for a given function gn we denote by ogn the set of

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Unformatted text preview: O(f(n)) if constants k and n0 exist such that A requires no more than k · f(n) time units to solve a problem of size n ≥ n0 Alternative definition of big O analysis [Cormen et al.] For a given function g(n), we denote by O(g(n)) the set of functions O(g(n)) = {f(n) | there exist positive constants k and n0 such that 0 ≤ f(n) ≤ k · g(n) for all n ≥ n0} •  Thus, big O gives an upper bound on a function f(n) to within a constant factor CPSC 223, 2009 7 Growth Rates Fairly typical growthrate functions •  Ordered from least to most expensive [Carrano, 2007] CPSC 223, 2009 8 4 9/23/10 Asymptotic Upper Bounds Big O notation can sometimes be misleading –  We are considering asymptotic upper bounds Any algorithm that is O(n) is also O(n2), since every function f(n) in O(n) is also in O(n2) … (n ≤ k1·n ≤ k2·n2) –  Are we analyzing the algorithm or the problem? The algorithm is a particular computational approach for solving the problem … e.g., selection sort is O(n2) The problem is the general task … e.g., what is the smallest upper bound for the complexity of sorting CPSC 223, 2009 9 CPSC 223  ­ ­ Fall 2010 10 Binary Search 5 9/23/10 Recurring Themes … •  What are common List ADT operations? –  Insert items into the collection (add) –  Remove items from the collection (delete) –  Search for elements of a collection (retrieve, lookup) –  … plus checking if empty, printing, etc. •  When analyzing data structures, we often are comparing the cost of each of these operations •  So how does “sorting” fit in? CPSC 223, 2009 11 Why Sorting … •  We often want to output elements in order •  Lookup (search) is more efficient if items are sorted –  for an unordered list O(n) … why? –  for a sorted list O(log n) … we’ll disc...
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