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2

Let $\sigma(S)$ denote the sum of all the elements in $S$ and $A_i = \{a_1, \dots, a_i\}$. Given, $i=0,\dots,n$ and $w \in \mathbb{Z}$, define $OPT[i,w]$ as the mazimum cardinality of a subset $S \cup S' \subseteq A_i$ where $S \cap S' = \emptyset$ and $\sigma(S) - 2\sigma(S') = w$. If no possible choice for $S$ and $S'$ exists, then let $OPT[i,w]=-\infty$....

1

What you want to prove is false. $T(n) = 16 T(n/4) + n^2\log^3 n$ has solution $T(n) = \Theta(n^2 \log^4 n)$. To see this notice that this recurrence fits in the general form $T(n) = a T(n/b) + f(n)$ once you set $a=16, b=4, f(n)=n^2\log^3 n$. Since $f(n) = n^2 \log^3 n = \Theta( n^{ \log_b a } \cdot \log^k n)$ for $k=3$, you can apply the Master theorem ...

-1

Perhaps this is a reference to Cobham's thesis https://en.wikipedia.org/wiki/Cobham%27s_thesis?wprov=sfla1. Usually the word used is "feasible" though.

3

There is no one true answer. It depends on context. The most common context is one where polynomial-time is taken as more or less synonymous with efficient, so if you had no further context, I would certainly guess "polynomial time". Polylogarithmic time is used only in very narrow contexts. In general, if you think your audience might not be sure about ...

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