ITEC 122 2008fall ibarland

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lect10a
log; analyzing binary search; mergesort

nod : N -> N, "number of digits":
returns the number of digits of a number.
  nod(999) = 3, nod(1000) = 4, nod (9999) = 4.
(nod(1)=1, so I guess nod(0)=0.)

Note that this actually depends on the base --
  so let nod10 be number-of-digits-if-written-base-10,
and nod2 be number-of-bits, nod16 is number-of-hex-digits, etc.

nod2(17) = ?


A few interesting properties of nod, roughly:
   nod(a*b) =~ ..
   nod(a+b) tends either ... or ...

   nod(10^k) = 
   nod(b^a) = 




Sound familiar?  Sure!  nod is like log.
  (More exactly: nod_k(n) = floor(log_k(n))+1)
To computer scientists, 
   'log' is *though of* as 'number of digits (plus 1)'.
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Play hi-lo.
  Run-time?

WHat is the run-time of
Mergesort?

A: O(n log(n)):
  you make log(n) recursive calls;
  each call also does O(n) work to 'unzip' and
  O(n) work to merge, for 2n log(n).

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