Problem:
Score : 600 points

Problem Statement

There is a tree with N vertices numbered 1 to N.
The i-th edge in this tree connects Vertex a_i and Vertex b_i, and the color and length of that edge are c_i and d_i, respectively.
Here the color of each edge is represented by an integer between 1 and N-1 (inclusive). The same integer corresponds to the same color, and different integers correspond to different colors.

Answer the following Q queries:

Query j (1 \leq j \leq Q): assuming that the length of every edge whose color is x_j is changed to y_j, find the distance between Vertex u_j and Vertex v_j. (The changes of the lengths of edges do not affect the subsequent queries.)

Constraints

2 \leq N \leq 10^5

1 \leq Q \leq 10^5

1 \leq a_i, b_i \leq N

1 \leq c_i \leq N-1

1 \leq d_i \leq 10^4

1 \leq x_j \leq N-1

1 \leq y_j \leq 10^4

1 \leq u_j < v_j \leq N

The given graph is a tree.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N Q
a_1 b_1 c_1 d_1
:
a_{N-1} b_{N-1} c_{N-1} d_{N-1}
x_1 y_1 u_1 v_1
:
x_Q y_Q u_Q v_Q

Output

Print Q lines. The j-th line (1 \leq j \leq Q) should contain the answer to Query j.

Sample Input 1

5 3
1 2 1 10
1 3 2 20
2 4 4 30
5 2 1 40
1 100 1 4
1 100 1 5
3 1000 3 4

Sample Output 1

130
200
60

The graph in this input is as follows:

Here the edges of Color 1 are shown as solid red lines, the edge of Color 2 is shown as a bold green line, and the edge of Color 4 is shown as a blue dashed line.

Query 1: Assuming that the length of every edge whose color is 1 is changed to 100, the distance between Vertex 1 and Vertex 4 is 100 + 30 = 130.

Query 2: Assuming that the length of every edge whose color is 1 is changed to 100, the distance between Vertex 1 and Vertex 5 is 100 + 100 = 200.

Query 3: Assuming that the length of every edge whose color is 3 is changed to 1000 (there is no such edge), the distance between Vertex 3 and Vertex 4 is 20 + 10 + 30 = 60. Note that the edges of Color 1 now have their original lengths.

Solution in Ruby (2.3.3) (judge verdict: Runtime Error)

def build_depth_info_subr(adjs, u, info)
  adjs[u].each do |v, (c, d)|
    next if info[v]
    info[v] = {}
    info[v][:parent] = u
    info[v][:depth] = info[u][:depth] + 1
    info[v][:distance] = info[u][:distance] + d

    build_depth_info_subr(adjs, v, info)
  end
end

def build_depth_info(adjs)
  info = {}
  u = 1 # root
  info[u] = {}
  info[u][:parent] = nil
  info[u][:depth] = 0
  info[u][:distance] = 0

  build_depth_info_subr(adjs, u, info)

  info.each do |u, r|
    r[:as] = [r[:parent]]
  end

  0.upto(30) do |i|
    found = nil
    info.each do |u, r|
      if r[:as][i]
        a = info[r[:as][i]][:as][i]
        found ||= a
        r[:as] << a
      end
    end
    break unless found
  end
  #info.each{|k,v| p [k, v]}
  
  info
end

def build_color_info_subr(adjs, info, qc, u, de, di) 
  adjs[u].each do |v, (c, d)|
    next if v == info[u][:parent]
    di[c] += d
    de[c] += 1
    #p [v, de, di] 
    if qc.key?(v)
      di2 = {}
      de2 = {}
      qc[v].each_key do |cc|
        di2[cc] = di[cc]
        de2[cc] = de[cc]
      end
      info[v][:colored_distance] = di2
      info[v][:colored_depth] = de2
    end
    build_color_info_subr(adjs, info, qc, v, de, di)
    di[c] -= d
    de[c] -= 1
  end
end

def build_color_info(adjs, info, qc)
  info[1][:colored_depth] = Hash.new(0)
  info[1][:colored_distance] = Hash.new(0)
  build_color_info_subr(adjs, info, qc, 1, Hash.new(0), Hash.new(0))
end

def nth_parent(u, n, info)
  i = 0
  while (2**i) <= n
    if (n & (2**i)) > 0
      u = info[u][:as][i]
      return u if !u
    end
    i += 1
  end
  return u
end

def lca(u, v, info)
  du = info[u][:depth]
  dv = info[v][:depth]
  if du < dv
    v = nth_parent(v, dv-du, info)
  else
    u = nth_parent(u, du-dv, info)    
  end

  return u if u == v

  x0 = 0
  x1 = du
  while x1-x0 > 1
    d = (x0 + x1)/2
    if nth_parent(u, d, info) == nth_parent(v, d, info)
      x1 = d
    else
      x0 = d
    end
  end

  d = x1

  return nth_parent(u, d, info)
end

def changed_distance(u, c, d, info)
  #p [:ch, u, info[u][:distance], info[u][:colored_distance][c], d*info[u][:colored_depth][c]]
  info[u][:distance] - info[u][:colored_distance][c] + d*info[u][:colored_depth][c]
end

n, q = gets.split.map(&:to_i)
adjs = {}
(n-1).times do
  a, b, c, d = gets.split.map(&:to_i)
  (adjs[a] ||= {})[b] = [c, d]
  (adjs[b] ||= {})[a] = [c, d]
end

info = build_depth_info(adjs)

queries = []
qc = {}
q.times do
  x, y, u, v = gets.split.map(&:to_i)

  m = lca(u, v, info)
  queries << [x, y, u, v, m]

  (qc[u] ||= {})[x] = true
  (qc[v] ||= {})[x] = true
  (qc[m] ||= {})[x] = true
end

build_color_info(adjs, info, qc)
if false
  puts
  info.each{|a| p a}
  puts
end

queries.each do |x, y, u, v, m|
  #p [x, y, u, v, m]
  cdu = changed_distance(u, x, y, info)
  cdv = changed_distance(v, x, y, info)
  cdm = changed_distance(m, x, y, info)
  
  len = cdu + cdv - 2*cdm
  puts len
end