CS1010 AY20/21 S1 Lecture 10

class: middle, center

# Lecture 10

### 25 October 2021

.smaller[
Admin Matters<br>
Unit 25: Tower of Hanoi<br>
Unit 26: Permutation<br>
Unit 27: N Queens<br>
]
---
class: middle

- Assignment 7: Due Wednesday 9 AM
- PE1 Grading: Almost done (only almost is not done)

---
class: middle

### Recursion

- Mostly linear recursion so far
- Can be easily written as loops
- What else can recursion do?

---
class: middle,center

## Tower of Hanoi

---
class: middle,wide

![start](figures/recursion/recursion.001.png)

---
class: middle,wide

![start](figures/recursion/recursion.002.png)

---
class: middle

[Let's play](https://www.mathplayground.com/logic_tower_of_hanoi.html)

---
class: middle

## Recursion

- Find a solution for trivial input (one disc)
- Assume we know how to move $k-1$ discs to another peg
- Solve the problem with $k$ discs

---
class: middle

Start with this:

![start](figures/recursion/recursion.001.png)

---
class: middle

Move $k-1$ disc from A to B (wishful thinking)

![start](figures/recursion/recursion.003.png)

---
class: middle

Move one disc from A to C

![start](figures/recursion/recursion.004.png)

---
class: middle

Move $k-1$ discs from B to C (wishful thinking)

![start](figures/recursion/recursion.005.png)

---
class: middle

### Abstraction

Representating the problem:
- discs as integers 1 to $k$
- pegs as characters `A` `B` `C`

---
class: top

### How many moves?

$$T(1) = 1$$
$$T(k) = 2T(k-1) + 1$$

---
class: middle,center

Iterative solution?

---
class: middle

### Generating Permutations

Given a string with no repetition, say `abc`, generate all permutations of 
the chars in the string:

```C
abc cba acb cab bac bca
```

---
class: middle

### Recursion

- Generate a permutation for trivial input (one-char string)
- Assume we know how to generate permutation for string of length $k-1$
- Generate permutation for string of length $k$

---
class: middle

To permutate `abc`:

- starts with `a`, permutate `bc`: `abc acb`
- starts with `b`, permutate `ac`: `bac bca`
- starts with `c`, permutate `ab`: `cab cba`

---
class: middle,wide

Example:

To permutate `abcd`:

- starts with `a`, permutate `bcd`: `abcd abdc acbd ..`
- starts with `b`, permutate `acd`: `bacd bacd bcad ..`
- starts with `c`, permutate `abd`: `cabd cadb cbad ..`
- starts with `d`, permutate `abc`: `dabc dacb cbac ..`

---
class: middle

To permutate `abc` when we starts with `d`:

- starts with `da`, permutate `bc`: `dabc dacb`
- starts with `db`, permutate `ac`: `dbac dbca`
- starts with `dc`, permutate `ab`: `dcab dcba `

---
class: middle

At any step, we have a fixed prefix, and permute
the suffix (recursively)

---
class: top

### Running Time

$$T(1) = 1$$
$$T(k) = kT(k-1)$$

---
class: middle,center

## N Queens

---
class: middle,center

![queens](figures/recursion/recursion.006.png)

---
class: middle,center

![queens](figures/recursion/recursion.007.png)

---
class: middle,center

Given $n \times n$ chess board, place $n$ queens so that they do not threaten each other.

---
class: top

### Recursion

- Find the solution for trivial input ($1 \times 1$ chessboard)
- Assume we know how to find a solution for a smaller chessboard ($n-1 \times n-1$)
- Find the solution for a $n \times n$ chessboard

Does not work!
---
class: middle

### We need the right representation of the problem

- There is one queen per row, one queen per column
- So the positions of the queens can be represented with the column ids

---
class: middle,center

![queens](figures/recursion/recursion.006.png)

---
class: middle,center

![queens](figures/recursion/recursion.007.png)

---
class: middle

### How to solve

- Generate all possible placement of $n$ queens
- Print all placements that are valid

---
class: middle,wide

### PE Advice

- Don't throw the "free" marks away (style, documentation, warnings)
- Don't write one gigantic function to solve the problem
- Read all questions, plan, and allocate time accordingly
- Solve incrementally (easy cases first, progressively towards harder ones)

---
class: bottom

.tiny[
Version: 1.0 Mon Oct 25 09:42:49 +08 2021
]