Data Structures and Algorithms Exam Review
1) Recurrences
1.1 10-mark: Solve T(n)=2T(n/2)+n, T(1)=c
T(n)=2T(n/2)+n
=4T(n/4)+2n
=8T(n/8)+3n
… after i steps: T(n)=2i T(n/2i) + i*n
Stop: n/2i=1 => i=log2 n
T(n)=n*T(1) + n log2 n = n*c + n log2 n
=> T(n)=Θ(n log n)
1.2 Master Theorem (Write As-Is)
T(n)=aT(n/b)+f(n), compare f(n) with nlogb a
- Case 1: f(n)=O(nlogb a – ε) => T(n)=Θ(nlogb a)
- Case 2: f(n)=Θ(nlogb a logk n) => T(n)=Θ(nlogb a logk+1 n)
- Case 3: f(n)=Ω(nlogb a + ε) and a f(n/b) ≤ c f(n) (c<1) => T(n)=Θ(f(n))
2)
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#include <sys/types.h>
#include <unistd.h>
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pid = getpid();
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