Algorithm.Search

This module contains a collection of generalized graph search algorithms, for when you don't want to explicitly represent your data as a graph. The general idea is to provide these algorithms with a way of generating "next" states (and associated information), a way of determining when you have found a solution, a way of pruning out "dead ends", and an initial state.

Documentation

bfs

Arguments

:: Ord state

=> (state -> [state])

Function to generate "next" states given a current state

-> [state -> Bool]

List of ways to prune search. These are predicates which, if True, are considered to indicate a "dead end".

-> (state -> Bool)

Predicate to determine if solution found. bfs returns a path to the first state for which this predicate returns True.

-> state

Initial state

-> Maybe [state]

First path found to a state matching the predicate, or Nothing if no such path exists.

bfs next prunes found initial performs a breadth-first search over a set of states, starting with initial, generating neighboring states with next, and pruning out any state for which a function in prunes returns True. It returns a path to a state for which found returns True. Returns Nothing if no path is possible.

Example: Making change problem

>>> :{
countChange target = bfs add_one_coin [(> target)] (== target) 0
  where
    add_one_coin amt = map (+ amt) coins
    coins = [25, 10, 5, 1]
:}

>>> countChange 67
Just [25,50,60,65,66,67]

dfs

Arguments

:: Ord state

=> (state -> [state])

Function to generate "next" states given a current state

-> [state -> Bool]

List of ways to prune search. These are predicates which, if True, are considered to indicate a "dead end".

-> (state -> Bool)

Predicate to determine if solution found. dfs returns a path to the first state for which this predicate returns True.

-> state

Initial state

-> Maybe [state]

First path found to a state matching the predicate, or Nothing if no such path exists.

dfs next prunes found initial performs a depth-first search over a set of states, starting with initial, generating neighboring states with next, and pruning out any state for which a function in prunes returns True. It returns a depth-first path to a state for which found returns True. Returns Nothing if no path is possible.

Example: Simple directed graph search

>>> import qualified Data.Map as Map

>>> graph = Map.fromList [(1, [2, 3]), (2, [4]), (3, [4]), (4, [])]

>>> dfs (graph Map.!) [] (== 4) 1
Just [3,4]

dijkstra

Arguments

:: (Num cost, Ord cost, Ord state)

=> (state -> [(cost, state)])

Function to generate list of incremental cost and neighboring states given the current state

-> [state -> Bool]

List of ways to prune search. These are predicates which, if True, are considered to indicate a "dead end".

-> (state -> Bool)

Predicate to determine if solution found. dijkstra returns the shortest path to the first state for which this predicate returns True.

-> state

Initial state

-> Maybe (cost, [(cost, state)])

dijkstra next prunes found initial performs a shortest-path search over a set of states using Dijkstra's algorithm, starting with initial, generating neighboring states and their incremental costs with next, and pruning out any state for which a function in prunes returns True. This will find the least-costly path from an initial state to a state for which found returns True. Returns Nothing if no path to a solved state is possible.

Example: Making change problem, with a twist

>>> :{
-- Twist: dimes have a face value of 10 cents, but are actually rare
-- misprints which are worth 10 dollars
countChange target = dijkstra add_one_coin [(> target)] (== target) 0
  where
    add_one_coin amt =
      map (\(true_val, face_val) -> (true_val, face_val + amt)) coin_values
    coin_values = [(25, 25), (1000, 10), (5, 5), (1, 1)]
:}

>>> countChange 67
Just (67,[(1,1),(1,2),(5,7),(5,12),(5,17),(25,42),(25,67)])

aStar

Arguments

:: (Num cost, Ord cost, Ord state)

=> (state -> [(cost, cost, state)])

Function which, when given the current state, produces a list whose elements are (incremental cost to reach neighboring state, estimate on remaining cost from said state, said state).

-> [state -> Bool]

List of ways to prune search. These are predicates which, if True, are considered to indicate a "dead end".

-> (state -> Bool)

Predicate to determine if solution found. aStar returns the shortest path to the first state for which this predicate returns True.

-> state

Initial state

-> Maybe (cost, [(cost, state)])

aStar next prunes found initial performs a best-first search using the A* search algorithm, starting with the state initial, generating neighboring states, their cost, and an estimate of the remaining cost with next, and pruning out any state for which a function in prunes returns True. This returns a path to a state for which found returns True. If the estimate is strictly a lower bound on the remaining cost to reach a solved state, then the returned path is the shortest path. Returns Nothing if no path to a solved state is possible.

Example: Path finding in taxicab geometry

>>> :{
neighbors (x, y) = [(x, y + 1), (x - 1, y), (x + 1, y), (x, y - 1)]
dist (x1, y1) (x2, y2) = abs (y2 - y1) + abs (x2 - x1)
nextTowards dest pos = map (\p -> (1, dist p dest, p)) (neighbors pos)
:}

>>> aStar (nextTowards (0, 2)) [(== (0, 1))] (== (0, 2)) (0, 0)
Just (4,[(1,(1,0)),(1,(1,1)),(1,(1,2)),(1,(0,2))])