COMPSCI 330 Design and Analysis of Algorithms Midterm 1 Review

Basic Algorithm Design Techniques Analyzing running time Divide and conquer Dynamic Programming Greedy Common Theme: To solve a large, complicated problem, break it into many smaller sub-problems. Design algorithm Proof of correctness

How to know the right technique? In exam: it will be fairly obvious. In reality: you need experience. Worst case can try all three.

Divide and conquer Use when you can partition problems into unrelated subproblems. Key step: How to merge the solutions. Analysis: Use recursion trees for running time (or guess and do induction) Usually use induction to prove correctness.

Dynamic Programming Use if all subproblems are highly related, and there are not many subproblems you need to solve. Define a table for your subproblems. Consider all possibilities in the last step, write the recursion function for the table. Usually use induction to prove correctness (correctness is often straightforward and you only need to go through the formalities) Try implement the basic algorithms we covered in class.

Greedy Use when you can choose an obvious action to reduce the problem size. Proof is crucial --- you will lose many points if you just give a greedy algorithm, even if it’s correct. Analysis: Try to find counterexamples If you cannot find counterexamples, try prove the correctness by “proof by contradiction” Try to argue OPTIMAL must look like your solution.

Dynamic Programming and Greedy Both require treating the problem as making a sequence of decisions. A greedy problem can often be solved using DP as well, but the DP algorithm might be slower. Once you tweak a greedy problem (by adding weights/costs etc.) often the original greedy algorithm fails and you need to do DP. Get used to finding counter-examples – if you succeed, it means the algorithm is wrong; if you fail, it often gives intuition for proof.

Recursions (a) (You will not see this kind of problem as it is difficult, but just wanted to use it as an example)

Recursions (b) 𝑇 𝑛 =𝐹 log 2 𝑛 𝐹 𝑘 =2𝑇 2 𝑘−1 + 2 𝑘

Dynamic Programming Alice is playing a game where she controls the in-game character to catch pancakes. The character moves on a stage of length n. The character's location can be described by a number x in {1,…,n}. Let xt be the position of the character at time t. At each time step, Alice can move the character to the left (xt+1=xt-1), right (xt+1=xt+1) or do nothing (xt+1=xt). Alice already knows the game very well, so she knows a sequence of pairs (ti; pi) - This means at time ti there will be a pancake at location pi. The character starts at location 1 at time 0, and the game goes until time m. Please design an algorithm to find out how many pancakes Alice can get.

Example n = 5, m = 5 Pancakes ((1; 1); (2; 2); (3; 4); (3; 5); (4; 3); (4; 4); (5; 2)), Optimal solution: 4 Can get (1,1), (2,2), (4, 3) and (5,2) simultaneously

Greedy After scheduling for classrooms, let's schedule meeting rooms. Suppose there are n groups of people requesting for meeting rooms at the same time, and there are currently m rooms available. Group i has ni people, and room j has cj capacity. For group i, they are satisfied with any meeting room j whose capacity cj is at least ni, the number of people in the group. Design an algorithm that tries to find a meeting room for as many groups as possible.

Exam Format is very similar to the practice exam It would be quite obvious what technique you should use. There will be an algorithm design question for divide and conquer but it is not difficult. Problems are not necessarily sorted in level of difficulty.