首页SupposewehaveT(n)≤c=O(1)foralln≤3,andforeveryn≥4,wehave ，T(n)<=T(n/4)+T(0.75*n) + c*n,Use Mathematical Induction to prove that T (n) = O(n log n) for all n ≥ 4.

SupposewehaveT(n)≤c=O(1)foralln≤3,andforeveryn≥4,wehave ，T(n)<=T(n/4)+T(0.75n) + cn,Use Mathematical Induction to prove that T (n) = O(n log n) for all n ≥ 4.

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证明：首先，假设T(n)<=c*n，当n=4时，我们有T(4)<=c*4，即T(4)<=O(1)；假设T(k)<=c*k*log k，当k>4时，由T(n)<=T(n/4) T(0.75*n)和T(n)<=c*n可得：T(n)<=(c/4)*n*log(n/4)+(3/4)*c*n<=c*n*(log n-log 4+3)<=c*n*log n，即T(n)<=c*n*log n，由归纳法可知T(n)<=c*n*log n 对于所有n >= 4。因此，T(n)=O(n log n) 对于所有n>=4。

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