Problem:


Score : 1200 points

Problem Statement

Snuke is playing with red and blue balls, placing them on a two-dimensional plane.

First, he performed N operations to place red balls. In the i-th of these operations, he placed RC_i red balls at coordinates (RX_i,RY_i).
Then, he performed another N operations to place blue balls. In the i-th of these operations, he placed BC_i blue balls at coordinates (BX_i,BY_i).
The total number of red balls placed and the total number of blue balls placed are equal, that is, \sum_{i=1}^{N} RC_i = \sum_{i=1}^{N} BC_i. Let this value be S.

Snuke will now form S pairs of red and blue balls so that every ball belongs to exactly one pair.
Let us define the score of a pair of a red ball at coordinates (rx, ry) and a blue ball at coordinates (bx, by) as |rx-bx| + |ry-by|.

Snuke wants to maximize the sum of the scores of the pairs.
Help him by finding the maximum possible sum of the scores of the pairs.

Constraints

1 \leq N \leq 1000

0 \leq RX_i,RY_i,BX_i,BY_i \leq 10^9

1 \leq RC_i,BC_i \leq 10

\sum_{i=1}^{N} RC_i = \sum_{i=1}^{N} BC_i

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
RX_1 RY_1 RC_1
RX_2 RY_2 RC_2
\vdots
RX_N RY_N RC_N
BX_1 BY_1 BC_1
BX_2 BY_2 BC_2
\vdots
BX_N BY_N BC_N

Output

Print the maximum possible sum of the scores of the pairs.

Sample Input 1

2
0 0 1
3 2 1
2 2 1
5 0 1

Sample Output 1

8

If we pair the red ball at coordinates (0,0) and the blue ball at coordinates (2,2), the score of this pair is |0-2| + |0-2|=4.
Then, if we pair the red ball at coordinates (3,2) and the blue ball at coordinates (5,0), the score of this pair is |3-5| + |2-0|=4.
Making these two pairs results in the total score of 8, which is the maximum result.

Sample Input 2

3
0 0 1
2 2 1
0 0 2
1 1 1
1 1 1
3 3 2

Sample Output 2

16

Snuke may have performed multiple operations at the same coordinates.

Sample Input 3

10
582463373 690528069 8
621230322 318051944 4
356524296 974059503 6
372751381 111542460 9
392867214 581476334 6
606955458 513028121 5
882201596 791660614 9
250465517 91918758 3
618624774 406956634 6
426294747 736401096 5
974896051 888765942 5
726682138 336960821 3
715144179 82444709 6
599055841 501257806 6
390484433 962747856 4
912334580 219343832 8
570458984 648862300 6
638017635 572157978 10
435958984 585073520 7
445612658 234265014 6

Sample Output 3

45152033546



Sample input:
2
0 0 1
3 2 1
2 2 1
5 0 1


Sample output:
8

Submission status: Runtime Error

Solution code:
import sys
from heapq import heappop, heappush


def min_cost_flow(links, s, t, flow, n2, INF, REV):
    remain = flow
    ans = 0

    potentials = [0] * n2  # to adjust negative edge for dijkstra
    predecessors_v = [0] * n2
    predecessors_i = [0] * n2

    while remain:
        distances = [INF] * n2
        distances[s] = 0
        q = [(0, s)]  # (cost, v, (p, v_idx_in_p))
        while q:
            cost, v = heappop(q)
            if cost > distances[v]:
                continue

            if v == t:
                break

            for i, (u, cap, uc, j) in enumerate(links[v]):
                if cap == 0:
                    continue
                new_cost = cost + uc + potentials[v] - potentials[u]
                if distances[u] <= new_cost:
                    continue
                distances[u] = new_cost
                predecessors_v[u] = v
                predecessors_i[u] = i
                heappush(q, (new_cost, u))

        # if distances[t] == INF:
        #     return -1

        for i in range(n2):
            potentials[i] += distances[i]

        min_cap = remain
        v = t
        forward_links = []
        backward_links = []
        while v != s:
            p = predecessors_v[v]
            i = predecessors_i[v]
            l = links[p][i]
            forward_links.append(l)
            if v != t:
                backward_links.append(links[v][l[3]])
            min_cap = min(min_cap, l[1])
            v = p
        for l in forward_links:
            l[1] -= min_cap
        for l in backward_links:
            l[1] += min_cap
        remain -= min_cap
        ans += min_cap * (REV * 2 - potentials[t])
    return ans


n = int(input())
lines = sys.stdin.readlines()
flow = 0
INF = 10 ** 18
REV = 10 ** 9
n2 = 2 * n + 6
t = 2 * n + 1
links = [[] for _ in range(n2)]  # [[tgt_node, capacity, unit_cost, idx_in_tgt], ]
for ri, line in enumerate(lines[:n], start=1):
    rx, ry, rc = map(int, line.split())
    flow += rc
    links[0].append([ri, rc, 0, 0])
    ki = n2 - 4
    kc = REV - (rx + ry)
    links[ri].append([ki, INF, kc, len(links[ki])])
    links[ki].append([ri, 0, -kc, len(links[ri]) - 1])
    ki = n2 - 3
    kc = REV - (-rx + ry)
    links[ri].append([ki, INF, kc, len(links[ki])])
    links[ki].append([ri, 0, -kc, len(links[ri]) - 1])
    ki = n2 - 2
    kc = REV - (rx - ry)
    links[ri].append([ki, INF, kc, len(links[ki])])
    links[ki].append([ri, 0, -kc, len(links[ri]) - 1])
    ki = n2 - 1
    kc = REV - (-rx - ry)
    links[ri].append([ki, INF, kc, len(links[ki])])
    links[ki].append([ri, 0, -kc, len(links[ri]) - 1])

for bi, line in enumerate(lines[n:], start=n + 1):
    bx, by, bc = map(int, line.split())
    links[bi].append([t, bc, 0, 0])
    ki = n2 - 4
    kc = REV - (-bx - by)
    links[ki].append([bi, INF, kc, len(links[bi])])
    links[bi].append([ki, 0, -kc, len(links[ki]) - 1])
    ki = n2 - 3
    kc = REV - (bx - by)
    links[ki].append([bi, INF, kc, len(links[bi])])
    links[bi].append([ki, 0, -kc, len(links[ki]) - 1])
    ki = n2 - 2
    kc = REV - (-bx + by)
    links[ki].append([bi, INF, kc, len(links[bi])])
    links[bi].append([ki, 0, -kc, len(links[ki]) - 1])
    ki = n2 - 1
    kc = REV - (bx + by)
    links[ki].append([bi, INF, kc, len(links[bi])])
    links[bi].append([ki, 0, -kc, len(links[ki]) - 1])

print(min_cost_flow(links, 0, t, flow, n2, INF, REV))

Submission status: Runtime Error

Solution code:
import math
N, X = map(int, input().split())
B = []
L = []
U = []
for i in range(N):
  b,l,u = map(int, input().split())
  B += [b]
  L += [l]
  U += [u]

behind = 0
for i in range(N):
  behind += L[i]*B[i]
  
gain_max = [(max(X*U[i] - B[i]*(U[i]-L[i]),X*L[i]),i) for i in range(N)]
studied = [0 for i in range(N)]
ans = 0

gain_max.sort()
takahashi_score = 0
while takahashi_score < behind:
  if takahashi_score + gain_max[-1][0] < behind:
    ans += X
    gain = gain_max.pop(-1)
    takahashi_score += gain[0]
    studied[gain[1]] = 1
  else:#残り１科目で勝てる！
    curr = X
    for i in range(N):
      if studied[i] == 0:
        curr = min(curr,min((behind - takahashi_score - B[i]*L[i])/U[i]+B[i],
                  (behind - takahashi_score - B[i]*L[i])/L[i]+B[i]))
    ans += math.ceil(curr)
    break
        
print(ans)