • Space • Time to execute an algorithm on a particular data set is system dependent. Why?

  8. Seconds? Microseconds? Nanoseconds? • Can’t use the above when talking about algorithms unless specific to a particular machine at a particular time. • Why not?

  9. What will we use? • Logical units that express the relationship between the size n of a data set and the amount of time t required to process the data • For algorithm X using dataset n, T(n) = amount of time needed to execute X using n. • Problem: Ordering of the values in n can affect T(n). What to do?

  10. How we measure resource usage • Three primary ways of mathematically discussing the amount of resources used by an algorithm • O(f(n)) • Ω(f(n)) • (f(n)) • What is ___(f(n)) ?

  11. The Definitions • T(N) = O(f(n)) if there are positive constants c and n0such that T(N) <= c*f(n) when N >= n0. • T(N) = Ω (g(n)) if there are positive constants c and n0such that T(N) >= c*g(n) when N >= n0. • T(N) = (h(n)) iffT(N) = O(h(n)) and T(N) = Ω (h(n))

  12. The Goal??? • To place a relative ordering on functions • To examine the relative rates of growth. • Example:

  13. Big - Oh • T(N) = O(f(n)) • What does this really mean? • Means: • T is big-O of f if there is a positive number c such that T is not larger than c*f for sufficiently large ns (for all ns larger than some number N) • In other words: • The relationship between T and f can be expressed by stating either that f(n) is an upper bound on the value of T(n) or that , in the long run, T grows at most as fast as f.

  14. Asymptotic Notation: Big-O Graphic Example

  15. Asymptotic Notation: Big-Oh Example: Show that 2n2 + 3n + 1 is O(n2). By the definition of Big-O 2n2 + 3n + 1 <= c*n2 for all n >= N. So we must find a c and N such that the inequality holds for all n > N. How?

  16. Asymptotic Notation: Big-O • Reality Check: • We’re interested in what happens to the number of ops needed to solve a problem as the size of the input increases toward infinity. • Not too interested in what happens with small data sets.

  17. Notes on Notation • Very common to see T(n) = O(f(n)) • Not completely accurate • not symmetric about = • Technically, O(f(n)) is a set of functions. • Set definition O(g(n)) = { f(n): there are constants c > 0, N>0 such that 0<=f(n)<=g(n) for all n > N.} • When we say f(n) = O(g(n)), we really mean thatf(n)∈ O(g(n)).

  18. Asymptotic Notation: Big-O Show that n2 = O(n3).

  19. Three Cases for Analysis • Best Case Analysis: • when the number of steps needed to complete an algorithm with a data set is minimized • e.g. Sorting a sorted list • Worst Case Analysis: • when the maximum number of steps possible with an algorithm is needed to solve a problem for a particular data set • Average Case Analysis:

  20. Example • The problem is SORTING (ascending) • Best Case: • Worst Case: • Average Case: Data Set: n = _______ 5 3 9 8

  21. T(N)? • For a particular algorithm, how do we determine T(N)? • Use a basic model of computation. • Instructions are executed sequentially • Standard simple instructions • Add, subtract, multiply, divide • Comparison, store, retrieve • Assumptions: • Takes one time unit, T(1) to do anything simple

  22. Determining Complexity • How can we determine the complexity of a particular algorithm? int sum(int* arr, int size) { int sum = 0; for (inti = 0; i < size; i++) sum += arr[i]; return sum; }

  23. ?

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