class: middle, center # Lecture 10 ### 28 October 2024 .smaller[ Admin Matters<br> Unit 25: Tower of Hanoi<br> Unit 26: Permutation<br> Unit 27: N Queens<br> ] --- ### Midterm and PE0 Twos Please file your request to remark by this Friday, 1 November 2024, 2359 --- class: middle, center ### No labs this week Happy Deepavali / Diwali! --- ### 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  --- class: middle,wide  --- class: middle,center [Let's play](https://www.mathplayground.com/logic_tower_of_hanoi.html) --- ### 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 --- Start with this:  --- Move $k-1$ disc from A to B (wishful thinking)  --- Move one disc from A to C  --- Move $k-1$ discs from B to C (wishful thinking)  --- ### Abstraction Representating the problem: - discs as integers 1 to $k$ - pegs as characters `A` `B` `C` --- ### How many moves? $T(1) = 1$ $T(k) = 2T(k-1) + 1$ --- #### Question: How about a different algorithm? 1. Move the top disk from peg `A` to peg `B`. -- 2. Move the remaining $k - 1$ disks from peg `A` to peg `C`. -- 3. Move the top disk from peg `B` to peg `C`. -- **Question:** Where did we go wrong? --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle,center Iterative solution? --- ### 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 ``` --- ### 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, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- class: middle, left  --- 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: wide Example: To permutate `abcd`: - start with `a`, permutate `bcd`: `abcd abdc acbd ..` - start with `b`, permutate `acd`: `bacd bacd bcad ..` - start with `c`, permutate `abd`: `cabd cadb cbad ..` - start with `d`, permutate `abc`: `dabc dacb cbac ..` --- To permutate `abc` when we start with `d`: - start with `da`, permutate `bc`: `dabc dacb` - start with `db`, permutate `ac`: `dbac dbca` - start with `dc`, permutate `ab`: `dcab dcba ` --- At any step, we have a fixed prefix, and permute the suffix (recursively) --- ### Running Time $T(1) = 1$ $T(k) = kT(k-1) + 1$ --- class: middle,center ## N Queens --- class: middle,center  --- class: middle,center  --- class: middle,center Given $n \times n$ chess board, place $n$ queens so that they do not threaten each other. --- ### 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! --- ### 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  --- class: middle,center  --- ### How to solve - Generate all possible placement of $n$ queens - Print all placements that are valid --- class: middle,center  --- class: middle,center  --- class: bottom .tiny[ Version: 1.0 Sat Oct 26 14:45:45 +08 2024 ]