$3.$In this hypothetical scenario, let's consider a city called AAAAA, where bank notes do not exist. Instead. coins of various denominations are in circulation, each with a unique weight. As an expert thief, you plan to rob a bank in this city. For this mission. you only have a bag to store all the looted coins, and it has a limited capacity to carry the weight.

Upon breaking into the bank and killing a dozen people on the way, you come across a table in the main vault that displays all the different denominations of coins, with one coin of each denomination arranged in a row. Being an experienced thief, you know the exact weight of each coin on the table.

Your primary objective is to maximize the total value of the stolen coins while making sure that you do not exceed the weight limit of the bag. If you store more coins than the bag's capacity, the strap will break. resulting in a failed operation. Therefore, you need to make wise choices and consider the weight of each coin while selecting the most valuable ones to steal.

By carefully choosing the coins to steal based on their denominations and weights, you can achieve the optimal outcome of maximizing the stolen money while ensuring the bag's weight limit is not exceeded.

Note:

Please note that the face value of a given coin is in no way related with its weight. So a Rs$.1$ coin may weigh more or lesser than a Rs$.5$ coin.

Also note that all the Rs$,1$ coins have the same weight.

Target$1$:

In developing the above algorithm which algorithm design techniques followed, and justify your answer how that design technique is suitable for this problem.

Is it possible to use both dynamic programming and branch and bound techniques to develop the algorithms? If possible, develop the algorithms. If not possible, give the justification for your answer.

Create your own simulation for the above problems, and analyse the developed algorithms for solving those problems.

Analyse the algorithms for their time and space complexities. Design a java application for the above processes.