From: Ruby Quiz Date: 2006-10-19T02:15:29+09:00 Subject: [SUMMARY] A* (#98) As was brought up in the discussion, the effectiveness of the A* algorithm is very dependent on the estimate function used to gage distance remaining to the goal. The quiz recommended the Manhattan Distance function for this role, but that's not very well suited to the kind of work we are doing here. Daniel Martin suggested an alternate implementation producing better results. Let's have a look at Daniel's code below. Daniel first posted a set of unit tests to verify code correctness. I won't go into all of those tests here, but let me show a couple. As always, in unit testing it's a good idea to begin by verifying functionality in some trivial case: # ... def test_simple_horizontal map = %q(@..X) solution = @solver.do_quiz_solution(map) assert_paths_equal(%q(####), solution) end # ... This test ensures that the solver can walk a straight path, with no obstacles between the starting point and the goal. Daniel's tests increase in difficulty, of course, and the final test is the one Daniel used to show how the Manhattan Distance gets into trouble: # ... def test_bad_distance map = %q(@.*.. ..~.. ..^.X).sub(/ +/,'') solution = @solver.do_quiz_solution(map) assert_paths_equal( %q(#?#.. .?~#. ..^.#), solution, "Got tripped up by manhattan distance") end # ... Here we see a neat feature of Daniel's custom assert_paths_equal() assertion (not shown). The question marks allow any output in those cells. The reasoning is that there are sometimes multiple correct paths. Here the first move must be to one of the question marks, but either will produce the same length path. The real test here is to ensure that the path goes over the forrest. The Manhattan Distance function can cause algorithms to favor the mountain route. Let's move on to the actual code now. One of the challenges the quiz didn't spend a lot of time on is that A* really needs a priority queue to manage the paths it is examining, always keeping the best path so far first in line. Daniel took the easy way out of this problem and just resorted the paths after each insert: # I suppose someone would think I should use a heap here. # I've found that the built-in sort method is much faster # than any heap implementation in ruby. As a plus, the logic # is easier to follow. class PriorityQueue def initialize @list = [] end def add(priority, item) # Add @list.length so that sort is always using Fixnum comparisons, # which should be fast, rather than whatever is comparison on `item' @list << [priority, @list.length, item] @list.sort! self end def <<(pritem) add(*pritem) end def next @list.shift[2] end def empty? @list.empty? end end The add() method is all of the magic here. When an item is add()ed, Daniel actually adds a three element Array to the queue and resorts the entire queue. The first element of the Array ensures that items are sorted by priority. When priorities match, the second item breaks ties by add order. The item never factors into the sort. When the item is retrieved with next(), the two extra sort fields are discarded. I have to side-step a moment here and address Daniel's first comment. To be clear, he is saying that using Ruby's sort() can be faster than a pure Ruby heap, since sort() is implemented in C. If both elements are correctly implemented in C, the heap should definitely out perform resorting. Finally, the Ruby heap can also be faster, as soon as significant input is involved: #!/usr/bin/env ruby -w require "sorted_array" # Daniel's PriorityQueue require "heap" # pure Ruby Heap, from Ruby Quiz #40 require "benchmark" DATA = Array.new(5_000) { rand(5_000) }.freeze Benchmark.bmbm(10) do |results| results.report("sorted_array:") do queue = PriorityQueue.new DATA.each { |n| queue.add(n, n) } queue.next until queue.empty? end results.report("ruby_heap:") do queue = Heap.new DATA.each { |n| queue.insert(n) } queue.extract until queue.empty? end end # >> Rehearsal ------------------------------------------------- # >> sorted_array: 33.950000 0.020000 33.970000 ( 33.972859) # >> ruby_heap: 0.450000 0.000000 0.450000 ( 0.449776) # >> --------------------------------------- total: 34.420000sec # >> # >> user system total real # >> sorted_array: 33.990000 0.010000 34.000000 ( 34.016562) # >> ruby_heap: 0.440000 0.000000 0.440000 ( 0.437217) OK, let's get back to Daniel's code: require 'enumerator' class Astar def do_quiz_solution(puzzle) @terrain = [] instr = "" puzzle.each {|rowstr| next if rowstr =~ /^\s*$/ rowstr.gsub!(/[^.@~X*^]/,'') instr += rowstr instr += "\n" row = [] rowstr.scan(/[.@~X*^]/) {|terrain| case terrain when /[.@X]/; row << 1 when /[*]/; row << 2 when /\^/; row << 3 when /~/; row << nil end } xind = rowstr.index('X') aind = rowstr.index('@') if (aind) @start = [@terrain.length, aind] end if (xind) @goal = [@terrain.length, xind] end @terrain << row } if do_find_path outarr = instr.split(/\n/) @path.each{|row,col| outarr[row][col]='#'} return outarr.join("\n") else return nil end end # ... This method manages the quiz process. First, an Array is created to hold the costs of each cell and a String to hold the passed map. The each() iterator cleans and dissects the input, filling both structures as it goes. It also locates the @start and @goal coordinates while it works. With the setup out of the way, a hand-off is made to do_find_path() which is the A* implementation. If a path is found, the indicated coordinates are marked on the original map String and returned. Otherwise, nil is returned. Here's the heart of the matter: # ... def do_find_path been_there = {} pqueue = PriorityQueue.new pqueue << [1,[@start,[],1]] while !pqueue.empty? spot,path_so_far,cost_so_far = pqueue.next next if been_there[spot] newpath = [path_so_far, spot] if (spot == @goal) @path = [] newpath.flatten.each_slice(2) {|i,j| @path << [i,j]} return @path end been_there[spot] = 1 spotsfrom(spot).each {|newspot| next if been_there[newspot] tcost = @terrain[newspot[0]][newspot[1]] newcost = cost_so_far + tcost pqueue << [newcost + estimate(newspot), [newspot,newpath,newcost]] } end return nil end # ... This is the A* algorithm in Daniel's code. It begins by establishing a Hash to track where it has been and a PriorityQueue to manage the paths being considered. From there the code dives into the search loop which terminates when the the PriorityQueue is empty?(), indicating that all move options have been exhausted. At each step through the loop, where we are, the path so far, and the cumlative cost are pulled out of the PriorityQueue. We skip ahead if we've been on this spot before. (The first path would have been shorter, since the PriorityQueue keeps them sorted.) The new path is built, including our new location, and we check to see if the @goal has been reached. When it has, the path is converted from to a list of coordinates and returned. If we're not at the goal, we need to extend the path one step in every direction. Here the helper method spotsfrom() generates the choices to iterate over. If we've been there before we can skip it, otherwise the new cost is calculated via the other helper method estimate() and added, with the new path, to the PriorityQueue. Let's have a look at those helper methods: # ... def estimate(spot) # quiz statement version # (spot[0] - @goal[0]).abs + (spot[1] - @goal[1]).abs # my version [(spot[0] - @goal[0]).abs, (spot[1] - @goal[1]).abs].max end def spotsfrom(spot) retval = [] vertadds = [0,1] horizadds = [0,1] if (spot[0] > 0) then vertadds << -1; end if (spot[1] > 0) then horizadds << -1; end vertadds.each{|v| horizadds.each{|h| if (v != 0 or h != 0) then ns = [spot[0]+v,spot[1]+h] if (@terrain[ns[0]] and @terrain[ns[0]][ns[1]]) then retval << ns end end }} retval end end The estimate() method is Daniel's improvement to distance calculation. Since diagonal moves are allowed to shorten two distances at once, we really just need to consider the longer distance, vertically or horizontally, to the goal. The other helper builds a list of neighbors by adding combinations of -1, 0, and 1 as offsets to the current location coordinates. The @terrain Array is used to reality check these coordinates as in-bounds and walkable, before they are added to the returned Array. The final solution is just an instance creation and interface method call away: if __FILE__ == $0 puts Astar.new.do_quiz_solution(ARGF) end My thanks to all the pathfinders who fiddled around with the quiz and to Daniel Martin especially for helping to clarify intent. Ruby Quiz for this week: Seek James out at RubyConf and introduce yourself!