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Unformatted text preview: ˛ . One iteration reduces the number of elements from n to ˛n , and i iterations reduces the number of elements to ˛ i n . At a leaf, there is just one remaining element, and so at a minimum-depth leaf of depth m , we have ˛ m n D 1 . Thus, ˛ m D 1=n . Taking logs, we get m lg ˛ D N lg n , or m D N lg n= lg ˛ . Similarly, maximum depth corresponds to always taking the larger part of the par-tition, i.e., keeping a fraction 1 N ˛ of the elements each time. The maximum depth M is reached when there is one element left, that is, when .1 N ˛/ M n D 1 . Thus, M D N lg n= lg .1 N ˛/ . All these equations are approximate because we are ignoring floors and ceilings....
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- Spring '10