Problem:
 Score : 600 points    Problem Statement Snuke, a water strider, lives in a rectangular pond that can be seen as a grid with H east-west rows and W north-south columns. Let (i,j) be the square at the i -th row from the north and j -th column from the west.  Some of the squares have a lotus leaf on it and cannot be entered.
The square (i,j) has a lotus leaf on it if c_{ij} is @ , and it does not if c_{ij} is . .  In one stroke, Snuke can move between 1 and K squares (inclusive) toward one of the four directions: north, east, south, and west.
The move may not pass through a square with a lotus leaf. Moving to such a square or out of the pond is also forbidden.  Find the minimum number of strokes Snuke takes to travel from the square (x_1,y_1) to (x_2,y_2) .
If the travel from (x_1,y_1) to (x_2,y_2) is impossible, point out that fact.      Constraints  1 \leq H,W,K \leq 10^6  H \times W \leq 10^6  1 \leq x_1,x_2 \leq H  1 \leq y_1,y_2 \leq W  x_1 \neq x_2 or y_1 \neq y_2 .  c_{i,j} is . or @ .  c_{x_1,y_1} =  .  c_{x_2,y_2} =  .  All numbers in input are integers.         Input Input is given from Standard Input in the following format:  H  W  K  x_1  y_1  x_2  y_2  c_{1,1}c_{1,2}  ..  c_{1,W}  c_{2,1}c_{2,2}  ..  c_{2,W}  :  c_{H,1}c_{H,2}  ..  c_{H,W}       Output Print the minimum number of strokes Snuke takes to travel from the square (x_1,y_1) to (x_2,y_2) , or print -1 if the travel is impossible.        Sample Input 1 3 5 2
3 2 3 4
.....
.@..@
..@..      Sample Output 1 5  Initially, Snuke is at the square (3,2) .
He can reach the square (3, 4) by making five strokes as follows:    From (3, 2) , go west one square to (3, 1) .    From (3, 1) , go north two squares to (1, 1) .    From (1, 1) , go east two squares to (1, 3) .    From (1, 3) , go east one square to (1, 4) .    From (1, 4) , go south two squares to (3, 4) .         Sample Input 2 1 6 4
1 1 1 6
......      Sample Output 2 2       Sample Input 3 3 3 1
2 1 2 3
.@.
.@.
.@.      Sample Output 3 -1

Solution (Python):

import sys
input = sys.stdin.readline
from collections import defaultdict, deque

def I(): return int(input())
def MI(): return map(int, input().split())
def LI(): return list(map(int, input().split()))

def main():
    h, w, k = MI()
    x1, y1, x2, y2 = MI()
    x1 -= 1
    y1 -= 1
    x2 -= 1
    y2 -= 1

    mylist = [0] * h

    
    for i in range(h):
        temp = input().rstrip()
        temp = [temp[j] for j in range(w)]
        mylist[i] = temp
    
    distance = [[-1 for i in range(w)] for j in range(h)]
    distance[x1][y1] = 0
    d = deque()
    start = (x1, y1)
    goal = [x2, y2]
    d.append(start)
    move = ([-1,0],[1,0],[0,1],[0,-1])

    while d:
        pos = d.popleft()
        x = pos[0]
        y = pos[1]
        if x == x2 and y == y2:
            print(distance[x2][y2])
            exit()
        dist = distance[x][y]
        for i in move:
            
            tempx = x
            tempy = y
            for l in range(k):
                tempx += i[0]
                tempy += i[1]
                if not ((tempx >= 0 and tempx <= h-1) and (tempy >= 0 and tempy <= w - 1)):
                    break
                elif mylist[tempx][tempy] == "." and distance[tempx][tempy] == -1:
                    distance[tempx][tempy] = dist + 1
                    d.append((tempx, tempy))
                elif dist + 1 <= distance[tempx][tempy] and mylist[tempx][tempy] == ".":
                    distance[tempx][tempy] = dist + 1
                else:
                    break
    print(-1)            
        
main()

Solution (Python):
from collections import deque
h,w,k=map(int,input().split())
x1,y1,x2,y2=map(lambda x:int(x)-1,input().split())
cmap=[input() for _ in range(h)]
inf=float('inf')
seen=[[[(inf,inf)]*4 for _ in range(w)] for _ in range(h)]
D=[(-1,0),(0,1),(1,0),(0,-1)]
q=deque()
for d in range(4):
  seen[x1][y1][d]=(1,0)
  q.append((x1, y1, d, 1, 0)) 
while q:
  x,y,d,c,k_=q.popleft()
  
  
  
  
  for nd in range(4):
    if nd==d:
      dx,dy=D[nd]
      nk=1 if k_==k else k_+1
      nc=c+1 if k_==k else c
    else:
      dx,dy=0,0
      nk=0
      nc=c+1
    nx=x+dx;ny=y+dy
    if 0<=nx<h and 0<=ny<w and cmap[nx][ny]!='@' and seen[nx][ny][nd]>(nc,nk):
      seen[nx][ny][nd]=(nc,nk)
      if nc==c: 
        q.appendleft((nx,ny,nd,nc,nk))
      else:
        q.append((nx,ny,nd,nc,nk))
ans=inf
for d in range(4):
  ans=min(ans,seen[x2][y2][d][0])
ans=-1 if ans==inf else ans
print(ans)

Solution (Python):
H,W,K=map(int,input().split())
x1,y1,x2,y2=map(lambda x:int(x)-1,input().split())

pond=[list(input()) for _ in range(H)]

inf=float("inf")
cost=[[inf]*W for _ in range(H)]
cost[x1][y1]=0
from heapq import heappop,heappush
hq=[(0,x1,y1)]

while hq:
    t,x,y=heappop(hq)
    if x==x2 and y==y2:
      print(t)
      exit()
    
    for i in range(1,K+1):
        if x-i<0 or pond[x-i][y]=="@" or cost[x-i][y]<t+1:
            break
        if cost[x-i][y]>t+1:
            cost[x-i][y]=t+1
            heappush(hq,(t+1,x-i,y))
    
    for i in range(1,K+1):
        if y+i>W-1 or pond[x][y+i]=="@" or cost[x][y+i]<t+1:
            break
        if cost[x][y+i]>t+1:
            cost[x][y+i]=t+1
            heappush(hq,(t+1,x,y+i))
    
    for i in range(1,K+1):
        if x+i>H-1 or pond[x+i][y]=="@" or cost[x+i][y]<t+1:
            break
        if cost[x+i][y]>t+1:
            cost[x+i][y]=t+1
            heappush(hq,(t+1,x+i,y))
    
    for i in range(1,K+1):
        if y-i<0 or pond[x][y-i]=="@" or cost[x][y-i]<t+1:
            break
        if cost[x][y-i]>t+1:
            cost[x][y-i]=t+1
            heappush(hq,(t+1,x,y-i))

print(-1)

Solution (Python):
H,W,K=map(int,input().split())
x1,y1,x2,y2=map(lambda x:int(x)-1,input().split())

pond=[]
for _ in range(H):
    pond.append(list(input()))

inf=10**7
cost=[[inf]*W for _ in range(H)]
cost[x1][y1]=0
from heapq import heapify,heappop,heappush
hq=[(0,x1,y1)]

while hq:
    t,x,y=heappop(hq)
    if x==x2 and y==y2:
      print(t)
      exit()
    
    for i in range(1,K+1):
        if x-i<0 or pond[x-i][y]=="@" or cost[x-i][y]<t+1:
            break
        if cost[x-i][y]>t+1:
            cost[x-i][y]=t+1
            heappush(hq,(t+1,x-i,y))
    
    for i in range(1,K+1):
        if y+i>W-1 or pond[x][y+i]=="@" or cost[x][y+i]<t+1:
            break
        if cost[x][y+i]>t+1:
            cost[x][y+i]=t+1
            heappush(hq,(t+1,x,y+i))
    
    for i in range(1,K+1):
        if x+i>H-1 or pond[x+i][y]=="@" or cost[x+i][y]<t+1:
            break
        if cost[x+i][y]>t+1:
            cost[x+i][y]=t+1
            heappush(hq,(t+1,x+i,y))
    
    for i in range(1,K+1):
        if y-i<0 or pond[x][y-i]=="@" or cost[x][y-i]<t+1:
            break
        if cost[x][y-i]>t+1:
            cost[x][y-i]=t+1
            heappush(hq,(t+1,x,y-i))

print(-1)

Solution (Python):
H,W,K=map(int,input().split())
x1,y1,x2,y2=map(lambda x:int(x)-1,input().split())

pond=[]
for _ in range(H):
    pond.append(list(input()))

inf=10**7
cost=[[inf]*W for _ in range(H)]
cost[x1][y1]=0
from heapq import heapify,heappop,heappush
hq=[(0,x1,y1)]

while hq:
    t,x,y=heappop(hq)
    
    for i in range(1,K+1):
        if x-i<0 or pond[x-i][y]=="@" or cost[x-i][y]<t+1:
            break
        if cost[x-i][y]>t+1:
            cost[x-i][y]=t+1
            heappush(hq,(t+1,x-i,y))
    
    for i in range(1,K+1):
        if y+i>W-1 or pond[x][y+i]=="@" or cost[x][y+i]<t+1:
            break
        if cost[x][y+i]>t+1:
            cost[x][y+i]=t+1
            heappush(hq,(t+1,x,y+i))
    
    for i in range(1,K+1):
        if x+i>H-1 or pond[x+i][y]=="@" or cost[x+i][y]<t+1:
            break
        if cost[x+i][y]>t+1:
            cost[x+i][y]=t+1
            heappush(hq,(t+1,x+i,y))
    
    for i in range(1,K+1):
        if y-i<0 or pond[x][y-i]=="@" or cost[x][y-i]<t+1:
            break
        if cost[x][y-i]>t+1:
            cost[x][y-i]=t+1
            heappush(hq,(t+1,x,y-i))

if cost[x2][y2]==inf:
    print(-1)
    exit()  
print(cost[x2][y2])

Solution (Python):
H,W,K=map(int,input().split())
x1,y1,x2,y2=map(lambda x:int(x)-1,input().split())

pond=[]
for _ in range(H):
    pond.append(list(input()))

inf=10**7
cost=[[inf]*W for _ in range(H)]
cost[x1][y1]=0
from heapq import heapify,heappop,heappush
hq=[(0,x1,y1)]

while hq:
    t,x,y=heappop(hq)
    
    for i in range(1,K+1):
        if x-i<0 or pond[x-i][y]=="@" or cost[x-i][y]<t+1:
            break
        if cost[x-i][y]>t+1:
            cost[x-i][y]=t+1
            heappush(hq,(t+1,x-i,y))
    
    for i in range(1,K+1):
        if y+i>W-1 or pond[x][y+i]=="@" or cost[x][y+i]<t+1:
            break
        if cost[x][y+i]>t+1:
            cost[x][y+i]=t+1
            heappush(hq,(t+1,x,y+i))
    
    for i in range(1,K+1):
        if x+i>H-1 or pond[x+i][y]=="@" or cost[x+i][y]<t+1:
            break
        if cost[x+i][y]>t+1:
            cost[x+i][y]=t+1
            heappush(hq,(t+1,x+i,y))
    
    for i in range(1,K+1):
        if y-i<0 or pond[x][y-i]=="@" or cost[x][y-i]<t+1:
            break
        if cost[x][y-i]>t+1:
            cost[x][y-i]=t+1
            heappush(hq,(t+1,x,y-i))

if cost[x2][y2]==inf:
    print(-1)
    exit()  
print(cost[x2][y2])

Solution (Python):
from heapq import heappop, heappush
import sys
input = sys.stdin.readline
H, W, K = map(int, input().split())
x1, y1, x2, y2 = map(lambda x: int(x)-1, input().split())
S = [input()[:-1] for _ in [0]*H]

def geth(x, y, z):
    return (x*W+y)*4+z

def push(x, y, z, d):
    h = geth(x, y, z)
    if D[h] <= d:
        return
    D[h] = d
    heappush(q, (d, h))

INF = 10**10
MAX = H*W*4
D = [INF]*MAX
dirs = [(-1, 0), (0, -1), (1, 0), (0, 1)]

q = []
for z in range(4):
    heappush(q, (0, geth(x1, y1, z)))

while q:
    d, h = heappop(q)
    z = h % 4
    x, y = divmod((h//4), W)

    if d > D[h]:
        continue
    D[h] = d

    for _z, (dx, dy) in enumerate(dirs):
        if _z == z:
            _x, _y = x+dx, y+dy
            if 0 <= _x < H and 0 <= _y < W and S[_x][_y] == '.':
                push(_x, _y, z, d+1)
        else:
            _h = geth(x, y, _z)
            _d = (d+K-1)//K*K
            push(x, y, _z, _d)

g = geth(x2, y2, 0)
ans = min(D[g:g+4])
if ans == INF:
    print(-1)
else:
    ans = (ans-1+K)//K
    print(ans)

Solution (Python):
from sys import stdin
def input():
    return stdin.readline().strip()

import heapq

def main():
    h, w, k = map(int, input().split())
    x1, y1, x2, y2 = map(int, input().split())

    
    c = [['@'] * (w+2)]
    for _ in range(h):
        i = input()
        l = ['@']
        for j in i:
            l.append(j)
        l.append('@')
        c.append(l)
    c.append(['@'] * (w+2))

    
    
    
    seen = [[-1] * (w+2) for _ in range(h+2)]

    
    todo = [(0, x1, y1, 0)]
    heapq.heapify(todo)

    
    dx = [-1, 0, 1, 0]
    dy = [0, 1, 0, -1]

    while len(todo) > 0:
        cost, xi, yi, direction = heapq.heappop(todo)

        
        if seen[xi][yi] != -1:
            if (seen[xi][yi]+k-1)//k * k > cost:
                xj, yj = xi + dx[direction], yi + dy[direction]
                if c[xj][yj] == '.':
                    heapq.heappush(todo, (cost+1, xj, yj, direction))
            continue
        seen[xi][yi] = cost

        
        if xi == x2 and yi == y2:
            print((cost+k-1)//k)
            exit()
        
        
        for j in range(4):
            xj, yj = xi + dx[j], yi + dy[j]
            if c[xj][yj] == '.':
                if j == direction:
                    if seen[xj][yj] == -1 or (seen[xj][yj]+k-1)//k * k > cost:
                        heapq.heappush(todo, (cost+1, xj, yj, j))
                elif seen[xj][yj] == -1:
                    heapq.heappush(todo, ((cost+k-1)//k * k + 1, xj, yj, j))

    print(-1)

main()

Solution (Python):
from sys import stdin
def input():
    return stdin.readline().strip()

import heapq

def main():
    h, w, k = map(int, input().split())
    x1, y1, x2, y2 = map(int, input().split())

    
    c = [['@'] * (w+2)]
    for _ in range(h):
        i = input()
        l = ['@']
        for j in i:
            l.append(j)
        l.append('@')
        c.append(l)
    c.append(['@'] * (w+2))

    
    
    
    seen = [[-1] * (w+2) for _ in range(h+2)]

    
    todo = [(0, x1, y1, 0)]
    heapq.heapify(todo)

    
    dx = [-1, 0, 1, 0]
    dy = [0, 1, 0, -1]

    while len(todo) > 0:
        cost, xi, yi, direction = heapq.heappop(todo)

        
        if seen[xi][yi] != -1:
            if (seen[xi][yi]+k-1)//k * k > cost:
                xj, yj = xi + dx[direction], yi + dy[direction]
                if c[xj][yj] == '.':
                    heapq.heappush(todo, (cost+1, xj, yj, direction))
            continue
        seen[xi][yi] = cost

        
        if xi == x2 and yi == y2:
            print((cost+k-1)//k)
            exit()
        
        
        for j in range(4):
            xj, yj = xi + dx[j], yi + dy[j]
            if c[xj][yj] == '.':
                if j == direction:
                    if seen[xj][yj] == -1 or (seen[xj][yj]+k-1)//k * k > cost:
                        heapq.heappush(todo, (cost+1, xj, yj, j))
                elif seen[xj][yj] == -1:
                    heapq.heappush(todo, ((cost+k-1)//k * k + 1, xj, yj, j))

    print(-1)

main()

Solution (Python):
from sys import stdin
def input():
    return stdin.readline().strip()

import heapq

def main():
    h, w, k = map(int, input().split())
    x1, y1, x2, y2 = map(int, input().split())

    
    c = [['@'] * (w+2)]
    for _ in range(h):
        i = input()
        l = ['@']
        for j in i:
            l.append(j)
        l.append('@')
        c.append(l)
    c.append(['@'] * (w+2))

    
    
    
    seen = [[-1] * (w+2) for _ in range(h+2)]

    
    todo = [(0, x1, y1, 0)]
    heapq.heapify(todo)

    
    dx = [-1, 0, 1, 0]
    dy = [0, 1, 0, -1]

    while len(todo) > 0:
        cost, xi, yi, direction = heapq.heappop(todo)

        
        if seen[xi][yi] != -1:
            if (seen[xi][yi]+k-1)//k * k > cost:
                xj, yj = xi + dx[direction], yi + dy[direction]
                if c[xj][yj] == '.':
                    heapq.heappush(todo, (cost+1, xj, yj, direction))
            continue
        seen[xi][yi] = cost

        
        if xi == x2 and yi == y2:
            print((cost+k-1)//k)
            exit()
        
        
        for j in range(4):
            xj, yj = xi + dx[j], yi + dy[j]
            if c[xj][yj] == '.':
                if j == direction:
                    if seen[xj][yj] == -1 or (seen[xj][yj]+k-1)//k * k > cost:
                        heapq.heappush(todo, (cost+1, xj, yj, j))
                elif seen[xj][yj] == -1:
                    heapq.heappush(todo, ((cost+k-1)//k * k + 1, xj, yj, j))

    print(-1)

main()

Solution (Python):
from sys import stdin
def input():
    return stdin.readline().strip()

import heapq

def main():
    h, w, k = map(int, input().split())
    x1, y1, x2, y2 = map(int, input().split())

    
    c = [['@'] * (w+2)]
    for _ in range(h):
        i = list('@' + input() + '@')
        c.append(i)
    c.append(['@'] * (w+2))

    
    
    
    seen = [[-1] * (w+2) for _ in range(h+2)]

    
    todo = [(0, x1, y1, 0)]
    heapq.heapify(todo)

    
    dx = [-1, 0, 1, 0]
    dy = [0, 1, 0, -1]

    while len(todo) > 0:
        cost, xi, yi, direction = heapq.heappop(todo)

        
        if seen[xi][yi] != -1:
            if (seen[xi][yi]+k-1)//k * k > cost:
                xj, yj = xi + dx[direction], yi + dy[direction]
                if c[xj][yj] == '.':
                    heapq.heappush(todo, (cost+1, xj, yj, direction))
            continue
        seen[xi][yi] = cost

        
        if xi == x2 and yi == y2:
            print((cost+k-1)//k)
            exit()
        
        
        for j in range(4):
            xj, yj = xi + dx[j], yi + dy[j]
            if c[xj][yj] == '.':
                if j == direction:
                    if seen[xj][yj] == -1 or (seen[xj][yj]+k-1)//k * k > cost:
                        heapq.heappush(todo, (cost+1, xj, yj, j))
                elif seen[xj][yj] == -1:
                    heapq.heappush(todo, ((cost+k-1)//k * k + 1, xj, yj, j))

    print(-1)

main()

Solution (Python):
from sys import stdin
def input():
    return stdin.readline().strip()

import heapq

def main():
    h, w, k = map(int, input().split())
    x1, y1, x2, y2 = map(int, input().split())

    
    c = [['@'] * (w+2)]
    for _ in range(h):
        i = list('@' + input() + '@')
        c.append(i)
    c.append(['@'] * (w+2))

    
    
    
    seen = [[-1] * (w+2) for _ in range(h+2)]

    
    todo = [(0, x1, y1, 0)]
    heapq.heapify(todo)

    
    dx = [-1, 0, 1, 0]
    dy = [0, 1, 0, -1]

    while len(todo) > 0:
        cost, xi, yi, direction = heapq.heappop(todo)

        
        if seen[xi][yi] != -1:
            if (seen[xi][yi]+k-1)//k * k >= cost:
                xj, yj = xi + dx[direction], yi + dy[direction]
                if c[xj][yj] == '.':
                    heapq.heappush(todo, (cost+1, xj, yj, direction))
            continue
        seen[xi][yi] = cost

        
        if xi == x2 and yi == y2:
            print((cost+k-1)//k)
            exit()
        
        
        for j in range(4):
            xj, yj = xi + dx[j], yi + dy[j]
            if c[xj][yj] == '.':
                if j == direction:
                    if seen[xj][yj] == -1 or (seen[xj][yj]+k-1)//k * k > cost:
                        heapq.heappush(todo, (cost+1, xj, yj, j))
                elif seen[xj][yj] == -1:
                    heapq.heappush(todo, ((cost+k-1)//k * k + 1, xj, yj, j))

    print(-1)

main()

Solution (Python):
from collections import defaultdict, deque, Counter
from heapq import heappush, heappop, heapify
import math
import bisect
import random
from itertools import permutations, accumulate, combinations, product
import sys
import string
from bisect import bisect_left, bisect_right
from math import factorial, ceil, floor, gcd
from operator import mul
from functools import reduce
from operator import mul
from pprint import pprint



sys.setrecursionlimit(2147483647)
INF = 10 ** 20
def LI(): return list(map(int, sys.stdin.buffer.readline().split()))
def I(): return int(sys.stdin.buffer.readline())
def LS(): return sys.stdin.buffer.readline().rstrip().decode('utf-8').split()
def S(): return sys.stdin.buffer.readline().rstrip().decode('utf-8')
def IR(n): return [I() for i in range(n)]
def LIR(n): return [LI() for i in range(n)]
def SR(n): return [S() for i in range(n)]
def LSR(n): return [LS() for i in range(n)]
def SRL(n): return [list(S()) for i in range(n)]
def MSRL(n): return [[int(j) for j in list(S())] for i in range(n)]
mod = 1000000007

h, w, k = LI()
sy, sx, gy, gx = LI()
cost = [[INF] * w for _ in range(h)]
cost[sy - 1][sx - 1] = 0
grid = SR(h)
q = deque([(sy - 1, sx - 1)])
while q:
    cy, cx = q.popleft()
    for i, j in ((1, 0), (0, 1), (-1, 0), (0, -1)):
        for l in range(1, k + 1):
            ny, nx = cy + i * l, cx + j * l
            if 0 <= nx < w and 0 <= ny < h and grid[ny][nx] == "." and cost[ny][nx] >= cost[cy][cx] + 1:
                if cost[ny][nx] == INF:
                    q.append((ny, nx))
                    cost[ny][nx] = cost[cy][cx] + 1
            else:
                break


print(cost[gy - 1][gx - 1] if cost[gy - 1][gx - 1] != INF else -1)

Solution (Python):
from collections import defaultdict, deque, Counter
from heapq import heappush, heappop, heapify
import math
import bisect
import random
from itertools import permutations, accumulate, combinations, product
import sys
import string
from bisect import bisect_left, bisect_right
from math import factorial, ceil, floor, gcd
from operator import mul
from functools import reduce
from operator import mul
from pprint import pprint



sys.setrecursionlimit(2147483647)
INF = 10 ** 20
def LI(): return list(map(int, sys.stdin.buffer.readline().split()))
def I(): return int(sys.stdin.buffer.readline())
def LS(): return sys.stdin.buffer.readline().rstrip().decode('utf-8').split()
def S(): return sys.stdin.buffer.readline().rstrip().decode('utf-8')
def IR(n): return [I() for i in range(n)]
def LIR(n): return [LI() for i in range(n)]
def SR(n): return [S() for i in range(n)]
def LSR(n): return [LS() for i in range(n)]
def SRL(n): return [list(S()) for i in range(n)]
def MSRL(n): return [[int(j) for j in list(S())] for i in range(n)]
mod = 1000000007

h, w, k = LI()
sy, sx, gy, gx = LI()
cost = [[INF] * w for _ in range(h)]
cost[sy - 1][sx - 1] = 0
grid = SR(h)
q = deque([(sy - 1, sx - 1)])
while q:
    cy, cx = q.popleft()
    for i, j in ((1, 0), (0, 1), (-1, 0), (0, -1)):
        for l in range(1, k + 1):
            ny, nx = cy + i * l, cx + j * l
            if 0 <= nx < w and 0 <= ny < h and grid[ny][nx] == "." and cost[ny][nx] >= cost[cy][cx] + 1:
                if cost[ny][nx] == INF:
                    q.append((ny, nx))
                cost[ny][nx] = cost[cy][cx] + 1
            else:
                break


print(cost[gy - 1][gx - 1] if cost[gy - 1][gx - 1] != INF else -1)

Solution (Python):
from collections import defaultdict, deque, Counter
from heapq import heappush, heappop, heapify
import math
import bisect
import random
from itertools import permutations, accumulate, combinations, product
import sys
import string
from bisect import bisect_left, bisect_right
from math import factorial, ceil, floor, gcd
from operator import mul
from functools import reduce
from operator import mul
from pprint import pprint



sys.setrecursionlimit(2147483647)
INF = 10 ** 20
def LI(): return list(map(int, sys.stdin.buffer.readline().split()))
def I(): return int(sys.stdin.buffer.readline())
def LS(): return sys.stdin.buffer.readline().rstrip().decode('utf-8').split()
def S(): return sys.stdin.buffer.readline().rstrip().decode('utf-8')
def IR(n): return [I() for i in range(n)]
def LIR(n): return [LI() for i in range(n)]
def SR(n): return [S() for i in range(n)]
def LSR(n): return [LS() for i in range(n)]
def SRL(n): return [list(S()) for i in range(n)]
def MSRL(n): return [[int(j) for j in list(S())] for i in range(n)]
mod = 1000000007

h, w, k = LI()
sy, sx, gy, gx = LI()
cost = [[INF] * w for _ in range(h)]
cost[sy - 1][sx - 1] = 0
grid = SR(h)
q = deque([(sy - 1, sx - 1)])
while q:
    cy, cx = q.popleft()
    for i, j in ((1, 0), (0, 1), (-1, 0), (0, -1)):
        for l in range(1, k + 1):
            ny, nx = cy + i * l, cx + j * l
            if 0 > nx or nx >= w or 0 > ny or ny >= h or grid[ny][nx] == "@":
                break
            if 0 <= cost[ny][nx] <= cost[cy][cx]:
                break
            if cost[ny][nx] == INF:
                q.append((ny, nx))
            cost[ny][nx] = cost[cy][cx] + 1


print(cost[gy - 1][gx - 1] if cost[gy - 1][gx - 1] != INF else -1)

Solution (Python):
import sys
from collections import deque
h,w,k = map(int, sys.stdin.readline().rstrip("\n").split())
sx,sy,gx,gy = map(int, sys.stdin.readline().rstrip("\n").split())
sx,sy,gx,gy = sx-1,sy-1,gx-1,gy-1
labyrinth = [line.rstrip("\n") for line in sys.stdin]
INF = float('inf')
footprint = [ [[(INF,INF)]*4 for j in range(w)] for i in range(h) ]
m = [(-1, 0), (0, 1), (1, 0), (0, -1)]
q = deque()
for d in range(4):
    footprint[sx][sy][d] = (1,0)
    q.append((sx,sy,d,1,0))
while q:
    x, y, d, c, kk = q.popleft()
    for nd in range(4):
        if nd == d:
            mx,my = m[d]
            nkk = 1 if kk == k else kk + 1
            nc = c + 1 if kk == k else c
        else:
            mx,my = 0,0
            nkk = 0
            nc = c + 1
        nx = x + mx
        ny = y + my
        if 0 <= nx < h and 0 <= ny < w and labyrinth[nx][ny] == '.' and footprint[nx][ny][nd] > (nc, nkk):
            footprint[nx][ny][nd] = (nc, nkk)
            if nc == c:
                q.appendleft((nx, ny, nd, nc, nkk))
            else:
                q.append((nx, ny, nd, nc, nkk))
ans = INF
for d in range(4):
    ans = min(ans, footprint[gx][gy][d][0])
ans = -1 if ans == INF else ans
print(ans)

Solution (Python):
import sys

input=sys.stdin.readline

H,W,K=map(int,input().rstrip().split())
x1,y1,x2,y2=map(int,input().split())
c=[input().rstrip() for i in range(H)]
s=W*(x1-1)+y1-1
g=W*(x2-1)+y2-1

import heapq
que = []
d = [float("inf")] * (4*H*W)
d[4*s] = 0
heapq.heapify(que)
heapq.heappush(que, (0, 4*s))

while len(que) != 0:
    cost, v = heapq.heappop(que)
    if d[v] < cost: continue

    for j in range(4):
        nv=v-v%4+(v%4+j)%4
        if d[nv]>d[v]-d[v]%K+K*(d[v]%K>0):
            d[nv]=d[v]-d[v]%K+K*(d[v]%K>0)
            heapq.heappush(que,(d[nv],nv))

    i=v//4
    h,w=i//W,i%W
    if v%4==0:
        if h!=0 and c[h-1][w]==".":
            if d[v-4*W]>d[v]+1:
                d[v-4*W]=d[v]+1
                heapq.heappush(que,(d[v-4*W],v-4*W))
    elif v%4==2:
        if h!=H-1 and c[h+1][w]==".":
            if d[v+4*W]>d[v]+1:
                d[v+4*W]=d[v]+1
                heapq.heappush(que,(d[v+4*W],v+4*W))
    elif v%4==1:
        if w!=W-1 and c[h][w+1]==".":
            if d[v+4]>d[v]+1:
                d[v+4]=d[v]+1
                heapq.heappush(que,(d[v+4],v+4))
    else:
        if w!=0 and c[h][w-1]==".":
            if d[v-4]>d[v]+1:
                d[v-4]=d[v]+1
                heapq.heappush(que,(d[v-4],v-4))

ans=d[4*g]
if ans>=10**15:
    print(-1)
else:
    print(ans//K+(ans%K>0))

Solution (Python):
from heapq import heappush, heappop
import sys
input = sys.stdin.readline

H, W, K = map(int, input().split())
x1, y1, x2, y2 = map(int, input().split())
c = [[i == '.' for i in input()] for _ in range(H)]

x1, y1, x2, y2 = x1 - 1, y1 - 1, x2 - 1, y2 - 1
INF = 1 << 60
dist = [INF] * (H * W * 2)

def pos(x, y, m) :
  return (x * W + y) * 2 + m
dist[pos(x1, y1, 0)] = dist[pos(x1, y1, 1)] = 0

q = [(0, pos(x1, y1, 0)), (0, pos(x1, y1, 1))]
while q :
  ccost, cpos = heappop(q)
  if ccost > dist[cpos] : continue
  cx = cpos // 2 // W
  cy = cpos // 2 % W
  cmode = cpos % 2
  dir = [
    [(-1, 0), (1, 0)],
    [(0, -1), (0, 1)]
  ]
  for dx, dy in dir[cmode] :
    nx, ny = cx + dx, cy + dy
    ncost = ccost + 1
    if 0 <= nx < H and 0 <= ny < W and c[nx][ny] and ncost < dist[pos(nx, ny, cmode)] :
      heappush(q, (ncost, pos(nx, ny, cmode)))
      dist[pos(nx, ny, cmode)] = ncost
      
  ncost = (ccost + K - 1) // K * K
  if ncost < dist[pos(cx, cy, cmode ^ 1)] :
    dist[pos(cx, cy, cmode ^ 1)] = ncost
    heappush(q, (ncost, pos(cx, cy, cmode ^ 1)))

ans = min(dist[pos(x2, y2, 0)], dist[pos(x2, y2, 1)])
if ans == INF :
  print(-1)
else :
  print((ans + K - 1) // K)

Solution (Python):
from collections import deque
import sys

read = sys.stdin.read
readline = sys.stdin.readline

h, w, k = map(int, readline().split())
x1, y1, x2, y2 = map(int, readline().split())
x1, y1, x2, y2 = x1 - 1, y1 - 1, x2 - 1, y2 - 1

field = [readline() for _ in range(h)]
costs = [[-1] * w for _ in range(h)]

q = deque([(x1, y1)])
costs[x1][y1] = 0

while q:
    x, y = q.popleft()
    for dx, dy in [(1, 0), (-1, 0), (0, 1), (0, -1)]:
        for i in range(1, k + 1):
            px, py = x + dx * i, y + dy * i
            if not(0 <= px < h and 0 <= py < w) or field[px][py] == '@':
                break
            if 0 <= costs[px][py] < costs[x][y] + 1:
                break
            if costs[px][py] == -1:
                q.append((px, py))
            costs[px][py] = costs[x][y] + 1

print(costs[x2][y2])

Solution (Python):


from heapq import heappop, heappush
import sys
input = sys.stdin.buffer.readline
INF = 10 ** 10



def main(H, W, K, x1, y1, x2, y2, mapdata):
    start = x1 * W + y1
    goal = x2 * W + y2

    NUM_DIR = 4
    distances = [(INF, 0)] * (H * W * NUM_DIR)
    key = start * NUM_DIR
    distances[key] = (0, 0)
    distances[key + 2] = (0, 0)

    shortest_path = [0] * (H * W * NUM_DIR)
    shortest_path[key] = 0

    queue = [((0, 0), key), ((0, 0), key + 2)]
    DIRS = [(1, 0), (-1, 0), (0, 1), (0, -1)]
    while queue:
        (d, frac), pos_dir = heappop(queue)
        pos, direction = divmod(pos_dir, NUM_DIR)
        x, y = divmod(pos, W)
        dist = distances[pos_dir]
        if dist < (d, frac):
            continue
        if pos == goal:
            break

        dx, dy = DIRS[direction]
        nx = x + dx
        ny = y + dy
        
        if mapdata[nx * W + ny]:
            if frac == 0:
                newd = d + 1
                newfrac = -K + 1
            else:
                newd = d
                newfrac = frac + 1
            newdist = (newd, newfrac)
            to = (nx * W + ny) * NUM_DIR + direction

            if distances[to] > newdist:
                
                distances[to] = newdist
                heappush(queue, (newdist, to))
                shortest_path[to] = pos_dir

        if direction < 2:
            
            dir = 2
            nx = x
            ny = y + 1
            if mapdata[nx * W + ny]:
                newdist = (d + 1, -K + 1)

                to = (nx * W + ny) * NUM_DIR + dir
                if distances[to] > newdist:
                    
                    distances[to] = newdist
                    heappush(queue, (newdist, to))
                    shortest_path[to] = pos_dir

            dir = 3
            nx = x
            ny = y - 1
            if mapdata[nx * W + ny]:
                newdist = (d + 1, -K + 1)

                to = (nx * W + ny) * NUM_DIR + dir
                if distances[to] > newdist:
                    
                    distances[to] = newdist
                    heappush(queue, (newdist, to))
                    shortest_path[to] = pos_dir
        if direction > 1:
            
            dir = 0
            nx = x + 1
            ny = y
            if mapdata[nx * W + ny]:
                newdist = (d + 1, -K + 1)

                to = (nx * W + ny) * NUM_DIR + dir
                if distances[to] > newdist:
                    
                    distances[to] = newdist
                    heappush(queue, (newdist, to))
                    shortest_path[to] = pos_dir

            dir = 1
            nx = x - 1
            ny = y
            if mapdata[nx * W + ny]:
                newdist = (d + 1, -K + 1)

                to = (nx * W + ny) * NUM_DIR + dir
                if distances[to] > newdist:
                    
                    distances[to] = newdist
                    heappush(queue, (newdist, to))
                    shortest_path[to] = pos_dir

    result = min(distances[goal * NUM_DIR:goal * NUM_DIR + NUM_DIR])
    if result[0] == INF:
        
        return -1
    else:
        return result[0]



if sys.argv[-1] == 'ONLINE_JUDGE':
    print("compiling")
    from numba.pycc import CC
    cc = CC('my_module')
    cc.export('main', 'i8(i8,i8,i8,i8,i8,i8,i8,b1[:])')(main)
    cc.compile()
    exit()
else:
    H, W, K = map(int, input().split())
    x1, y1, x2, y2 = map(int, input().split())

    from my_module import main
    import numpy as np
    C = np.zeros((H + 2, W + 2), np.bool_)
    data = np.frombuffer(
        sys.stdin.buffer.read(),
        'S1')
    
    data = data.reshape(H, -1)
    
    data = data[:, : W] == b'.'
    
    C[1: -1, 1: -1] = data
    C = C.ravel()
    H += 2
    W += 2

    print(main(H, W, K, x1, y1, x2, y2, C))