CLRs Chapter II Study Questions

Study Questions 2-1

A) The time required for a table of K K ak2+bk+c Ak^2+bk+c, wherein A,b,c a,b,c is constant, for NK \frac NK such tables need Nk⋅ (AK2+BK+C) \frac nk⋅ (ak^2+bk+c), i.e. Θ (NK) \ Theta (NK).

b) If the merging of 2 tables is extended to merge all tables, then Θ (n⋅nk) =θ (n2k) \theta (N⋅\frac nk) =\theta (\frac {n^2}{k}) will be required (each time a minimum element is selected from the NK \frac NK table to be placed into the final table, a heavy Complex n n times).
To reach Θ (N⋅LGNK) \theta (N⋅\lg {\frac NK}), we merge 22, that is, merging two tables at a time, then merging the results of two tables and merging the results of the other two tables again 22 merges, each layer merging requires Θ (n) \theta (n), one with LG (n/k ) \lg (n/k) height, so the total time is Θ (N⋅LG (n/k)) \theta (N⋅\LG (n/k)).

c) when k=θ (LGN) K=\theta (\lg N), there is Θ (NK+N⋅LG (n/k)) =θ (NK+NLGN−NLGK) =θ (2NLGN−NLGLGN