Can be a rule there to transform the Top-Down to Bottom-Up ?

i solve this problem using memoization recursive function (UVa 10306 — e-Coins) , but when i try bottom up i don't know how to make all possible sum pairs in 2d table using their rows and columns indexes , any ideas ? thanks in advance.

And i still asking myself until i become completely mad , how i detect the overlap or the building formula of the dp table in the non classical dynamic programming problem even though i solves the classical ones easily , i really have problem with this what should i do ? :/

is it possible to solve it by a recursive knapsack algorithm ? it take Time limit exceed when i memo it in 2d array , and if it is impossible how can i solve it by 1d subset sum ? thanks in advance :D

i wish that anyone explains to me in simple way the algorithm of the Longest Increasing Subsequence in O(nlogn) time , cuz i actually read all online articles about it and found no one explain it well , thanks in advance :)