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Time complexity

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Basic info about how to use time complexity and how to get the time of each program to solve the problem.

Basic info about how to use time complexity and how to get the time of each program to solve the problem.

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  • 1. CS 213Data Structures and AlgorithmsSecond Semester, 2011-2012
  • 2. Sum of Squaresint sumOfSquares(int n){ int sum = 0; for(int i=1; i<=n; i++) sum = sum + i*i; return sum;} T(n) = 5n + 4
  • 3. Selection SortGiven the following numbers in an array: 53 23 10 34 2 17 2 23 10 34 53 17 2 10 23 34 53 17 2 10 17 34 53 23 2 10 17 23 53 34 2 10 17 23 34 53
  • 4. Still on Selection Sortvoid selectionSort(int A[], int n){ for(int i=0; i<n; i++){ int min = i; for(int j=i+1; j<n; j++){ if(A[j] < A[min]) min = j; What is this function’s T(n)? What } we need is to compute for T(n) int temp = A[i]; from the inner loop going A[i] = A[min]; outwards. A[min] = temp; We need to use the summation } notation to solve the T(n).}
  • 5. The Summation Notation = x1 + x2 + x3 + x4
  • 6. Still on the summation notation4 n∑ i = 1+2+3+4 = 10 ∑i = n ( n +1) 2 i =1i=1 5 n∑ 3 = 3+3+3+3+3 = 15 ∑c =nc =15i=1 i=1
  • 7. Exercise for(int i=1; i<=5; i++) cout<<endl; for(int i=1; i<=n; i++) cout<<endl; for(int i=3; i<=m; i++) cout<<endl; for(int i=1; i<=6; i++) for(int j=1; j<=8; j++) cout<<endl; for(int i=1; i<=n; i++) for(int j=1; j<=n; j++) cout<<endl; for(int i=1; i<=m; i++) for(int j=1; j<=m; j++) cout<<endl; for(int i=1; i<=n; i++) for(int j=i; j<=n; j++) cout<<endl;
  • 8. Going back to the Selection Sortvoid selectionSort(int A[], int n){ for(int i=0; i<n; i++){ n n n int min = i; for(int j=i+1; j<n; j++){ ∑ ∑ c =∑ c(n-i+1-1) if(A[j] < A[min]) i=1 j=i+1 i=1 min = j; } n n int temp = A[i]; A[i] = A[min]; ∑ c(n-i) = c n-i∑ A[min] = temp; i=1 i=1 } n n n} ∑ n-i = ∑ n - ∑ i i=1 i=1 i=1
  • 9. Time ComplexityA function that maps problem size into the time required to solve the problem.Typically, we are interested in the inherent complexity of computing the solution to problems in a particular class.
  • 10. Lower Bound We might want to know how fast we can sort a list of n items, initially in an arbitrary order, regardless of the algorithm used. What is sought here is the lower bound, L(n), on sorting, a property of the sorting problem and not of any particular algorithm. This says that no algorithm can do the job in fewer than L(n) time units for arbitrary inputs.
  • 11. Upper Bound We might also like to know how long it would take to sort such list using a known algorithm with a worst-case input. What is sought here is the upper bound, U(n), which says that for arbitrary inputs we can always sort in time at most U(n).
  • 12. Goal of Time Complexity Analysis While there are apparently two complexity functions for problems, L(n) and U(n), the ultimate goal is to make these two bounds coincide. This is the optimal algorithm which has L(n) = U(n). For some of the problems, this goal has not been realized yet!
  • 13. InvitationConsider this, CS 213, as you journey into finding optimal solutions to classes of problems!Who knows, you might win a million dollars ($$$) from Claymath Foundation!
  • 14. Upper Bound ComplexityThere are two ways in analyzing this bound: Counting instructions Solving recurrencesBoth are used to find the worst case of an algorithm.
  • 15. Big O-Notation (O(g(n)))The O-notation is used to describe the worst-case running time of an algorithm.O(n) means that the growth of the running time of the algorithm is a function of n.O-notation computes for the upper bound.
  • 16. O-Notation DefinedO(g(n)) = {f(n): ∃ c>0, n0>0 s.t. 0 ≤ f(n) ≤ cg(n) for all n ≥ n0}.Example: Check if (n2/2) – 3n ∈ O(n2) (n2/2) – 3n ≤ cn2 ½ - 3/n ≤ c Choosing c = ½, n0 = 6 proves the claim.
  • 17. Another Example3n2 - 100n + 6 ∈ O(n2) ? 3n2 - 100n + 6 ≤ cn2 3 – 100/n + 6/n2 ≤ c At this point, we have to choose a c>0 and an n0 What values will prove our claim?
  • 18. Lower Bound Complexity This is the more difficult of the bounds. There is no algorithm to analyze. Ω(g(n)) is used to describe the lower bound of the running time of the algorithm or minimum possible running time of the algorithm.
  • 19. Ω-NotationΩ(g(n)) = {f(n): ∃ c>0, n0>0 s.t. 0 ≤ cg(n) ≤ f(n) for all n ≥ n0}.Example: Check if (n2/2) – 3n ∈ Ω(n2) cn2 ≤ (n2/2) – 3n c ≤ ½ - 3/n Choosing c = 1/14, n0 = 7 proves the claim.
  • 20. Another Example Check if 3n2 - 100n + 6 ∈ Ω(n)  cn ≤ 3n2 - 100n + 6  c ≤ 3n – 100 + 6/n At this point we need to find a c and an n0 that will prove our claim. What values of c and n0 will suffice the inequality??
  • 21. θ-NotationUsed to denote that the lower and upper bounds of the running time of the algorithm is tight, i.e. the growth rate of the upper and lower bounds are the same.
  • 22. θ-Notation Definedθ(g(n)) = {f(n): ∃ c1>0, c2>0, n0>0 s.t. 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n ≥ n0}.f(n) ∈ θ(g(n)) if f(n) ∈ O(g(n)) and f(n) ∈ Ω(g(n))
  • 23. Complexity ClassesDescription O-notationconstant O(1)logarithmic O(log n)linear O(n)n log n O(n log n)quadratic O(n2)cubic O(n3)polynomial O(nk), k≥1exponential O(an), a>1
  • 24. Growth rate of complexity classes class n=2 n=16 n=256 n=1024 1 1 1 1 1log n 1 4 8 10n 2 16 256 1024n log n 2 64 2048 10240n^2 4 256 65536 1048576n^3 8 4096 16777216 1.07E+092^n 4 65536 1.16E+77 1.8E+308
  • 25. Graph of the Growth Rates 50 n^2 45 40 n log n 35 2^n 30 n^3 25 20 15 10 n 5 log n 0 1 2 3 4 5 6 7 8 9 10 11log n 0 1 1.58 2 2.32 2.58 2.81 3 3.17 3.32 3.46n 1 2 3 4 5 6 7 8 9 10 11n log n 0 2 4.75 8 11.61 15.51 19.65 24 28.53 33.22 38.05n^2 1 4 9 16 25 36 49 64 81 100 121n^3 1 8 27 64 125 216 343 512 729 1000 13312^n 2 4 8 16 32 64 128 256 512 1024 2048
  • 26. Bigger N 2000 1800 1600 1400 n^3 1200 1000 800 600 2^n 400 200 n^2 0 n log n n n log 1 2 3 4 5 6 7 8 9 10 11log n 0 1 1.58 2 2.32 2.58 2.81 3 3.17 3.32 3.46n 1 2 3 4 5 6 7 8 9 10 11n log n 0 2 4.75 8 11.61 15.51 19.65 24 28.53 33.22 38.05n^2 1 4 9 16 25 36 49 64 81 100 121n^3 1 8 27 64 125 216 343 512 729 1000 13312^n 2 4 8 16 32 64 128 256 512 1024 2048
  • 27. Still on the Graph 2000 1800 1600 1400 n^3 1200 10*n^2 1000 800 20*n log n 600 50*n 2^n 400 100*log n 200 0 1 2 3 4 5 6 7 8 9 10 11100*log n 0 100 158.5 200 232.19 258.5 280.74 300 316.99 332.19 345.9450*n 50 100 150 200 250 300 350 400 450 500 55020*n log n 0 40 95.1 160 232.19 310.2 393.03 480 570.59 664.39 761.0710*n^2 10 40 90 160 250 360 490 640 810 1000 1210n^3 1 8 27 64 125 216 343 512 729 1000 13312^n 2 4 8 16 32 64 128 256 512 1024 2048

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