Lecture 12 Ticket

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In class we saw the “Master Theorem” for solving recurrences:

Master Theorem. Suppose the running time \(T(n)\) of a (recursively defined) method satisfies \(T(n) = a T(n / b) + f(n)\). Define \(c = \log_b a\). Then:

1. If \(f(n) = O(n^d)\) for \(d < c\) then \(T(n) = O(n^c)\)
2. If \(f(n) = \Theta(n^c \log^k n)\) then \(T(n) = O(n^c \log^{k+1} n)\)
3. If \(f(n) = \Omega(n^d)\) for \(d > c\), then \(T(n) = O(f(n))\)

For the following methods, apply the Master Theorem to derive a bound on the running time of the method. In particular, for each method you should compute the values \(a, b, c,\) and the function \(f\) to determine which (if any) of the three conditions above apply. In both cases, the size \(n\) of the input is \(n = j - i\).

1. Binary Search.

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    BinarySearch(a, val, i, j):
      if j = i then return false
      if j - i = 1 then return a[i] = val
      m <- (j + i) / 2
      if a[m] > val then
        return BinarySearch(a, val, i, m)
      else
        return BinarySearch(a, val, m, j)
      endif
   

2. Merge Sort. Assume that the Merge procedure runs in time \(O(n)\).

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      MergeSort(a, i, j):
        if j - i = 1 then
          return
        endif
        m <- (i + j) / 2
        MergeSort(a,i,m)
        MergeSort(a,m,j)
        Merge(a,i,m,j)