Skip list

The skip list is a nifty tool to implement time lines or Z-index for sprites.

Warning, hardcore paragraph: (indexable) skip list is an ordered, probabilistic data structure with O(log(n)) insertions, deletions and search, and O(n*log(n)) space usage. Their main advantage over binary trees is that they're very easy to implement ~= 100 LOC in Lua.

The only restriction is that it's elements have to be comparable with < and <=.

Here's a Lua implementation.

The MIT License
 
Original Python version Copyright (c) 2009 Raymond Hettinger 
see http://code.activestate.com/recipes/576930/
 
Lua conversion + extensions Copyright (c) 2010 Pierre-Yves Gérardy
 
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
 
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
 
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
 
 
do 
local log, floor, ceil, min, random 
= math.log, math.floor, math.ceil, math.min, math.random
 
local makeNode = function(value,size)
	return {
		value=value, 
		next={}, 
		width={}, 
		size=size
	}
end
local End ={}
_End = End
local NIL = makeNode(End,0)
 
local insert = function(self,value)
	local node, chain, stepsAtLevel = self.head, {}, {}
	for i=1, self.maxLevel do stepsAtLevel[i]=0 end
	for level = self.maxLevel, 1, -1 do
		while node.next[level] ~= NIL and node.next[level].value <= value do
			stepsAtLevel[level] = ( stepsAtLevel[level] or 0 ) + node.width[level]
			node = node.next[level]
			--print(level, stepsAtLevel[level],value)
		end
		chain[level]=node
	end
 
	local nodeLevel = min( self.maxLevel, - floor(log(random()) / log(2) ) )
	local newNode = makeNode( value,  nodeLevel)
	local steps, prevNode = 0
	for level= 1, nodeLevel do
		prevNode = chain[level]
		newNode.next[level] = prevNode.next[level]
		prevNode.next[level] = newNode
		newNode.width[level] = prevNode.width[level] - steps
		prevNode.width[level] = steps + 1
		steps = steps + stepsAtLevel[level]
	end
	for level = nodeLevel + 1, self.maxLevel do
		chain[level].width[level] = chain[level].width[level] +1
	end
	self.size = self.size + 1
end
 
local delete = function(self,value)
	-- find first node on each level where node.next[levels].value >= value
 
	node, chain = self.head, {}
	for level = self.maxLevel, 1, -1 do
		while node.next[level] ~= NIL and node.next[level].value < value do
			node = node.next[level]
		end
		chain[level] = node
	end
	if value ~= chain[1].next[1].value then
		return nil, "value not found: "..value
	end
 
	-- remove one link at each level
	nodeLevel = chain[1].next[1].size
	for level = 1, nodeLevel do
		prevnode = chain[level]
		prevnode.width[level] = prevnode.width[level] + prevnode.next[level].width[level] - 1
		prevnode.next[level] = prevnode.next[level].next[level]
	end
	for level = nodeLevel+1, self.maxLevel do
		chain[level].width[level] = chain[level].width[level] - 1
	end
	self.size = self.size - 1
	return true --success
end
 
 
local first = function(self)
	return self.head.next[1].value
end
 
local pop=function (self)
	if self.size == 0 then return nil, "Trying to pop an empty list" end
 
	local node, head = self.head.next[1], self.head
	for level = 1, node.size do
		head.next[level]=node.next[level]
		head.width[level]=node.width[level]
	end
	for level = node.size + 1, self.maxLevel do
		head.width[level] = head.width[level] -1
	end
	self.size = self.size - 1
	return node.value
end
 
-- get the value of the node at index i ( O( log( n ) ) )
 
local islMT = {
	__index = function(self,i)
		if i > self.size then return end
		local node = self.head
		for level=self.maxLevel, 1, -1 do
			-- print(node.width[level],i)
			while node.width[level] <= i do
				i = i - node.width[level]
				node = node.next[level]
			end
		end
	return node.value
	end,
	__newindex = function(self,k,v) 
		if k==0 then self:insert(v)
		else error("use SkipList[0]=el or SkipList:insert(el)")
		end
	end,
	}
 
local tostring = function (self)
	local t = {}
	for k,v in self:iter() do table.insert(t,v) end
	return "( "..table.concat(t,", ").. " )"
end
 
local iter = function (self)
	local node, size = self.head.next[1] , self.size
	count = 0
	return function()
		value=node.value
		node = node.next[1]
		count = count+1
		return count <= size and count or nil, value 
	end
end
 
function makeSkipList(expected_size)
	local maxLevel = floor( log(expected_size) / log(2) ) -- FIXME?? * or / ?
	local head = makeNode("HEAD",maxLevel)
	for i=1,maxLevel do 
		head.next[i] = NIL
		head.width[i] = 1
	end
 
	return setmetatable( {
		size = 0,
		head = head,
		maxLevel = maxLevel,
		insert = insert,
		delete = delete,
		first = first,
		tostring = tostring,
		iter=iter,
		pop = pop
		}, islMT )
end
 
end

The API by example:

math.randomseed(os.time())
 
-- Creation:
 
insdel = makeSkipList(8) -- you must pass an estimation of the length of the list for better performance.
 
-- insertions
s = "YeeHoyeE" 
for i=1,#s do
	 insdel:insert(s:sub(i,i)) 
	print(insdel:tostring()) 
end
print()
 
--  indexing
print(insdel[4])
print(insdel[8])
print(insdel[12]) -- out of bounds --> nil
print()
 
-- iterate over the list.
for k,v in insdel:iter() do 
	print(k,v) 
end
print()
 
-- remove elements
print(insdel:delete"Z") --> not found
 
for i=1,#s do 
	insdel:delete(s:sub(i,i)) 
	print(insdel:tostring()) 
end
 
-- alternate insertion syntax, pop() and first()
insdel:insert("e")
insdel[0]=("g")
 
print( insdel:first() ) -- returns the first element.
print( insdel:pop() ) -- returns the first element and removes it from the list
print( insdel:first() )
print( insdel.size )
print( insdel:pop() )
print( insdel:pop() ) -- attempt to pop an empty list --> nil + error message.
print()
 
-------------------------------
-- ASCII representation of the structure:
 
l = makeSkipList(9)
s = "SweetLOVE"
 
for i=1,#s do local e = s:sub(i,i) l:insert(e) end
 
for level = l.maxLevel, 1, -1 do
	local line='Level '..level..": "
	node = l.head
	---[[
	while node.value ~= _End do
		line=line..node.value..node.width[level].." "
		for i=1,node.width[level]-1 do line=line.."   " end
		node = node.next[level]
	end
	--]]
 
	print(line.."NIL")
end

Output:

( Y )
( Y, e )
( Y, e, e )
( H, Y, e, e )
( H, Y, e, e, o )
( H, Y, e, e, o, y )
( H, Y, e, e, e, o, y )
( E, H, Y, e, e, e, o, y )
 
e
y
nil
 
1       E
2       H
3       Y
4       e
5       e
6       e
7       o
8       y
 
nil     value not found: Z
( E, H, e, e, e, o, y )
( E, H, e, e, o, y )
( E, H, e, o, y )
( E, e, o, y )
( E, e, y )
( E, e )
( E )
(  )
e
e
g
1
g       
nil     Trying to pop an empty list
 
 
Level 3: HEAD4          S6                NIL
Level 2: HEAD1 E3       S1 V2    e2    w1 NIL
Level 1: HEAD1 E1 L1 O1 S1 V1 e1 e1 t1 w1 NIL