CS 206 Introduction to Computer Science II

Transcription

1 CS 206 Introduction to Computer Science II 03 / 09 / 2018 Instructor: Michael Eckmann

2 Today s Topics Questions? Comments? More examples Change making algorithm Greedy algorithm Recursive implementation Michael Eckmann - Skidmore College - CS Spring 2018

3 What do we know about recursion?

4 We know what a tree is. Can anyone give a recursive definition of a tree?

5 We know what a tree is. Can anyone give a recursive definition of a tree? A tree is empty or it has a root connected to 0 or more subtrees. (Note a subtree, taken on its own, is a tree.)

6 Let's write the recursive depth method in the BST class

7 Change making algorithms. Problem: have some amount of money for which you need to make change in the fewest coins possible. You have unlimited numbers of coins C 1... C N each with different values. example: make change for.63 and the coins you have are C 1 =.01, C 2 =.05, C 3 =.10, and C 4 =.25 only. ideas?

8 Change making algorithms. Problem: have some amount of money for which you need to make change in the fewest coins possible. You have unlimited numbers of coins C 1... C N each with different values. example: make change for.63 and the coins you have are C 1 =.01, C 2 =.05, C 3 =.10, and C 4 =.25 only. The algorithm that works for these denominations is a greedy algorithm (that is, one that makes an optimal choice at each step to achieve the optimal solution to the whole problem.) Let's write it in Java.

9 What if : make change for.63 and the coins you have are C 1 =.01, C 2 =.05, C 3 =.10, C 4 =.21 and C 5 =.25 only. A 21 cent piece comes into the picture.

10 What if : make change for.63 and the coins you have are C 1 =.01, C 2 =.05, C 3 =.10, C 4 =.21 and C 5 =.25 only. A 21 cent piece comes into the picture. The greedy algorithm doesn't work in this case because the minimum is 3 coins all of C 4 =.21 whereas the greedy algorithm would yield 2.25's, 1.10 and 3.01's for a total of 6 coins. We always assume we have.01 coin to guarantee a way to make change.

11 So, we want to create a way to solve the minimum # of coins problem with arbitrary coin denominations. A recursive strategy is: BASE CASE: If the change K, we're trying to make is exactly equal to a coin denomination, then we only need 1 coin, which is the least # of coins. RECURSIVE STEP: Otherwise, for all possible values of i, split the amount of change K into two sets i and K- i. Solve these two sets recursively and when done with all the i's, keep the minimum sum of the number of coins among all the i's.

12 split the total into parts and solve those parts recursively. e.g..63 = = = = = Each time through, save the least number of coins.

13 split the total into parts and solve those parts recursively. e.g..63 = = = = = Each time through, save the least number of coins. The base case of the recursion is when the change we are making is equal to one of the coins hence 1 coin. Otherwise recurse. Why is this bad?

14 split the total into parts and solve those parts recursively. e.g..63 = = = = = Each time through, save the least coins. The base case of the recursion is when the change we are making is equal to one of the coins hence 1 coin. Otherwise recurse. Why is this bad? Let's see (let's try to make change for some amounts with a Java implementation of this.)

15 The major problem with that change making algorithm is that it makes so many recursive calls and it duplicates work already done. Example anyone? An idea called memoization which is used in dynamic programming is a good solution in this case. The basic idea is instead of making recursive calls to figure out something that we already figured out we compute it once and save the value in a table for lookup later.

16 Table lookup We used this idea for the fibonacci numbers. Let's talk about how you could alter the change making algorithm to save results of smaller problems that continually are needed. Then instead of recomputing, we look it up. e.g. If we already computed how many coins it takes to make say 33 cents, we should save that so the next time it is needed, it is not recomputed, but instead just looked up.