Problem:
Score : 500 points

Problem Statement

There are K items placed on a grid of squares with R rows and C columns. Let (i, j) denote the square at the i-th row (1 \leq i \leq R) and the j-th column (1 \leq j \leq C). The i-th item is at (r_i, c_i) and has the value v_i.

Takahashi will begin at (1, 1), the start, and get to (R, C), the goal. When he is at (i, j), he can move to (i + 1, j) or (i, j + 1) (but cannot move to a non-existent square).

He can pick up items on the squares he visits, including the start and the goal, but at most three for each row. It is allowed to ignore the item on a square he visits.

Find the maximum possible sum of the values of items he picks up.

Constraints

1 \leq R, C \leq 3000

1 \leq K \leq \min(2 \times 10^5, R \times C)

1 \leq r_i \leq R

1 \leq c_i \leq C

(r_i, c_i) \neq (r_j, c_j) (i \neq j)

1 \leq v_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

R C K
r_1 c_1 v_1
r_2 c_2 v_2
:
r_K c_K v_K

Output

Print the maximum possible sum of the values of items Takahashi picks up.

Sample Input 1

2 2 3
1 1 3
2 1 4
1 2 5

Sample Output 1

8

He has two ways to get to the goal:

Visit (1, 1), (1, 2), and (2, 2), in this order. In this case, the total value of the items he can pick up is 3 + 5 = 8.

Visit (1, 1), (2, 1), and (2, 2), in this order. In this case, the total value of the items he can pick up is 3 + 4 = 7.

Thus, the maximum possible sum of the values of items he picks up is 8.

Sample Input 2

2 5 5
1 1 3
2 4 20
1 2 1
1 3 4
1 4 2

Sample Output 2

29

We have four items in the 1-st row. The optimal choices are as follows:

Visit (1, 1) (1, 2), (1, 3), (1, 4), (2, 4), and (2, 5), in this order, and pick up all items except the one on (1, 2). Then, the total value of the items he picks up will be 3 + 4 + 2 + 20 = 29.

Sample Input 3

4 5 10
2 5 12
1 5 12
2 3 15
1 2 20
1 1 28
2 4 26
3 2 27
4 5 21
3 5 10
1 3 10

Sample Output 3

142

Solution in Lua (LuaJIT 2.1.0) (judge verdict: Accepted)

local read = setmetatable({}, {__index = function(t, k) local a = {} for i=1,#k do table.insert(a, '*'..string.sub(k, i, i)) end local r = io.read local u = table.unpack or unpack return function() return r(u(a)) end end})
read.N = function(N) local t={} for i=1,N do t[i]=read.n() end return t end
string.totable = function(s) local t={} local u=string.sub for i=1,#s do t[i] = u(s, i, i) end return t end
string.split = function(s) local t={} for w in string.gmatch(s, "[^%s]+") do table.insert(t, w) end return (table.unpack or unpack)(t) end
local function array(dimensi0n, default_val) assert(type(default_val) ~= 'table') local n=dimensi0n local m={}if default_val~=nil then m[1]={__index=function()return default_val end}end for i=2,n do m[i]={__index=function(p, k)local c=setmetatable({},m[i-1])rawset(p,k,c)return c end}end return setmetatable({},m[n])end
local function tostringxx(o, depth) depth = depth or 0 if depth > 10 then return "<too deep!>" end if o == _G then return "<_G>" end local indent0 = (" "):rep((depth) * 2) local indent1 = (" "):rep((depth+1) * 2) local indent2= (" "):rep((depth+2) * 2) if type(o) == 'table' then local keys = {} local types = {} for k in pairs(o) do types[type(k)] = true table.insert(keys, k) end local types_count = 0 local lasttype for k in pairs(types) do types_count = types_count + 1 lasttype = k end if types_count == 1 and (lasttype == 'string' or lasttype == 'number') then table.sort(keys) end local inside = {} for i=1,#keys do local k = keys[i] local v = o[k] if type(k) == 'string' then k = string.format('%q', k) end table.insert(inside, indent1 .. '['..tostring(k)..'] = ' .. tostringxx(v, depth + 1)) end return '{\n' .. table.concat(inside, ',\n') .. '\n' .. indent0 .. '}' else if type(o) == 'string' then o = string.format('%q', o) end return tostring(o) end end
local function richtraceback() local x = 2 while true do local info = debug.getinfo(x) if not info then break end local fname = '<' .. info.short_src .. ":" .. info.linedefined .. ">" if info.name then fname = info.name end print(info.short_src .. ":" .. info.currentline .. ": in " .. ("%q"):format(fname)) print(" LOCALS:") local p = 1 while true do local name, val = debug.getlocal(x,p) if not name then break end print(" " .. name .. ": " .. tostringxx(val, 3)) p = p + 1 end print(" UPVALUES:") for p=1,info.nups do local name, val = debug.getupvalue(info.func,p) if not name then break end print(" " .. name .. ": " .. tostringxx(val, 3)) end x = x + 1 end end
local function myassert(b) if not b then richtraceback() error("assertion failed") end end 
--
local max = math.max

local R, C, K = read.nnn()
local item = array(2,0)
for i=1,K do
    local r,c,v = read.nnn()
    item[r][c] = v
end

local dp,dp_next = array(2,0),array(2,0)
local function chmax1(c,g,val)
    dp[c][g] = max(dp[c][g], val)
end
local function chmax2(c,g,val)
    dp_next[c][g] = max(dp_next[c][g], val)
end

dp[1][0] = 0
dp[1][1] = item[1][1]
for r=1,R do
    for c=1,C do
        for g=0,3 do
            dp_next[c][g] = 0
        end
        local rval = item[r][c+1]
        local dval = item[r+1][c]
        for g=0,3 do
            local curval = dp[c][g]
            chmax1(c+1,g+1,curval+rval)
            chmax1(c+1,g,curval)
            chmax2(c,1,curval+dval)
            chmax2(c,0,curval)
        end
    end
    dp, dp_next = dp_next, dp
end
print(max(dp[C][0],dp[C][1],dp[C][2],dp[C][3]))