Assignment: Dynamic Programming

1. Solve Dynamic Programming Problem and Compare with Na?ve approach

You are playing a puzzle. A random number N will be given, you have blocks of length 1 unit

and 2 units. You need to arrange the blocks back to back such that you get a total length of N

units. In how many distinct ways can you arrange the blocks for given N.

a. Write a description of approach to solve it using Dynamic Programming paradigm

b. Implement a function blockpuzzle_dp(N) that solves this problem using Dynamic

Programming using either top-down or bottom-up approach

c. Write pseudocode for the brute force approach

d. Compare the time complexity of both the approaches

e. Write the recurrence formula for the problem

Name your file BlockPuzzle.py

Example 1:

Input: N=2, Result: 2

Explanation: There are two ways. (1+1 , 2)

Example 2:

Input: N=3, Result: 3

Explanation: There are three ways (1+1+1, 1+2, 2+1)

2. Solve a problem using top-down and bottom-up approaches of Dynamic Programming

technique

Two players A & B are playing a game. The rules of the game are:

At the start one number N will be given. The player who starts would have to pick a

number i such that 0<i<N, the condition is that N%i == 0. The second player would pick a

number j from N-i, satisfying the condition 0<j<(N-i). And the game goes on until there is no

more possibility of making any selection. Each player would play in turns, and A always starts

the game. Assume both players play optimally.

Given a number N return if A would win the game or not.

Example 1:

Input: N=2, Result: True

Explanation: A choses 1, and B has no more numbers to chose

Example 2:

Input: N=3, Result: False

Explanation: A choses 1, B choses 1, and A has no more numbers to chose

a. Implement a solution to this problem using Top-down Approach of Dynamic

Programming, name your function game_topdown(N)

b. Implement a solution to this problem using Bottom-up Approach of Dynamic

Programming, name your function game_bottomup(N)

c. Explain your approach to solve this problem. How is your top-down approach different

from the bottom-up approach?

d. What is the time complexity and Space complexity using Top-down Approach

e. What is the time complexity and Space complexity using Bottom-up Approach

f. Write the subproblem and recurrence formula for your approach

Name your file Game.py

Debriefing (required!): --------------------------

Report:

1. Approximately how many hours did you spend on this assignment?

2. Would you rate it as easy, moderate, or difficult?

3. How deeply do you feel you understand the material it covers (0%–100%)?

4. Any other comments?

Note: ‘Debriefing’ section is intended to help us calibrate the assignments.

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