本文介绍如何使用动态规划解决矩阵中从任意位置到达指定位置的最短路径问题，考虑有限次数的跳跃操作。通过字典和三维数组实现，并提供速度优化和内存节省策略。
[算法-动态规划]求可跳转情况下的最小路径
Description
简单来说，我们给定了一个XY大小的矩阵，每个entry上都是一些非负数，我们同时也给定了一组跃迁数，但是我们必须按顺序使用他们，当然也可以不用，我们想通过这些跃迁数找到到（X-1,Y-1）的最短路径
input
output
sample input and output
6 4
1 0 3 7 2 5
8 3 7 6 9 8
9 7 8 2 1 1
3 2 9 1 7 8
1 1
16
# fly right, fly right, jump right, jump up, fly right, fly up, for a total of 1+0+3+2+1+1+8=16
思路
我们使用动态规划的思想，首先我们到了一个点有四个选择，往右跳，往右走，往下走，往下跳。首先我们先从第一个jump开始，我们可以选择跳与不跳，这样就可以将所有不用跳的到（i,j）最短距离和一部分只用一次跳的距离更新（不一定是最优的，可能会在下个循环里中的往右走或往下走更新）
我们使用字典来实现，里面的key(i,j,s)表示到（i,j）并且利用s次跃迁的最小值。初始化的时候将输入的矩阵的左上角存入。并且使用exception来防止某些不可能出现的(i,j,s）
最后，我们对（X-1,Y-1,s）进行遍历，选取其中的最小值并且将它return
速度优化
当我们在某次更新（i,j）是没有用到jump时，那么按照规定，这之后的jump我们也没办法使用，所以这时候我们可以直接break，这样会缩短差不多2/3的时间
内存优化
对于不会出现的key选择不添加入字典（使用exception）
实现（使用字典）
if __name__ == '__main__':
width, height = map(int, input().split())
input_matrix = [[0 for i in range(width)]for j in range(height)]
for i in range(height):
input_matrix[i] = list(map(int, input().split()))
jumps = list(map(int, input().split()))
#jumps.insert(0, 0)
jumps.append(0)
len_jump = len(jumps)
for i in range(len_jump):
if jumps[i]>=width-1 & jumps[i]>= height-1:
jumps = jumps[0:i]
len_jump = len(jumps)
dp = dict()
dp[(0,0,0)] = input_matrix[0][0]
det_row = 0 # if i + det_row > width not need to jump anymore 0 can jump
det_col = 0 # 0 can jump, 1 can't jump
det = 0
det_break_jump = 0
det_jump = 1
mini = -1
det_mini = 0
for s in range(len_jump):
det_row = 0
if s==1:
mini = dp[(height-1,width-1,0)]
det_mini = 1
if s>1:
mini = min(mini,dp[height-1,width-1,s-1])
#else:
#mini = min(mini, dp[(i,j,s-1)])
#else:
if (det_jump == 0):
break
det_jump = 0
for i in range(height):
det_col = 0
if i + jumps[s] + 1 > height - 1: # can't jump row for now
det_row = 1
for j in range(width):
if j + jumps[s] + 1 > width - 1: # col can't jump for now
det_col = 1
try:
dp[(i, j, s)] == -2
except:
det = 1
else:
if det_col == 1: # col can't jump
if det_row == 0: # row can jump
if j == width - 1: # col can't walk
# if i < height - jumps[s] - 1: # can jump
try:
dp[(i+1,j,s)] == -2
except:
dp[(i+1,j,s)] = dp[(i,j,s)] + input_matrix[i+1][j]
else:
dp[(i+1,j,s)] = min(dp[(i+1,j,s)],dp[(i,j,s)] + input_matrix[i+1][j])
try:
dp[(i+jumps[s]+1,j,s+1)] == -2
except:
dp[(i + jumps[s] + 1, j, s + 1)] = dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
else:
if dp[(i+jumps[s]+1,j,s+1)] >= dp[(i,j,s)] + input_matrix[i+jumps[s]+1][j]:
dp[(i+jumps[s]+1,j,s+1)] = dp[(i,j,s)] + input_matrix[i+jumps[s]+1][j]
det_jump = 1
else: # col can walk
try:
dp[(i+1,j,s)] == -2
except:
dp[(i+1,j,s)] = dp[(i,j,s)] + input_matrix[i+1][j]
else:
dp[(i+1,j,s)] = min(dp[(i+1,j,s)],dp[(i,j,s)] + input_matrix[i+1][j])
try:
dp[(i,j+1,s)] == -2
except:
dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
try:
dp[(i+jumps[s]+1,j,s+1)] == -2
except:
dp[(i + jumps[s] + 1, j, s + 1)] = dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
else:
if dp[(i+jumps[s]+1,j,s+1)] >= dp[(i,j,s)] + input_matrix[i+jumps[s]+1][j]:
dp[(i+jumps[s]+1,j,s+1)] = dp[(i,j,s)] + input_matrix[i+jumps[s]+1][j]
det_jump = 1
else: # det_row = 1 row can't jump
if j == width - 1: # col can't walk
if i != height - 1: # row can walk
try:
dp[(i+1,j,s)] == -2
except:
dp[(i+1,j,s)] = dp[(i,j,s)] + input_matrix[i+1][j]
else:
dp[(i+1,j,s)] = min(dp[(i+1,j,s)],dp[(i,j,s)] + input_matrix[i+1][j])
else: # col can walk
if i != height - 1: # row can walk
try:
dp[(i+1,j,s)] == -2
except:
dp[(i+1,j,s)] = dp[(i,j,s)] + input_matrix[i+1][j]
else:
dp[(i+1,j,s)] = min(dp[(i+1,j,s)],dp[(i,j,s)] + input_matrix[i+1][j])
try:
dp[(i, j + 1, s)] == -2
except:
dp[(i, j + 1, s)] = dp[(i, j, s)] + input_matrix[i][j + 1]
else:
dp[(i, j + 1, s)] = min(dp[(i, j + 1, s)],
dp[(i, j, s)] + input_matrix[i][j + 1])
else: # row can't walk
try:
dp[(i,j+1,s)] == -2
except:
dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
else: # det_col = 0, col can jump
if det_row == 0:# row can jump
try:
dp[(i + 1, j, s)] == -2
except:
dp[(i + 1, j, s)] = dp[(i, j, s)] + input_matrix[i + 1][j]
else:
dp[(i + 1, j, s)] = min(dp[(i + 1, j, s)], dp[(i, j, s)] + input_matrix[i + 1][j])
try:
dp[(i, j + 1, s)] == -2
except:
dp[(i, j + 1, s)] = dp[(i, j, s)] + input_matrix[i][j + 1]
else:
dp[(i, j + 1, s)] = min(dp[(i, j + 1, s)], dp[(i, j, s)] + input_matrix[i][j + 1])
try:
dp[(i + jumps[s] + 1, j, s + 1)] == -2
except:
dp[(i + jumps[s] + 1, j, s + 1)] = dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
else:
if dp[(i + jumps[s] + 1,j,s+1)] >= dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]:
dp[(i + jumps[s] + 1, j, s + 1)] = dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
try:
dp[(i, j + jumps[s] + 1, s + 1)] == -2
except:
dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j+ jumps[s] + 1]
det_jump = 1
else:
if dp[(i, j + jumps[s] + 1, s + 1)] >= dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]:
dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]
det_jump = 1
else: # det_row = 1 row can't jump
if i == height - 1: # row can't walk
try:
dp[(i, j + 1, s)] == -2
except:
dp[(i, j + 1, s)] = dp[(i, j, s)] + input_matrix[i][j + 1]
else:
dp[(i, j + 1, s)] = min(dp[(i, j + 1, s)], dp[(i, j, s)] + input_matrix[i][j + 1])
try:
dp[(i, j + jumps[s] + 1, s + 1)] == -2
except:
dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]
det_jump = 1
else:
if dp[(i, j + jumps[s] + 1, s + 1)] >= dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]:
dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]
det_jump = 1
else: # row can walk
try:
dp[(i + 1, j, s)] == -2
except:
dp[(i + 1, j, s)] = dp[(i, j, s)] + input_matrix[i + 1][j]
else:
dp[(i + 1, j, s)] = min(dp[(i + 1, j, s)], dp[(i, j, s)] + input_matrix[i + 1][j])
try:
dp[(i, j + 1, s)] == -2
except:
dp[(i, j + 1, s)] = dp[(i, j, s)] + input_matrix[i][j + 1]
else:
dp[(i, j + 1, s)] = min(dp[(i, j + 1, s)], dp[(i, j, s)] + input_matrix[i][j + 1])
try:
dp[(i, j + jumps[s] + 1, s + 1)] == -2
except:
dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j+ jumps[s] + 1]
det_jump = 1
else:
if dp[(i, j + jumps[s] + 1, s + 1)] >= dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]:
dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j + jumps[s] + 1]
det_jump = 1
# renew near element
#for i in range(height):
# for j in range(width):
mini = dp[(height-1,width-1,0)]
for i in range(len_jump ):
try:
if mini > dp[(height-1,width-1,i)]:
mini = dp[(height-1,width-1,i)]
except:
det = 0
print(mini)
实现（使用三维数组）
import copy
if __name__ == '__main__':
width, height = map(int, input().split())
input_matrix = [[-1 for i in range(width)]for j in range(height)]
for i in range(height):
input_matrix[i] = list(map(int, input().split()))
jumps = list(map(int, input().split()))
#jumps.insert(0, 0)
jumps.append(0)
len_jump = len(jumps)
for i in range(len_jump):
if jumps[i]>=width-1 & jumps[i]>= height-1:
jumps = jumps[0:i]
len_jump = len(jumps)
dp_plus = []
for k in range(len_jump):
dp_plus.append([])
for j in range(height):
dp_plus[k].append([])
for i in range(width):
dp_plus[k][j].append(-1)
#dp_plus = [[[0 for i in range(width)]for j in range(height)]for k in range(len_jump)]
#for i in range(height):
# input_matrix_buffer = copy.deepcopy(input_matrix[i])
# dp_plus[0][i] = input_matrix_buffer
#dp = dict()
#dp[(0,0,0)] = input_matrix[0][0]
dp_plus[0][0][0] = input_matrix[0][0]
det_row = 0 # if i + det_row > width not need to jump anymore 0 can jump
det_col = 0 # 0 can jump, 1 can't jump
det = 0
det_break_jump = 0
det_jump = 1
for s in range(len_jump):
det_row = 0
if (det_jump == 0 ):
break
det_jump = 0
for i in range(height):
det_col = 0
if i + jumps[s] + 1 > height - 1: # can't jump row for now
det_row = 1
for j in range(width):
if j + jumps[s] + 1 > width - 1: # col can't jump for now
det_col = 1
if dp_plus[s][i][j] != -1:
if det_col == 1: # col can't jump
if det_row == 0: # row can jump
if j == width - 1: # col can't walk
# if i < height - jumps[s] - 1: # can jump
if dp_plus[s][i+1][j] == -1:
dp_plus[s][i+1][j] = dp_plus[s][i][j] + input_matrix[i+1][j]
#dp[(i+1,j,s)] = dp[(i,j,s)] + input_matrix[i+1][j]
else:
dp_plus[s][i + 1][j] = min(dp_plus[s][i+1][j],dp_plus[s][i][j]+input_matrix[i+1][j])
#dp[(i+1,j,s)] = min(dp[(i+1,j,s)],dp[(i,j,s)] + input_matrix[i+1][j])
if s!= len_jump-1:
if dp_plus[s+1][i+jumps[s]+1][j] == -1:
dp_plus[s + 1][i + jumps[s] + 1][j] = dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][j]
#dp[(i+jumps[s]+1,j,s+1)] == -2
det_jump = 1
else:
if dp_plus[s+1][i+jumps[s]+1][j] >= dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][j]:
dp_plus[s + 1][i + jumps[s] + 1][j] = dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
else: # col can walk
if dp_plus[s][i+1][j] == -1:
dp_plus[s][i + 1][j] = dp_plus[s][i][j] + input_matrix[i + 1][j]
#dp[(i+1,j,s)] == -2
#except:
#dp[(i+1,j,s)] = dp[(i,j,s)] + input_matrix[i+1][j]
else:
dp_plus[s][i+1][j] = min(dp_plus[s][i+1][j],dp_plus[s][i][j] + input_matrix[i + 1][j])
#try:
#dp[(i,j+1,s)] == -2
if dp_plus[s][i][j+1] == -1:
dp_plus[s][i][j+1] = dp_plus[s][i][j] + input_matrix[i][j+1]
#except:
#dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
#dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
dp_plus[s][i][j + 1] = min(dp_plus[s][i][j+1],dp_plus[s][i][j] + input_matrix[i][j+1])
#try:
#dp[(i+jumps[s]+1,j,s+1)] == -2
if s != len_jump - 1:
if dp_plus[s+1][i+jumps[s]+1][j] == -1:
dp_plus[s + 1][i + jumps[s] + 1][j] = dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][j]
#except:
#dp[(i + jumps[s] + 1, j, s + 1)] = dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
else:
if dp_plus[s + 1][i + jumps[s] + 1][j] >= dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][j]:
dp_plus[s + 1][i + jumps[s] + 1][j] = min(dp_plus[s + 1][i + jumps[s] + 1][j],dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][j])
det_jump = 1
else: # det_row = 1 row can't jump
if j == width - 1: # col can't walk
if i != height - 1: # row can walk
if dp_plus[s][i+1][j] == -1:
dp_plus[s][i + 1][j] = dp_plus[s][i][j] + input_matrix[i+1][j]
else:
dp_plus[s][i+1][j] = min(dp_plus[s][i+1][j],dp_plus[s][i][j] + input_matrix[i+1][j])
else: # col can walk
if i != height - 1: # row can walk
if dp_plus[s][i + 1][j] == -1:
dp_plus[s][i + 1][j] = dp_plus[s][i][j] + input_matrix[i + 1][j]
else:
dp_plus[s][i + 1][j] = min(dp_plus[s][i + 1][j],
dp_plus[s][i][j] + input_matrix[i + 1][j])
if dp_plus[s][i][j + 1] == -1:
dp_plus[s][i][j + 1] = dp_plus[s][i][j] + input_matrix[i][j + 1]
# except:
# dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
# dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
dp_plus[s][i][j + 1] = min(dp_plus[s][i][j + 1],
dp_plus[s][i][j] + input_matrix[i][j + 1])
else: # row can't walk
if dp_plus[s][i][j + 1] == -1:
dp_plus[s][i][j + 1] = dp_plus[s][i][j] + input_matrix[i][j + 1]
# except:
# dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
# dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
dp_plus[s][i][j + 1] = min(dp_plus[s][i][j + 1],
dp_plus[s][i][j] + input_matrix[i][j + 1])
else: # det_col = 0, col can jump
if det_row == 0:# row can jump
if dp_plus[s][i + 1][j] == -1:
dp_plus[s][i + 1][j] = dp_plus[s][i][j] + input_matrix[i + 1][j]
else:
dp_plus[s][i + 1][j] = min(dp_plus[s][i + 1][j],
dp_plus[s][i][j] + input_matrix[i + 1][j])
if dp_plus[s][i][j + 1] == -1:
dp_plus[s][i][j + 1] = dp_plus[s][i][j] + input_matrix[i][j + 1]
# except:
# dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
# dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
dp_plus[s][i][j + 1] = min(dp_plus[s][i][j + 1],
dp_plus[s][i][j] + input_matrix[i][j + 1])
if s != len_jump - 1:
if dp_plus[s + 1][i + jumps[s] + 1][j] == -1:
dp_plus[s + 1][i + jumps[s] + 1][j] = dp_plus[s][i][j] + input_matrix[i + jumps[s] + 1][
j]
# except:
# dp[(i + jumps[s] + 1, j, s + 1)] = dp[(i, j, s)] + input_matrix[i + jumps[s] + 1][j]
det_jump = 1
else:
if dp_plus[s + 1][i + jumps[s] + 1][j] >= dp_plus[s][i][j] + \
input_matrix[i + jumps[s] + 1][j]:
dp_plus[s + 1][i + jumps[s] + 1][j] = min(dp_plus[s + 1][i + jumps[s] + 1][j],
dp_plus[s][i][j] +
input_matrix[i + jumps[s] + 1][j])
det_jump = 1
#try:
#dp[(i, j + jumps[s] + 1, s + 1)] == -2
if dp_plus[s+1][i][j+jumps[s]+1] == -1:
#except:
#dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j+ jumps[s] + 1]
dp_plus[s + 1][i][j + jumps[s] + 1] = dp_plus[s][i][j] + input_matrix[i][j + jumps[s] + 1]
det_jump = 1
else:
if dp_plus[s + 1][i][j + jumps[s] + 1] >= dp_plus[s][i][j] + input_matrix[i][j + jumps[s] + 1]:
dp_plus[s + 1][i][j + jumps[s] + 1] = dp_plus[s][i][j] + input_matrix[i][j + jumps[s] + 1]
det_jump = 1
else: # det_row = 1 row can't jump
if i == height - 1: # row can't walk
if dp_plus[s][i][j + 1] == -1:
dp_plus[s][i][j + 1] = dp_plus[s][i][j] + input_matrix[i][j + 1]
# except:
# dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
# dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
dp_plus[s][i][j + 1] = min(dp_plus[s][i][j + 1],
dp_plus[s][i][j] + input_matrix[i][j + 1])
if s != len_jump - 1:
if dp_plus[s + 1][i][j + jumps[s] + 1] == -1:
# except:
# dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j+ jumps[s] + 1]
dp_plus[s + 1][i][j + jumps[s] + 1] = dp_plus[s][i][j] + input_matrix[i][
j + jumps[s] + 1]
det_jump = 1
else:
if dp_plus[s + 1][i][j + jumps[s] + 1] >= dp_plus[s][i][j] + input_matrix[i][
j + jumps[s] + 1]:
dp_plus[s + 1][i][j + jumps[s] + 1] = dp_plus[s][i][j] + input_matrix[i][
j + jumps[s] + 1]
det_jump = 1
else: # row can walk
if dp_plus[s][i + 1][j] == -1:
dp_plus[s][i + 1][j] = dp_plus[s][i][j] + input_matrix[i + 1][j]
else:
dp_plus[s][i + 1][j] = min(dp_plus[s][i + 1][j],
dp_plus[s][i][j] + input_matrix[i + 1][j])
if dp_plus[s][i][j + 1] == -1:
dp_plus[s][i][j + 1] = dp_plus[s][i][j] + input_matrix[i][j + 1]
# except:
# dp[(i,j+1,s)] = dp[(i,j,s)] + input_matrix[i][j+1]
else:
# dp[(i,j+1,s)] = min(dp[(i,j+1,s)],dp[(i,j,s)] + input_matrix[i][j+1])
dp_plus[s][i][j + 1] = min(dp_plus[s][i][j + 1],
dp_plus[s][i][j] + input_matrix[i][j + 1])
if s != len_jump - 1:
if dp_plus[s + 1][i][j + jumps[s] + 1] == -1:
# except:
# dp[(i, j + jumps[s] + 1, s + 1)] = dp[(i, j, s)] + input_matrix[i][j+ jumps[s] + 1]
dp_plus[s + 1][i][j + jumps[s] + 1] = dp_plus[s][i][j] + input_matrix[i][
j + jumps[s] + 1]
det_jump = 1
else:
if dp_plus[s + 1][i][j + jumps[s] + 1] >= dp_plus[s][i][j] + input_matrix[i][
j + jumps[s] + 1]:
dp_plus[s + 1][i][j + jumps[s] + 1] = dp_plus[s][i][j] + input_matrix[i][
j + jumps[s] + 1]
det_jump = 1
# renew near element
#for i in range(height):
# for j in range(width):
mini = dp_plus[0][height-1][width-1]
for i in range(len_jump):
if dp_plus[i][height-1][width-1] != -1:
if mini > dp_plus[i][height-1][width-1]:
mini = dp_plus[i][height-1][width-1]
print(mini)
sample dictionary
对于sample，最后我们会得到这样的一个dict
{(0, 0, 0): 1, (1, 0, 0): 9, (0, 1, 0): 1, (2, 0, 1): 10, (0, 2, 1): 4, (1, 1, 0): 4, (0, 2, 0): 4, (2, 1, 1): 8, (0, 3, 1): 8, (1, 2, 0): 11, (0, 3, 0): 11, (2, 2, 1): 12, (0, 4, 1): 6, (1, 3, 0): 17, (0, 4, 0): 13, (2, 3, 1): 12, (0, 5, 1): 11, (1, 4, 0): 22, (0, 5, 0): 18, (2, 4, 1): 13, (1, 5, 0): 26, (2, 5, 1): 14, (2, 0, 0): 18, (3, 0, 1): 12, (1, 2, 1): 11, (2, 1, 0): 11, (3, 1, 1): 6, (1, 3, 1): 10, (2, 2, 0): 19, (3, 2, 1): 15, (1, 4, 1): 15, (2, 3, 0): 19, (3, 3, 1): 13, (1, 5, 1): 19, (2, 4, 0): 20, (3, 4, 1): 20, (2, 5, 0): 21, (3, 5, 1): 22, (3, 0, 0): 21, (3, 1, 0): 13, (3, 2, 0): 22, (3, 3, 0): 20, (3, 4, 0): 27, (3, 5, 0): 29, (2, 2, 2): 12, (0, 4, 2): 6, (2, 3, 2): 10, (0, 5, 2): 11, (2, 4, 2): 7, (2, 5, 2): 8, (3, 2, 2): 20, (1, 4, 2): 15, (3, 3, 2): 7, (1, 5, 2): 18, (3, 4, 2): 14, (3, 5, 2): 16, (1, 4, 3): 15, (0, 5, 3): 11, (1, 5, 3): 19, (2, 4, 3): 11, (2, 5, 3): 8, (3, 2, 3): 21, (2, 3, 3): 14, (3, 3, 3): 11, (3, 4, 3): 14, (3, 5, 3): 16}
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