mapgen_rivers/terrainlib_lua/rivermapper.lua

-- rivermapper.lua

-- This file provide functions to construct the river tree from an elevation model.
-- Based on a research paper:
--
--     Cordonnier, G., Bovy, B., and Braun, J.:
--     A versatile, linear complexity algorithm for flow routing in topographies with depressions,
--     Earth Surf. Dynam., 7, 549–562, https://doi.org/10.5194/esurf-7-549-2019, 2019.
--
-- Big thanks to them for releasing this paper under a free license ! :)

-- The algorithm here makes use of most of the paper's concepts, including the Planar Borůvka algorithm.
-- Only flow_local and accumulate_flow are custom algorithms.


local function flow_local_semirandom(plist)
	-- Determines how water should flow at 1 node scale.
	-- The straightforward approach would be "Water will flow to the lowest of the 4 neighbours", but here water flows to one of the lower neighbours, chosen randomly, but probability depends on height difference.
	-- This makes rivers better follow the curvature of the topography at large scale, and be less biased by pure N/E/S/W directions.
	-- 'plist': array of downward height differences (0 if upward)
	local sum = 0
	for i=1, #plist do
		sum = sum + plist[i] -- Sum of probabilities
	end

	if sum == 0 then
		return 0
	end
	local r = math.random() * sum
	for i=1, #plist do
		local p = plist[i]
		if r < p then
			return i
		end
		r = r - p
	end
	return 0
end

-- Maybe implement more flow methods in the future?
local flow_methods = {
	semirandom = flow_local_semirandom,
}

-- Applies all steps of the flow routing, to calculate flow direction for every node, and lake surface elevation.
-- It's quite a hard piece of code, but we will go step by step and explain what's going on, so stay with me and... let's goooooooo!
local function flow_routing(dem, dirs, lakes, method) -- 'dirs' and 'lakes' are optional tables to reuse for memory optimization, they may contain any data.
	method = method or 'semirandom'
	local flow_local = flow_methods[method] or flow_local_semirandom

	dirs = dirs or {}
	lakes = lakes or {}

	-- Localize for performance
	local tremove = table.remove
	local mmax = math.max

	local X, Y = dem.X, dem.Y
	dirs.X = X
	dirs.Y = Y
	lakes.X = X
	lakes.Y = Y
	local i = 1
	local dirs2 = {}
	for i=1, X*Y do
		dirs2[i] = 0
	end

	----------------------------------------
	-- STEP 1: Find local flow directions --
	----------------------------------------
	-- Use the local flow function and fill the flow direction tables
	local singular = {}
	for y=1, Y do
		for x=1, X do
			local zi = dem[i]
			local plist = { -- Get the height difference of the 4 neighbours (and 0 if uphill)
				y<Y and mmax(zi-dem[i+X], 0) or 0, -- Southward
				x<X and mmax(zi-dem[i+1], 0) or 0, -- Eastward
				y>1 and mmax(zi-dem[i-X], 0) or 0, -- Northward
				x>1 and mmax(zi-dem[i-1], 0) or 0, -- Westward
			}

			local d = flow_local(plist)
			-- 'dirs': Direction toward which water flow
			-- 'dirs2': Directions from which water comes
			dirs[i] = d
			if d == 0 then -- If water can't flow from this node, add it to the list of singular nodes that will be resolved later
				singular[#singular+1] = i
			elseif d == 1 then
				dirs2[i+X] = dirs2[i+X] + 1
			elseif d == 2 then
				dirs2[i+1] = dirs2[i+1] + 2
			elseif d == 3 then
				dirs2[i-X] = dirs2[i-X] + 4
			elseif d == 4 then
				dirs2[i-1] = dirs2[i-1] + 8
			end
			i = i + 1
		end
	end

	--------------------------------------
	-- STEP 2: Compute basins and links --
	--------------------------------------
	-- Now water can flow until it reaches a singular node (which is in most cases the bottom of a depression)
	-- We will calculate the drainage basin of every singular node (all the nodes from which the water will flow in this singular node, directly or indirectly), make an adjacency list of basins, and find the lowest pass between each pair of adjacent basins (they are potential lake outlets)
	local nbasins = #singular
	local basin_id = {}
	local links = {}
	local basin_links

	-- Function to analyse a link between two nodes
	local function add_link(i1, i2, b1, isY)
		-- i1, i2: coordinates of two nodes
		-- b1: basin that contains i1
		-- isY: whether the link is in Y direction
		local b2
		-- Note that basin number #0 represents the outside of the map; or if the coordinate is inside the map, means that the basin number is uninitialized.
		if i2 == 0 then -- If outside the map
			b2 = 0
		else
			b2 = basin_id[i2]
			if b2 == 0 then -- If basin of i2 is not already computed, skip
				return
			end
		end
		if b2 ~= b1 then -- If these two nodes don't belong to the same basin, we have found a link between two adjacent basins
			local elev = i2 == 0 and dem[i1] or mmax(dem[i1], dem[i2]) -- Elevation of the highest of the two sides of the link (or only i1 if b2 is map outside)
			local l2 = basin_links[b2]
			if not l2 then
				l2 = {}
				basin_links[b2] = l2
			end
			if not l2.elev or l2.elev > elev then -- If this link is lower than the lowest registered link between these two basins, register it as the new lowest pass
				l2.elev = elev
				l2.i = mmax(i1,i2)
				l2.is_y = isY
				l2[1] = b2
				l2[2] = b1
			end
		end
	end

	for i=1, X*Y do
		basin_id[i] = 0
	end

	for ib=1, nbasins do
		-- Here we will recursively search upstream from the singular node to determine its drainage basin
		local queue = {singular[ib]} -- Start with the singular node, then this queue will be filled with water donors neighbours
		basin_links = {}
		links[#links+1] = basin_links
		while #queue > 0 do
			local i = tremove(queue)
			basin_id[i] = ib
			local d = dirs2[i] -- Get the directions water is coming from

			-- Iterate through the 4 directions
			if d >= 8 then -- River coming from the East
				d = d - 8
				queue[#queue+1] = i+1
			-- If no river is coming from the East, we might be at the limit of two basins, thus we need to test adjacency.
			elseif i%X > 0 then
				add_link(i, i+1, ib, false)
			else -- If the eastern neighbour is outside the map
				add_link(i, 0, ib, false)
			end

			if d >= 4 then -- River coming from the South
				d = d - 4
				queue[#queue+1] = i+X
			elseif i <= X*(Y-1) then
				add_link(i, i+X, ib, true)
			else
				add_link(i, 0, ib, true)
			end

			if d >= 2 then -- River coming from the West
				d = d - 2
				queue[#queue+1] = i-1
			elseif i%X ~= 1 then
				add_link(i, i-1, ib, false)
			else
				add_link(i, 0, ib, false)
			end

			if d >= 1 then -- River coming from the North
				queue[#queue+1] = i-X
			elseif i > X then
				add_link(i, i-X, ib, true)
			else
				add_link(i, 0, ib, true)
			end
		end
	end
	dirs2 = nil

	links[0] = {}
	local nlinks = {}
	for i=0, nbasins do
		nlinks[i] = 0
	end

	-- Iterate through pairs of adjacent basins, and make the links reciprocal
	for ib1=1, #links do
		for ib2, link in pairs(links[ib1]) do
			if ib2 < ib1 then
				links[ib2][ib1] = link
				nlinks[ib1] = nlinks[ib1] + 1
				nlinks[ib2] = nlinks[ib2] + 1
			end
		end
	end

	-----------------------------------------------------
	-- STEP 3: Compute minimal spanning tree of basins --
	-----------------------------------------------------
	-- We've got an adjacency list of basins with the elevation of their links.
	-- We will build a minimal spanning tree of the basins (where costs are the elevation of the links). As demonstrated by Cordonnier et al., this finds the outlets of the basins, where water would naturally flow. This does not tell in which direction water is flowing, however.
	-- We will use a version of Borůvka's algorithm, with Mareš' optimizations to approach linear complexity (see paper).
	-- The concept of Borůvka's algorithm is to take elements and merge them with their lowest neighbour, until all elements are merged.
	-- Mareš' optimizations mainly consist in skipping elements that have over 8 links, until extra links are removed when other elements are merged.
	-- Note that for this step we are only working on basins, not grid nodes.
	local lowlevel = {}
	for i, n in pairs(nlinks) do
		if n <= 8 then
			lowlevel[i] = links[i]
		end
	end

	local basin_graph = {}
	for n=1, nbasins do
		-- Iterate in lowlevel but its contents may change during the loop
		-- 'next' called with only one argument always returns an element if table is not empty
		local b1, lnk1 = next(lowlevel)
		lowlevel[b1] = nil

		local b2
		local lowest = math.huge
		local lnk1 = links[b1]
		local i = 0
		-- Look for lowest link
		for bn, bdata in pairs(lnk1) do
			i = i + 1
			if bdata.elev < lowest then
				lowest = bdata.elev
				b2 = bn
			end
		end

		-- Add link to the graph, in both directions
		local bound = lnk1[b2]
		local bb1, bb2 = bound[1], bound[2]
		if not basin_graph[bb1] then
			basin_graph[bb1] = {}
		end
		if not basin_graph[bb2] then
			basin_graph[bb2] = {}
		end
		basin_graph[bb1][bb2] = bound
		basin_graph[bb2][bb1] = bound

		-- Merge basin b1 into b2
		local lnk2 = links[b2]
		-- First, remove the link between b1 and b2
		lnk1[b2] = nil
		lnk2[b1] = nil
		nlinks[b2] = nlinks[b2] - 1
		-- When the number of links is changing, we need to check whether the basin can be added to / removed from 'lowlevel'
		if nlinks[b2] == 8 then
			lowlevel[b2] = lnk2
		end
		-- Look for basin 1's neighbours, and add them to basin 2 if they have a lower pass
		for bn, bdata in pairs(lnk1) do
			local lnkn = links[bn]
			lnkn[b1] = nil

			if lnkn[b2] then -- If bassin bn is also linked to b2
				nlinks[bn] = nlinks[bn] - 1 -- Then bassin bn is losing a link because it keeps only one link toward b1/b2 after the merge
				if nlinks[bn] == 8 then
					lowlevel[bn] = lnkn
				end
			else -- If bn was linked to b1 but not to b2
				nlinks[b2] = nlinks[b2] + 1 -- Then b2 is gaining a link to bn because of the merge
				if nlinks[b2] == 9 then
					lowlevel[b2] = nil
				end
			end

			if not lnkn[b2] or lnkn[b2].elev > bdata.elev then -- If the link b1-bn will become the new lowest link between b2 and bn, redirect the link to b2
				lnkn[b2] = bdata
				lnk2[bn] = bdata
			end
		end
	end

	--------------------------------------------------------------
	-- STEP 4: Orient basin graph, and grid nodes inside basins --
	--------------------------------------------------------------
	-- We will finally solve those freaking singular nodes.
	-- To orient the basin graph, we will consider that the ultimate basin water should flow into is the map outside (basin #0). We will start from it and recursively walk upstream to the neighbouring basins, using only links that are in the minimal spanning tree. This gives the flow direction of the links, and thus, the outlet of every basin.
	-- This will also give lake elevation, which is the highest link encountered between map outside and the given basin on the spanning tree.
	-- And within each basin, we need to modify flow directions to connect the singular node to the outlet.
	local queue = {[0] = -math.huge}
	local basin_lake = {}
	for n=1, nbasins do
		basin_lake[n] = 0
	end
	local reverse = {3, 4, 1, 2, [0]=0}
	for n=1, nbasins do
		local b1, elev1 = next(queue) -- Pop from queue
		queue[b1] = nil
		basin_lake[b1] = elev1
		-- Iterate through b1's neighbours (according to the spanning tree)
		for b2, bound in pairs(basin_graph[b1]) do
			-- Make b2 flow into b1
			local i = bound.i -- Get the coordinate of the link (which is the basin's outlet)
			local dir = bound.is_y and 3 or 4 -- And get the direction (S/E/N/W)
			if basin_id[i] ~= b2 then
				dir = dir - 2
				-- Coordinate 'i' refers to the side of the link with the highest X/Y position. In case it is in the wrong basin, take the other side by decrementing by one row/column.
				if bound.is_y then
					i = i - X
				else
					i = i - 1
				end
			elseif b1 == 0 then
				dir = 0
			end

			-- Use the flow directions computed in STEP 2 to find the route from the outlet position to the singular node, and reverse this route to make the singular node flow into the outlet
			-- This can make the river flow uphill, which may seem unnatural, but it can only happen below a lake (because outlet elevation defines lake surface elevation)
			repeat
				-- Assign i's direction to 'dir', and get i's former direction
				dir, dirs[i] = dirs[i], dir
				-- Move i by following its former flow direction (downstream)
				if dir == 1 then
					i = i + X
				elseif dir == 2 then
					i = i + 1
				elseif dir == 3 then
					i = i - X
				elseif dir == 4 then
					i = i - 1
				end
				-- Reverse the flow direction for the next node, which will flow into i
				dir = reverse[dir]
			until dir == 0 -- Stop when reaching the singular node

			-- Add basin b2 into the queue, and keep the highest link elevation, that will define the elevation of the lake in b2
			queue[b2] = mmax(elev1, bound.elev)
			-- Remove b1 from b2's neighbours to avoid coming back to b1
			basin_graph[b2][b1] = nil
		end
		basin_graph[b1] = nil
	end

	-- Every node will be assigned the lake elevation of the basin it belongs to.
	-- If lake elevation is lower than ground elevation, it simply means that there is no lake here.
	for i=1, X*Y do
		lakes[i] = basin_lake[basin_id[i]]
	end

	-- That's it!
	return dirs, lakes
end


local function accumulate(dirs, waterq)
	-- Calculates the river flow by determining the surface of the catchment area for every node
	-- This means: how many nodes will give their water to that given node, directly or indirectly?
	-- This is obtained by following rivers downstream and summing up the flow of every tributary, starting with a value of 1 at the sources.
	-- This will give non-zero values for every node but only large values will be considered to be rivers.
	waterq = waterq or {}
	local X, Y = dirs.X, dirs.Y

	local ndonors = {}
	local waterq = {X=X, Y=Y}
	for i=1, X*Y do
		ndonors[i] = 0
		waterq[i] = 1
	end

	-- Calculate the number of direct donors
	for i1=1, X*Y do
		local i2
		local dir = dirs[i1]
		if dir == 1 then
			i2 = i1+X
		elseif dir == 2 then
			i2 = i1+1
		elseif dir == 3 then
			i2 = i1-X
		elseif dir == 4 then
			i2 = i1-1
		end
		if i2 then
			ndonors[i2] = ndonors[i2] + 1
		end
	end

	for i1=1, X*Y do
		-- Find sources (nodes that have no donor)
		if ndonors[i1] == 0 then
			local i2 = i1
			local dir = dirs[i2]
			local w = waterq[i2]
			-- Follow the water flow downstream: move 'i2' to the next node according to its flow direction
			while dir > 0 do
				if dir == 1 then
					i2 = i2 + X
				elseif dir == 2 then
					i2 = i2 + 1
				elseif dir == 3 then
					i2 = i2 - X
				elseif dir == 4 then
					i2 = i2 - 1
				end
				-- Increment the water quantity of i2
				w = w + waterq[i2]
				waterq[i2] = w

				-- Stop on an unresolved confluence (node with >1 donors) and decrease the number of remaining donors
				-- When the ndonors of a confluence has decreased to 1, it means that its water quantity has already been incremented by its tributaries, so it can be resolved like a standard river section. However, do not decrease ndonors to zero to avoid considering it as a source.
				if ndonors[i2] > 1 then
					ndonors[i2] = ndonors[i2] - 1
					break
				end
				dir = dirs[i2]
			end
		end
	end

	return waterq
end

return {
	flow_routing = flow_routing,
	accumulate = accumulate,
	flow_methods = flow_methods,
}