Argue that the solution to the recurrence T (n) = T (n/3) + T (2n/3) + cn, where c is a constant, is Ω(n lg n) by appealing to a recursion tree.
Answer to relevant QuestionsUse a recursion tree to give an asymptotically tight solution to the recurrence T(n) = T(αn) + T((1 - α)n) + cn, where α is a constant in the range 0 0 is also a constant.Let A[1 .. n] be an array of n distinct numbers. If i < j and A[i] > A[j], then the pair (i, j) is called an inversion of A. (See Problem 2-4 for more on inversions.) Suppose that each element of A is chosen randomly, ...Show that an n-element heap has height [lg n].Show that quick sort's best-case running time is Ω (n lg n).In the algorithm SELECT, the input elements are divided into groups of 5. Will the algorithm work in linear time if they are divided into groups of 7? Argue that SELECT does not run in linear time if groups of 3 are used.
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