Petya has a simple graph (that is, a graph without loops or multiple edges) consisting of $n$ vertices and $m$ edges.

The weight of the $i$-th vertex is $a_i$.

The weight of the $i$-th edge is $w_i$.

A subgraph of a graph is some set of the graph vertices and some set of the graph edges. The set of edges must meet the condition: both ends of each edge from the set must belong to the chosen set of vertices. 

The weight of a subgraph is the sum of the weights of its edges, minus the sum of the weights of its vertices. You need to find the maximum weight of subgraph of given graph. The given graph does not contain loops and multiple edges.


-----Input-----

The first line contains two numbers $n$ and $m$ ($1 \le n \le 10^3, 0 \le m \le 10^3$) - the number of vertices and edges in the graph, respectively.

The next line contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$) - the weights of the vertices of the graph.

The following $m$ lines contain edges: the $i$-e edge is defined by a triple of integers $v_i, u_i, w_i$ ($1 \le v_i, u_i \le n, 1 \le w_i \le 10^9, v_i \neq u_i$). This triple means that between the vertices $v_i$ and $u_i$ there is an edge of weight $w_i$. It is guaranteed that the graph does not contain loops and multiple edges.


-----Output-----

Print one integer — the maximum weight of the subgraph of the given graph.


-----Examples-----
Input
4 5
1 5 2 2
1 3 4
1 4 4
3 4 5
3 2 2
4 2 2

Output
8

Input
3 3
9 7 8
1 2 1
2 3 2
1 3 3

Output
0



-----Note-----

In the first test example, the optimal subgraph consists of the vertices ${1, 3, 4}$ and has weight $4 + 4 + 5 - (1 + 2 + 2) = 8$. In the second test case, the optimal subgraph is empty.

Difficulty: VERY_HARD
Tags: Graph algorithms

Flows and cuts

Starter code:

Solutions:
from collections import deque

def addedge(u, v, value):
	nonlocal e
	a = [v, value, None]
	b = [u, 0, a]
	a[2] = b
	e[u].append(a)
	e[v].append(b)
inf = 2 * 10 ** 12
ans = 0
(n, m) = list(map(int, input().split()))
e = [[] for i in range(n + m + 2)]
a = tuple(map(int, input().split()))
(S, T) = (0, m + n + 1)
for i in range(1, m + 1):
	(u, v, w) = list(map(int, input().split()))
	ans += w
	addedge(i, u + m, inf)
	addedge(i, v + m, inf)
	addedge(S, i, w)
for i in range(m + 1, T):
	addedge(i, T, a[i - m - 1])
lvl = None

def bfs():
	nonlocal e, lvl
	lvl = [0] * (n + m + 2)
	q = deque([0])
	while q:
		node = q.popleft()
		for edge in e[node]:
			if edge[0] != 0 and lvl[edge[0]] == 0 and edge[1]:
				lvl[edge[0]] = lvl[node] + 1
				q.append(edge[0])

def dfs(node, maxdelta):
	nonlocal e, lvl
	if node == T:
		return maxdelta
	delta = 0
	for edge in e[node]:
		if lvl[edge[0]] == lvl[node] + 1 and edge[1]:
			tmp = dfs(edge[0], min(maxdelta, edge[1]))
			if tmp > 0:
				edge[1] -= tmp
				edge[2][1] += tmp
				maxdelta -= tmp
				delta += tmp
			if maxdelta == 0:
				break
	return delta
flow = 0
while 1:
	bfs()
	tmp = dfs(0, inf)
	if tmp == 0:
		break
	flow += tmp
ans -= flow
print(ans)


class edge(object):

	def __init__(self, ne, to, fl):
		self.ne = ne
		self.to = to
		self.fl = fl

def add(x, y, z):
	global tot
	tot += 1
	e.append(edge(he[x], y, z))
	he[x] = tot

def addedge(x, y, z):
	add(x, y, z)
	add(y, x, 0)

def bfs():
	global deep
	deep = [0 for i in range(T + 1)]
	q = []
	q.append(S)
	deep[S] = 1
	while len(q) > 0:
		x = q[0]
		del q[0]
		i = he[x]
		while i:
			y = e[i].to
			if deep[y] == 0 and e[i].fl != 0:
				deep[y] = deep[x] + 1
				q.append(y)
			i = e[i].ne
	return deep[T] != 0

def dfs(x, flow):
	global deep
	if x == T or flow == 0:
		return flow
	used = 0
	i = he[x]
	while i:
		y = e[i].to
		if deep[y] == deep[x] + 1 and e[i].fl != 0:
			now = dfs(y, min(flow - used, e[i].fl))
			used += now
			e[i].fl -= now
			e[i ^ 1].fl += now
			if flow == used:
				break
		i = e[i].ne
	if used == 0:
		deep[x] = -1
	return used

def dinic():
	res = 0
	while bfs():
		res += dfs(S, INF)
	return res
(n, m) = map(int, input().split())
ans = 0
weight = [0] + list(map(int, input().split()))
e = [0, 0]
tot = 1
S = n + m + 1
T = S + 1
he = [0 for i in range(T + 1)]
INF = 1000000007
for i in range(1, n + 1):
	addedge(S, i, weight[i])
for i in range(1, m + 1):
	(x, y, w) = map(int, input().split())
	addedge(n + i, T, w)
	addedge(x, n + i, INF)
	addedge(y, n + i, INF)
	ans += w
ans -= dinic()
print(ans)


import sys

class Graph:
	verticies = {}
	nodesCount = 0

	class Vertex:

		def __init__(self, label, endPoint=None):
			self.label = label
			self.edges = []
			self.visitedToken = 0
			self.endPoint = endPoint

	class Edge:
		residual = None

		def __init__(self, from_, to_, isResidual, maxCapacity):
			self.from_ = from_
			self.to_ = to_
			self.isResidual = isResidual
			self.capacity = maxCapacity
			self.flow = 0

		def augment(self, bootleneck):
			self.flow += bootleneck
			self.residual.flow -= bootleneck

		def remainingCapacity(self):
			return self.capacity - self.flow

	def addEdge(self, from_, to_, capacity):
		from_ = self.verticies[from_]
		to_ = self.verticies[to_]
		if from_.endPoint and from_.endPoint != to_:
			from_ = from_.endPoint
		main = self.Edge(from_, to_, False, capacity)
		residual = self.Edge(to_, from_, True, 0)
		main.residual = residual
		residual.residual = main
		from_.edges.append(main)
		to_.edges.append(residual)

	def addVertex(self, label, *args):
		self.nodesCount += 1
		self.verticies[label] = self.Vertex(label)

	def maxFlow(self, f, t):
		f = self.verticies[f]
		t = self.verticies[t]
		visitedToken = 1
		flow = 0

		def dfs(node, bootleneck=sys.maxsize):
			node.visitedToken = visitedToken
			bootleneck_backup = bootleneck
			if node == t:
				return bootleneck
			for edge in node.edges:
				if edge.remainingCapacity() == 0 or edge.to_.visitedToken == visitedToken:
					continue
				bootleneck = dfs(edge.to_, min(bootleneck, edge.remainingCapacity()))
				if bootleneck:
					edge.augment(bootleneck)
					return bootleneck
				else:
					bootleneck = bootleneck_backup
			return 0
		while True:
			bootleneck = dfs(f)
			if not bootleneck:
				break
			flow += bootleneck
			visitedToken += 1
		return flow
g = Graph()
(n, m) = map(int, input().split())
vv = list(map(int, input().split()))
for i in range(n + m + 2):
	g.addVertex(i)
for (i, v) in enumerate(vv):
	g.addEdge(m + i + 1, n + m + 1, v)
s = 0
for i in range(1, m + 1):
	(a, b, c) = map(int, input().split())
	s += c
	g.addEdge(0, i, c)
	g.addEdge(i, a + m, c)
	g.addEdge(i, b + m, c)
print(s - g.maxFlow(0, n + m + 1))