Jump!
February 3, 2017
Our solution visits each cell in order:
(define (jump xs)
(let loop ((reachable 0) (xs xs))
(if (null? xs) #t
(let ((next (max (- reachable 1) (car xs))))
(if (zero? next) #f (loop next (cdr xs)))))))
From any given cell, the maximum reachable position reachable is the greater of one less than the reachable of the previous cell, or the value in the current cell. If reachable goes to 0 before reaching the end of the array, the game is unwinnable. Here are three winnable games and three unwinnable games:
> (jump '(1 3 3 0 0 2 0 4 1 3 0)) #t > (jump '(2 0 3 5 0 0 3 0 0 6 3)) #t > (jump '(3 1 4 3 5 0 0 0 3 1 0)) #t > (jump '(4 4 0 3 0 1 2 2 1 0 0)) #f > (jump '(3 2 0 1 4 3 1 0 0 0 0)) #f > (jump '(5 2 0 0 1 0 6 0 0 2 9)) #f
You can run the program at http://ideone.com/aPTEbc.
Programming Praxis 
(defun winnablep (jumps) (labels ((winnable-position-p (position) (when (<= (length jumps) position) (return-from winnablep t)) (let ((jumpable (aref jumps position))) (when (zerop jumpable) (return-from winnable-position-p nil)) (loop for jump downfrom jumpable to 1 if (winnable-position-p (+ position jump)) do (return-from winnablep t)) nil))) (winnable-position-p 0))) (winnablep #(2 0 3 5 0 0 3 0 0 3 1 0)) ; --> t (winnablep #(2 0 3 1 0 0 3 0 0 3 1 0)) ; --> nil (winnablep #(1)) ; --> t (winnablep #(0)) ; --> nil (winnablep #()) ; --> t (defun winnable-path (jumps) (labels ((winnable-path-from-position (position) (if (<= (length jumps) position) '(t) (let ((jumpable (aref jumps position))) (if (zerop jumpable) nil (loop for jump downfrom jumpable to 1 for path = (winnable-path-from-position (+ position jump)) if path do (return (cons jump path)) finally (return nil))))))) (winnable-path-from-position 0))) (winnable-path #(2 0 3 6 0 0 3 0 0 3 1 0)) ; --> (2 1 6 3 t) (winnable-path #(2 0 3 5 0 0 3 0 0 3 1 0)) ; --> (2 1 3 3 3 t) (winnable-path #(2 0 3 1 0 0 3 0 0 3 1 0)) ; --> nil (winnable-path #(1)) ; --> (1 t) (winnable-path #(0)) ; --> nil (winnable-path #()) ; --> (t)In winnablep, return-from is used to shortcut the exit when the result is known.
On the other hand, in winnable-path, we use O(log(n)) stack space to avoid allocating O(n) temporary cons cells, and instead we cons the path only once we found it.
Two solutions in Python. The first models the problem as finding the shortest path in a DAG (function dag_sp is omitted). You have to set up the weights for all possible jumps.
def winnable(jumps): 'model as shortest path in a DAG and solve' n = len(jumps) W = {i: {} for i in range(n+1)} for label, jump in enumerate(jumps): W[label].update({min(label+j+1, n): 1 for j in range(jump)}) return dag_sp(W, s=0, t=n) < float('inf') def winnable2(jumps): jumps = list(jumps) + [0] # add entry for target reach = 0 for index, jump in enumerate(jumps): if index > reach: return False reach = max(reach, index + jump) return TrueA Haskell version. If we’re looking at an empty list then either the initial list was empty, which we treat as a win, or we’ve made it to the end of a non-empty list, which is by definition a win. If we ever land on a 0 then we’re on a losing path. Otherwise, if any of the paths within n jumps of our current position is a winner then we’ve won.
Here’s another Haskell program that prints out all winning paths. It has a similar structure to the previous program.
Here is a simple solution in Ruby. It uses the Directed Acyclic Graph approach. The class definition is there for readability only and I could of just as well used a hash.
class JumpNode
attr_accessor :val
attr_accessor :winner
attr_reader :links
def initialize(value)
@val = value
@links = []
@winner = false
end
def add_link(link)
@links << link
end
end
def init_graph(cells)
Array.new(cells.length) { |i| JumpNode.new(cells[i]) }
end
def to_jump_graph(cells)
jump_graph = init_graph(cells)
jump_graph.each_index do |node_index|
node = jump_graph[node_index]
if node.val + node_index >= cells.length
node.winner = true
else
(1..node.val).each { |i| node.add_link(jump_graph[node_index + i]) }
end
end
end
def can_reach_end(start_node)
return true if start_node.winner
return false if start_node.links.empty?
start_node.links.each { |link| return true if can_reach_end(link) }
false
end
def winnable?(cells)
return false if cells.nil? || cells.length.zero?
can_reach_end(to_jump_graph(cells)[0])
end
$ irb -r ‘./jump.rb’
irb(main):001:0> winnable?([2, 0, 3, 5, 0, 0, 3, 0, 0, 3, 1, 0])
=> true
irb(main):002:0> winnable?([0, 1, 3])
=> false
irb(main):003:0> winnable?([0, 5, 3, 1, 1])
=> false
irb(main):004:0> winnable?([1, 0, 3, 1, 1])
=> false
irb(main):005:0> winnable?([1, 2, 3, 0, 0])
=> true
irb(main):006:0> winnable?([1, 2, 3, 1, 0])
=> true
irb(main):007:0> winnable?([1, 2, 0, 1, 0])
=> false
irb(main):008:0> winnable?([2, 2, 0, 1, 0])
=> false
irb(main):009:0> winnable?([2, 2, 0, 1, 1])
=> true
irb(main):010:0> winnable?([2, 2, 0, 1, 6])
=> true
irb(main):011:0> winnable?([8, 2, 0, 1, 6])
=> true
irb(main):012:0> winnable?([3, 2, 2, 0, 0])
=> false
irb(main):013:0> winnable?([3, 2, 2, 0, 1])
=> true
class JumpNode attr_accessor :val attr_accessor :winner attr_reader :links def initialize(value) @val = value @links = [] @winner = false end def add_link(link) @links << link end end def init_graph(cells) Array.new(cells.length) { |i| JumpNode.new(cells[i]) } end def to_jump_graph(cells) jump_graph = init_graph(cells) jump_graph.each_index do |node_index| node = jump_graph[node_index] if node.val + node_index >= cells.length node.winner = true else (1..node.val).each { |i| node.add_link(jump_graph[node_index + i]) } end end end def can_reach_end(start_node) return true if start_node.winner return false if start_node.links.empty? start_node.links.each { |link| return true if can_reach_end(link) } false end def winnable?(cells) return false if cells.nil? || cells.length.zero? can_reach_end(to_jump_graph(cells)[0]) end