Course1_Week1_ProgrammingHomeWork
Exercise 1: Pascal’s Triangle
The following pattern of numbers is called Pascal’s triangle.
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 ...
The numbers at the edge of the triangle are all 1, and each number inside the triangle is the sum of the two numbers above it. Write a function that computes the elements of Pascal’s triangle by means of a recursive process.
Do this exercise by implementing the pascal function in Main.scala, which takes a column c and a row r, counting from 0 and returns the number at that spot in the triangle. For example, pascal(0,2)=1,pascal(1,2)=2 and pascal(1,3)=3.
def pascal(c: Int, r: Int): Int
Exercise 2: Parentheses Balancing
Write a recursive function which verifies the balancing of parentheses in a string, which we represent as a List[Char] not a String. For example, the function should return true for the following strings:
(if (zero? x) max (/ 1 x))
I told him (that it’s not (yet) done). (But he wasn’t listening)
The function should return false for the following strings:
:-)
())(
The last example shows that it’s not enough to verify that a string contains the same number of opening and closing parentheses.
Do this exercise by implementing the balance function in Main.scala. Its signature is as follows:
def balance(chars: List[Char]): Boolean
There are three methods on List[Char] that are useful for this exercise:
- chars.isEmpty: Boolean returns whether a list is empty
- chars.head: Char returns the first element of the list
- chars.tail: List[Char] returns the list without the first element
Hint: you can define an inner function if you need to pass extra parameters to your function.
Testing: You can use the toList method to convert from a String to aList[Char]: e.g. "(just an) example".toList.
Exercise 3: Counting Change
Write a recursive function that counts how many different ways you can make change for an amount, given a list of coin denominations. For example, there are 3 ways to give change for 4 if you have coins with denomination 1 and 2: 1+1+1+1, 1+1+2, 2+2.
Do this exercise by implementing the countChange function inMain.scala. This function takes an amount to change, and a list of unique denominations for the coins. Its signature is as follows:
def countChange(money: Int, coins: List[Int]): Int
Once again, you can make use of functions isEmpty, head and tail on the list of integers coins.
Code
/** * Exercise 1 */ def pascal(c: Int, r: Int): Int = { // 满足条件始终返回1 if (c == 0 || r == c) 1 else pascal(c - 1, r - 1) + pascal(c, r - 1) } /** * Exercise 2 */ def balance(chars: List[Char]): Boolean = { @scala.annotation.tailrec def loop(chars: List[Char], cnt: Int): Boolean = { if (cnt < 0) false else if (chars.isEmpty && cnt == 0) true else { if (chars.head == '(') loop(chars.tail, cnt + 1) else if (chars.head == ')') loop(chars.tail, cnt - 1) else loop(chars.tail, cnt) } } loop(chars, 0) } /** * Exercise 3 */ def countChange(money: Int, coins: List[Int]): Int = { if (money < 0 || coins.isEmpty) 0 else if (money == 0) 1 else // 很好的形式,值得借鉴 countChange(money - coins.head, coins) + countChange(money, coins.tail) }
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