mirror of
https://github.com/Brandon-Rozek/website.git
synced 2024-12-24 00:11:02 +00:00
66 lines
1.9 KiB
Markdown
66 lines
1.9 KiB
Markdown
---
|
|
title: Backtracking
|
|
showthedate: false
|
|
---
|
|
|
|
This algorithm tries to construct a solution to a problem one piece at a time. Whenever the algorithm needs to decide between multiple alternatives to the part of the solution it *recursively* evaluates every option and chooses the best one.
|
|
|
|
|
|
## How to Win
|
|
To beat any *non-random perfect information* game you can define a Backtracking algorithm that only needs to know the following.
|
|
- A game state is good if either the current player has already won or if the current player can move to a bad state for the opposing player.
|
|
- A game state is bad if either the current player has already lost or if every available move leads to a good state for the opposing player.
|
|
|
|
```
|
|
PlayAnyGame(X, player)
|
|
if player has already won in state X
|
|
return GOOD
|
|
if player has lost in state X
|
|
return BAD
|
|
for all legal moves X -> Y
|
|
if PlayAnyGame(y, other player) = Bad
|
|
return GOOD
|
|
return BAD
|
|
```
|
|
|
|
In practice, most problems have an enormous number of states not making it possible to traverse the entire game tree.
|
|
|
|
## Subset Sum
|
|
|
|
For a given set, can you find a subset that sums to a certain value?
|
|
|
|
```
|
|
SubsetSum(X, T):
|
|
if T = 0
|
|
return True
|
|
else if T < 0 or X is empty
|
|
return False
|
|
else
|
|
x = any element of X
|
|
with = SubsetSum(X \ {x}, T - x)
|
|
without = SubsetSum(X \ {x}, T)
|
|
return (with or without)
|
|
```
|
|
|
|
X \ {x} denotes set subtraction. It means X without x.
|
|
|
|
```
|
|
ConstructSubset(X, i, T):
|
|
if T = 0
|
|
return empty set
|
|
if T < 0 or n = 0
|
|
return None
|
|
Y = ConstructSubset(X, i - 1, T)
|
|
if Y does not equal None
|
|
return Y
|
|
Y = ConstructSubset(X, i - 1, T - X[i])
|
|
if Y does not equal None
|
|
return Y with X[i]
|
|
return None
|
|
```
|
|
|
|
## Big Idea
|
|
|
|
Backtracking algorithms are used to make a *sequence of decisions*.
|
|
|
|
When we design a new recursive backtracking algorithm, we must figure out in advance what information we will need about past decisions in the middle of the algorithm.
|