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



Sample input:
2 2 3
1 1 3
2 1 4
1 2 5


Sample output:
8

Submission status: Accepted

Solution code:
"""
dp[i][j]　=> i行目j列目がゴールの場合の最大値
"""
def main():
    import sys
    input = sys.stdin.readline
    R,C,K = map(int,input().split())
    items = [[0]*(C) for _ in range(R)]

    for _ in range(K):
        r,c,v = map(int,input().split())
        items[r-1][c-1] = v

    dp = [[0]*4 for _ in range(C)]
    dp[0][1] = items[0][0]
    for i in range(R):
        for j in range(C):
            v = items[i][j]
            if i==0 and j == 0:
                continue

            if i!=0:
                #上から入る場合のシミュレート
                dp[j][0] = max(dp[j])
                if v:
                    dp[j][1] = dp[j][0]+v

            if j!=0:
                #左から入る場合のシミュレート
                dp[j][0] = max(dp[j][0],dp[j-1][0])
                if v:
                    #itemが存在する場合の処理
                    for k in range(1,4):
                        dp[j][k] = max(dp[j][k],dp[j-1][k-1]+v,dp[j-1][k])
                else:
                    for k in range(1,4):
                        dp[j][k] = max(dp[j][k],dp[j-1][k])
    print(max(dp[-1]))
main()

Submission status: Accepted

Solution code:
R,C,K=map(int,input().split())
point=[[0]*C for _ in range(R)]
for _ in range(K):
    r,c,v=map(int,input().split())
    point[r-1][c-1]=v

dp=[[[0]*C for _ in range(R)] for _ in range(4)]
dp[1][0][0]=point[0][0]
for i in range(R):
    for j in range(C-1):
        for k in range(4):
            dp[k][i][j+1]=max(dp[k][i][j+1],dp[k][i][j])
            if k!=0:
                dp[k][i][j+1]=max(dp[k][i][j+1],dp[k-1][i][j]+point[i][j+1])
    if i==R-1:
        break
    for j in range(C):
        mx=0
        for k in range(4):
            mx=max(dp[k][i][j],mx)
        dp[0][i+1][j]=mx
        dp[1][i+1][j]=mx+point[i+1][j]
ans=0
for i in range(4):
    ans=max(ans,dp[i][R-1][C-1])
print(ans)

Submission status: Accepted

Solution code:
import numpy as np
import numba

@numba.njit("(i8,i8,i8,i8[:,:])", cache=True)
def solve(R, C, K, RCV):
    # dp[y][x][n]
    M = np.zeros((R+1, C+1), dtype=np.int64)
    for i in range(K):
        r, c, v = RCV[i]
        M[r, c] = v
    dp = np.zeros((R+1, C+1, 4), dtype=np.int64)
    for y in range(1, R+1):
        for x in range(1, C+1):
            for n in range(4):
                if n == 0:
                    dp[y, x, n] = max(
                        dp[y, x-1, 0],
                        dp[y-1, x, :].max()
                    )
                if n > 0:
                    dp[y, x, n] = max(
                        dp[y, x-1, n],
                        dp[y, x-1, n-1] + M[y, x],
                        dp[y-1, x, :].max() + M[y, x]
                    )
    ans = dp[R, C].max()
    print(ans)

import sys
def main():
    R, C, K = map(int, input().split())
    RCV = np.array(sys.stdin.read().split(), dtype=np.int64).reshape(K, 3)
    solve(R, C, K, RCV)

main()

Submission status: Accepted

Solution code:
from numba import jit
import numpy as np
@jit
def main():
  R,C,K=map(int,input().split())
  scores=np.zeros((R,C),np.int64)
  for _ in range(K): #各マスにアイテム配置
    r,c,v=map(int,input().split())
    scores[r-1][c-1]=v
  dp=np.zeros((R+1,C+1,4),np.int64) #後のために余分に作っておく、scoresの配列は別なのでindexに注意
  for i in range(1,R+1):
    for j in range(1,C+1):
      for k in range(4):
        dp[i][j][k]=max(dp[i][j-1][k],dp[i-1][j][3]) #同じ行ですでに3つ拾ってから次の行にくるのかただ隣のマスに進むか
      for k in range(3,0,-1):
        dp[i][j][k]=max(dp[i][j][k],dp[i][j][k-1]+scores[i-1][j-1]) #そのマスでアイテムを拾うかどうか
  return dp[R][C][3]
print(main())

Submission status: Accepted

Solution code:
h,w,k=map(int,input().split())
dp=[[[0]*(w+2) for _ in range(h+2)] for _ in range(4)]
f=[[0]*(w+1) for _ in range(h+1)]
for _ in range(k):
  r,c,v=map(int,input().split())
  f[r][c]=v
for i in range(1,h+1):
  for j in range(1,w+1):
    for k in range(3,-1,-1):
      if k:dp[k][i][j]=max(dp[k][i][j],dp[k-1][i][j]+f[i][j])
      dp[0][i+1][j]=max(dp[0][i+1][j],dp[k][i][j])
      dp[k][i][j+1]=max(dp[k][i][j+1],dp[k][i][j])
print(max(dp[k][h][w] for k in range(4)))

Submission status: Accepted

Solution code:
import sys

input = sys.stdin.readline
R, C, k = map(int, input().split())
xs = [[0] * C for _ in range(R)]
for _ in range(k):
    r, c, v = map(int, input().split())
    xs[r - 1][c - 1] = v

dp = [[[0] * (C + 1) for _ in range(R + 1)] for _ in range(4)]
for i in range(R):
    for j in range(C):
        for k in range(2, -1, -1):
            dp[k + 1][i][j] = max(dp[k + 1][i][j], dp[k][i][j] + xs[i][j])
        for k in range(4):
            dp[k][i][j + 1] = max(dp[k][i][j + 1], dp[k][i][j])
            dp[0][i + 1][j] = max(dp[0][i + 1][j], dp[k][i][j])
ans = 0
for k in range(4):
    ans = max(ans, dp[k][R - 1][C - 1])
print(ans)

Submission status: Accepted

Solution code:
R, C, k = map(int, input().split())
xs = [[0] * C for _ in range(R)]
for _ in range(k):
    r, c, v = map(int, input().split())
    xs[r - 1][c - 1] = v

dp = [[[0] * (C + 1) for _ in range(R + 1)] for _ in range(4)]
for i in range(R):
    for j in range(C):
        for k in range(2, -1, -1):
            dp[k + 1][i][j] = max(dp[k + 1][i][j], dp[k][i][j] + xs[i][j])
        for k in range(4):
            dp[k][i][j + 1] = max(dp[k][i][j + 1], dp[k][i][j])
            dp[0][i + 1][j] = max(dp[0][i + 1][j], dp[k][i][j])
ans = 0
for k in range(4):
    ans = max(ans, dp[k][R - 1][C - 1])
print(ans)

Submission status: Accepted

Solution code:
# from sys import stdin
# input = stdin.readline
(R,C,K) = map(int, input().split())
rcv = [list(map(int, input().split())) for _ in range(K)]
 
grids = [[0]*C for _ in range(R)]
 
for r,c,v in rcv:
    grids[r-1][c-1] = v
 
dp = [0] * C

for r in range(R):
#     print(dp)
    ndp = [0] * C
    elm = [0] * 4
    for c in range(C):
        elm[0] = max(elm[0], dp[c])
        value = grids[r][c]
        if value:
            nelm = [0] * 4
            nelm[0] = elm[0]
#             print(f'{value=}')
#             print(f'{ elm=}')
#             print(f'{nelm=}')
            for i in range(3)[::-1]:
                nelm[i + 1] = max(elm[i + 1], elm[i] + value)
#             print(f'{value=}')
#             print(f'{ elm=}')
#             print(f'{nelm=}')
            elm = nelm
        ndp[c] = max(elm)
    dp = ndp
 

print(dp[-1])

Submission status: Accepted

Solution code:
R, C, K = map(int, input().split())
xy = [[0] * (C + 1) for _ in range(R + 1)]
for _ in range(K) :
  r, c, v = map(int, input().split())
  xy[r - 1][c - 1] = v

dp = [[[0] * (C + 1) for _ in range(R + 1)] for _ in range(4)]
for cy in range(R + 1) :
  for cx in range(C + 1) :
    if cx < C :
      dp[0][cy][cx + 1] = max(dp[0][cy][cx + 1], dp[0][cy][cx])
      dp[1][cy][cx + 1] = max(dp[1][cy][cx + 1], dp[0][cy][cx] + xy[cy][cx], dp[1][cy][cx])
      dp[2][cy][cx + 1] = max(dp[2][cy][cx + 1], dp[1][cy][cx] + xy[cy][cx], dp[2][cy][cx])
      dp[3][cy][cx + 1] = max(dp[3][cy][cx + 1], dp[2][cy][cx] + xy[cy][cx], dp[3][cy][cx])
    if cy < R :
      dp[0][cy + 1][cx] = max(dp[i][cy][cx] for i in range(3)) + xy[cy][cx]
      dp[0][cy + 1][cx] = max(dp[0][cy + 1][cx], dp[3][cy][cx])

print(max(dp[i][-1][-1] for i in range(4)))

Submission status: Accepted

Solution code:
import sys

readline = sys.stdin.readline

def solve():
    R, C, K = map(int, readline().split())
    items = [[0] * C for i in range(R)]
    for i in range(K):
        r,c,k = map(int, readline().split())
        items[r - 1][c - 1] = k
    
    dp = [[[0] * C for i in range(R)] for j in range(4)]

    for i in range(R):
        for j in range(C):
            for k in range(4):
                if i == 0 and j == 0:
                    if k == 0:
                        dp[k][i][j] = 0
                    else:
                        dp[k][i][j] = items[i][j]
                elif i == 0:
                    if k == 0:
                        dp[k][i][j] = dp[k][i][j - 1]
                    else:
                        dp[k][i][j] = max(dp[k - 1][i][j - 1] + items[i][j], dp[k][i][j - 1])
                elif j == 0:
                    if k == 0:
                        dp[k][i][j] = dp[3][i - 1][j]
                    else:
                        dp[k][i][j] = dp[3][i - 1][j] + items[i][j]
                else:
                    if k == 0:
                        dp[k][i][j] = max(dp[3][i - 1][j], dp[k][i][j - 1])
                    else:
                        dp[k][i][j] = max(
                            dp[3][i - 1][j] + items[i][j],
                            dp[k - 1][i][j - 1] + items[i][j],
                            dp[k][i][j - 1]
                        )

    print(dp[-1][-1][-1])

solve()

Submission status: Accepted

Solution code:
## coding: UTF-8


R,C,K = map(int,input().split())
rcv = []
for i in range(K):
    rcv.append(list(map(int,input().split())))

item = [0] * (R*C+1)
for i in range(K):
    index = (rcv[i][0] - 1) * C + (rcv[i][1] - 1) 
    #available[index] = True
    item[index] = rcv[i][2]

#print(item)



def solver(R,C,K):
    empty = []
    prev = []
    for i in range(C):
        empty.append([0] * 4)
        prev.append([0] * 4)
    
    for i in range(R):
        #print('prev', prev)
        now = []
        #ret = []
        zero = max(prev[0])
        now.append([zero, zero+item[i*C], 0, 0]) #一番左
        for j in range(1, C):
            index = i * C + j
            point = item[index]
            #print(i,j,index)
            ret = []
            left = now[-1]
            zero = max(left[0], max(prev[j])) #左からor上から
            ret.append(zero) #0
            one = max(zero + point, left[1]) #左からor上からきてアイテム取得 or 左から一個持った状態で拾わない
            two = max(left[1] + point, left[2])
            three = max(left[2] + point, left[3])
            ret.append(one)
            ret.append(two)
            ret.append(three)
            now.append(ret)
        prev = now
        #print(now)
    
    print(max(now[-1]))


solver(R,C,K)

Submission status: Accepted

Solution code:
from collections import defaultdict
import sys


def input(): return sys.stdin.readline().rstrip()


r, c, K = map(int, input().split())

dp = [[[0]*(c+1) for _ in range(r+1)] for i in range(4)]

# dp[N][i][j] := (i,j) にN個目でたどりつく方法
d = [[0 for i in range(c+1)] for j in range(r+1)]

for _ in range(K):
    r_, c_, v = map(int, input().split())
    d[r_][c_] = v
for i in range(1, r+1):
    for j in range(1, c+1):
        v = d[i][j]
        if v > 0:
            maxi = max(dp[k][i-1][j] for k in range(4))
            dp[0][i][j] = max(maxi, dp[0][i][j-1])
            dp[1][i][j] = max(dp[1][i][j-1], dp[0][i][j-1]+v, maxi+v)
            dp[2][i][j] = max(dp[2][i][j-1], dp[1][i][j-1]+v)
            dp[3][i][j] = max(dp[3][i][j-1], dp[2][i][j-1]+v)

        elif v == 0:
            maxi = max(dp[k][i-1][j] for k in range(4))
            dp[0][i][j] = max(dp[0][i][j-1], maxi)
            dp[1][i][j] = dp[1][i][j-1]
            dp[2][i][j] = dp[2][i][j-1]
            dp[3][i][j] = dp[3][i][j-1]


ans = max(dp[k][r][c] for k in range(4))

print(ans)

Submission status: Accepted

Solution code:
r, c, k = map(int, input().split())

atlas = [[0]*c for i in range(r)]
dp = [[[0]*c for i in range(r)] for j in range(4)]

for i in range(k):
  x, y, v = map(int, input().split())
  atlas[x-1][y-1] = v
  
def max_sum(r, c, atlas):
  global dp

  if atlas[0][0] > 0:
    dp[1][0][0] = atlas[0][0]
    
  for i in range(r):
    for j in range(c):
      for h in range(4):
        
        if j+1 < c:
          dp[h][i][j+1] = max(dp[h][i][j+1], dp[h][i][j])
          if atlas[i][j+1] > 0 and h < 3:
            dp[h+1][i][j+1] = max(dp[h+1][i][j+1], dp[h][i][j]+atlas[i][j+1])
            
        if i+1 < r:
          dp[0][i+1][j] = max(dp[0][i+1][j], dp[h][i][j])
          if atlas[i+1][j] > 0:
            dp[1][i+1][j] = max(dp[1][i+1][j], dp[h][i][j]+atlas[i+1][j])
          
            
  return dp

max_sum(r, c, atlas)

ans = 0
for i in range(4):
  ans = max(ans, dp[i][r-1][c-1])
  
print(ans)

Submission status: Accepted

Solution code:
def main():
    import sys
    from copy import deepcopy
    input = sys.stdin.readline
    R, C, K = [int(x) for x in input().strip().split()]
    G = [[0] * (C+1) for r in range(R+1)]
    for k in range(K):
        r, c, v = [int(x) for x in input().strip().split()]
        G[r][c] = v
    # for r in range(R+1):
    #     print(G[r])
    ans = [[[0] * 4 for _ in range(C+1)] for _ in range(2)]
    for r in range(1, R+1):
        for c in range(1, C+1):
            ans[r%2][c][0] = max(ans[(r-1)%2][c][-1], ans[r%2][c-1][0])
            for i in range(1, 4):
                ans[r%2][c][i] = max(ans[(r-1)%2][c][3], ans[r%2][c-1][i], ans[(r-1)%2][c][3]+G[r][c], ans[r%2][c-1][i-1]+G[r][c])

    print(max(ans[R%2][-1]))

if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
import sys
import copy
from collections import deque

sys.setrecursionlimit(10**6)
input = sys.stdin.readline
#read = sys.stdin.buffer.read

def main():
    global field
    R, C, K = map(int, input().split())
    field = []
    for _ in range(R):
        tmp = [0] * C
        field.append(tmp)
    DP1 = copy.deepcopy(field)
    DP2 = copy.deepcopy(field)
    DP3 = copy.deepcopy(field)

    for _ in range(K):
        r, c, k = map(int, input().split())
        field[r-1][c-1] = k
    
    i = 0
    DP1[0][0] = field[0][0]
    for j in range(1, C):
        DP3[i][j] = max(DP2[i][j-1] + field[i][j], DP3[i][j-1])
        DP2[i][j] = max(DP1[i][j-1] + field[i][j], DP2[i][j-1])
        DP1[i][j] = max(field[i][j], DP1[i][j-1])

    for i in range(1, R):
        for j in range(C):
            DP3[i][j] = max(DP2[i][j-1] + field[i][j], DP3[i][j-1])
            DP2[i][j] = max(DP1[i][j-1] + field[i][j], DP2[i][j-1])
            DP1[i][j] = max(field[i][j], DP1[i][j-1], max(DP1[i-1][j], DP2[i-1][j], DP3[i-1][j]) + field[i][j])

    ans = max(DP3[R-1][C-1], DP2[R-1][C-1], DP1[R-1][C-1])
    print(ans)


if __name__ == "__main__":
    main()

Submission status: Accepted

Solution code:
from itertools import accumulate
R, C, K = map(int, input().split())
vss = [[None]*C for _ in range(R)]
for _ in range(K):
    r, c, v = map(int, input().split())
    vss[r-1][c-1] = v

dp = []

for y in range(R):
    ndp = []
    for x in range(C):
        tmp = []
        uemax = 0
        if y > 0:
            uemax = max(dp[x])
        if vss[y][x] is None:
            if x > 0:
                left = ndp[-1]
                tmp.append(left[0] if left[0] > uemax else uemax)
                tmp.append(left[1])
                tmp.append(left[2])
                tmp.append(left[3])
            else:
                tmp.append(uemax)
                tmp.append(0)
                tmp.append(0)
                tmp.append(0)
        else:
            v = vss[y][x]
            if x > 0:
                left = ndp[-1]
                tmp.append(left[0] if left[0] > uemax else uemax)
                tmp.append(max(left[0]+v, left[1], uemax+v))
                tmp.append(left[1]+v if left[1]+v > left[2] else left[2])
                tmp.append(left[2]+v if left[2]+v > left[3] else left[3])
            else:
                tmp.append(uemax)
                tmp.append(uemax + v)
                tmp.append(0)
                tmp.append(0)
        ndp.append(tmp)
    dp = ndp

print(max(dp[-1]))