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Algorithm Analysis Algorithm Analysis Lectures 3 & 4 Resources Data Structures & Algorithms Analysis in C++ (MAW): Chap. 2 Introduction to Algorithms (Cormen, Leiserson, & Rivest): Chap.1 Algorithms Theory & Practice (Brassard & Bratley): Chap. 1
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Algorithms An algorithm is a well-defined computational procedure that takes some value or a set of values, as input and produces some value, or a set of values as output. Or, an algorithm is a well-specified set of instructions to be solve a problem.
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Efficiency of Algorithms Empirical –Programming competing algorithms and trying them on different instances Theoretical –Determining mathematically the quantity of resources (execution time, memory space, etc) needed by each algorithm
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Analyzing Algorithms Predicting the resources that the algorithm requires: Computational running time Memory usage Communication bandwidth The running time of an algorithm Number of primitive operations on a particular input size Depends on –Input size (e.g. 60 elements vs. 70000) –The input itself ( partially sorted input for a sorting algorithm)
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Order of Growth The order (rate) of growth of a running time –Ignore machine dependant constants –Look at growth of T(n) as n – notation Drop low-order terms Ignore leading constants E.g. –3n 3 + 90n 2 – 2n +5 = (n 3 )
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Mathematical Background
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Definitions: –T(N) = O(f(N)) iff c and n 0 T(N) c.f(N) when N n 0 –T(N) = (g(N)) iff c and n 0 T(N) c.g(N) when N n 0 –T(N) = (h(N)) iff T(N) = O(h(N)) and T(N) = (h(N))
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Mathematical Background Definitions: –T(N) = o(f(N)) iff c and n 0 T(N) c.f(N) when N n 0 –T(N) = (g(N)) iff c and n 0 T(N) c.g(N) when N n 0
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Mathematical Background Rules: –If T 1 (N) = O(f(N)) and T 2 (N) = O(g(N)) then a) T 1 (N) + T 2 (N) = max( O(f(N)),O(g(N)) b) 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 ) –Log k N = O(N) for any constant k.
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More … 1.3n 3 + 90n 2 – 2n +5 = O(n 3 ) 2.2n 2 + 3n +1000000 = (n 2 ) 3.2n = o(n 2 ) ( set membership) 4.3n 2 = O(n 2 ) tighter (n 2 ) 5.n log n = O(n 2 ) 6.True or false: –n 2 = O(n 3 ) –n 3 = O(n 2 ) –2 n+1 = O(2 n ) –(n+1)! = O(n!)
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Ranking by Order of Growth 1 nn log nn 2 n k (3/2) n 2 n (n)!(n+1)!
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Running time calculations Rule 1 – For Loops The running time of a for loop is at most the running time of the statement inside the for loop (including tests) times the number of iterations Rule 2 – Nested Loops Analyze these inside out. The total running time of a statement inside a group of nested loops is the running time of the statement multiplied by the product of the sizes of all the loops
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Running time calculations: Examples Example 1: sum = 0; for (i=1; i <=n; i++) sum += n; Example 2: sum = 0; for (j=1; j<=n; j++) for (i=1; i<=j; i++) sum++; for (k=0; k<n; k++) A[k] = k;
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