🈁 🧖 🤰🏻 डोमिनोज़ फ़र्श 💃🏼 💆🏾 🚣🏽

गणित की पहली दिलचस्प समस्याओं में से एक जिसका मैंने सामना किया था, इस प्रकार तैयार की गई थी: "दो विपरीत कोने की कोशिकाओं को एक बिसात से काट दिया गया था, क्या बाकी को" डोमिनोज़ "में काटा जा सकता है - दो कोशिकाओं के आंकड़े आम तौर पर एक साथ?" । Fermat की महान प्रमेय के विपरीत, इसका एक बहुत ही सरल सूत्रीकरण है, इसका एक सरल, सुरुचिपूर्ण, लेकिन स्पष्ट समाधान नहीं है (यदि आप समस्या का समाधान जानते हैं, तो इसे दाईं ओर के आंकड़े पर लागू करने का प्रयास करें)।

इस लेख में मैं कई एल्गोरिदम के बारे में बात करूंगा जो डोमिनोज़ के साथ एक विमान पर एक मनमाना सेलुलर आकृति को कवर कर सकते हैं, यदि संभव हो तो, उन स्थितियों को ढूंढें जहां यह असंभव है और संभावित डोमिनोज़ झुकाव की संख्या पर विचार करें।

नोटा बेने! इस लेख की सामग्री इस ज्यूपिटर-नोटबुक का एक छोटा संस्करण है , जो लेख में दिखाई देने वाले सभी चित्रों और एनिमेशनों को इस लैपटॉप से कोड द्वारा उत्पन्न किया जाता है (हालांकि गितुब पूर्वावलोकन में कोई एनिमेशन नहीं होगा)। वैसे, हेडर से छवियों को भी अजगर / matplotlib का उपयोग करके उत्पन्न किया जाता है

प्रतिपादन के लिए हेल्पर कोड, यह काम में आएगा

import matplotlib.pyplot as plt
from matplotlib import colors as mcolors
colors = dict(mcolors.BASE_COLORS, **mcolors.CSS4_COLORS)

# Sort colors by hue, saturation, value and name.
by_hsv = sorted((tuple(mcolors.rgb_to_hsv(mcolors.to_rgba(color)[:3])), name)
                for name, color in colors.items())
names = [name for hsv, name in by_hsv if name not in {'black', 'k', 'w', 'white', 'crimson', 'royalblue', 'limegreen', 'yellow', 'orange'}]
import random
random.shuffle(names)
names = ['crimson', 'royalblue', 'limegreen', 'yellow', 'orange', *names]
names.append('red')
names.append('white')
names.append('black')

def fill_cell(i, j, color, ax):
    ax.fill([i, i, i + 1, i + 1, i], [j, j + 1, j + 1, j, j], color=color)

def draw_filling(filling):
    if filling is not None:
        n = len(filling)
        m = len(filling[0])
        fig = plt.figure(figsize=(m * 0.75, n * 0.75))

        ax = fig.add_axes([0, 0, 1, 1])
        ax.get_xaxis().set_visible(False)
        ax.get_yaxis().set_visible(False)      

        for name, spine in ax.spines.items():
            spine.set_visible(False)
            spine.set_visible(False)

        for i, row in enumerate(filling):
            i = n - i - 1
            for j, cell in enumerate(row):
                fill_cell(j, i, names[cell], ax)

        for i in range(n + 1):
            ax.plot([0, m], [i, i], color='black')
        for i in range(m + 1):
            ax.plot([i, i], [0, n], color='black')
        plt.close(fig)
        return fig
    else:
        return None

मूल समस्या को हल करने से पहले, आगे बढ़ें

ऐसा करना असंभव है, और इसके लिए एक सुंदर और सरल व्याख्या है:

30 32
,

प्रोफ़ाइल द्वारा डायनामिक प्रोग्रामिंग

, , .

, - , . , ,

F_{n + 1} = F_{n} + F_{n - 1} .

$F_{n+1}=F_n+F_{n-1}.$

"" $C_n^k$ ,

C_{n}^{k} = \frac{n}{k} C_{n - 1}^{k - 1},

$C_n^k=\frac{n}{k}C_{n-1}^{k-1},$

C_{n}^{k} = C_{n - 1}^{k} + C_{n - 1}^{k - 1} .

$C_n^k=C_{n-1}^k+C_{n-1}^{k-1}.$

" $n\times m$ ?" . , , , .

: $(k-1)\times m$ , $k$ $j$ , , $k+1$ $j$ , . — $m$ : $i$ - , , $2^m$ ( — , — ).

$[0, 2^m)$ ( $m$ ).

$n\times m$ $k=n$ , $j=0$ .

$g(k, j, m, p)$ , $g(k, m, m, p)=g(k+1, 0, m, p)$ $g(k, j, m, p)$ $g(k, j-1, m, \cdot)$ , $j$ $j+1$ — : ,

,

. , , , — . , , .

( , ). ( — , — ), , -

tiling = [
    '........',
    '........',
    '........',
    '........',
    '........',
    '........',
    '........',
    '........',
]

def count_tilings(tiling):
    n = len(tiling)
    m = len(tiling[0])
    if ((n + 1) * m * 2 ** m) <= 10000000:
        dp = [[[(0 if k != 0 or j != 0 or mask != 0 else 1) for mask in range(2 ** m)] for j in range(m)] for k in range(n + 1)]
    for k in range(n):
        for j in range(m):
            for mask in range(2 ** m):
                #  
                if k < n - 1 and tiling[k][j] == '.' and tiling[k + 1][j] == '.' and (mask &  (1 << j)) == 0:
                    dp[k + ((j + 1) // m)][(j + 1) % m][mask + (1 << j)] += dp[k][j][mask]
                #  
                if j < m - 1 and tiling[k][j] == '.' and tiling[k][j + 1] == '.' and (mask & (3 << j)) == 0:
                    dp[k + ((j + 1) // m)][(j + 1) % m][mask + (2 << j)] += dp[k][j][mask]
                #  
                if ((1 << j) &  mask) != 0 or tiling[k][j] != '.':
                    dp[k + ((j + 1) // m)][(j + 1) % m][(mask | (1 << j)) - (1 << j)] += dp[k][j][mask]
    return dp

dp = count_tilings(tiling)
print(dp[8][0][0])

12988816

, . , $n\times 2$ — .

tiling_fib = [
    '..',
    '..',
    '..',
    '..',
    '..',
    '..',
    '..',
    '..'
]
dp = count_tilings(tiling_fib)
for i in range(8):
    print(dp[i][0][0], end=' ')

1 1 2 3 5 8 13 21

, ,

tiling_no_corners_opposite = [
    '.......#',
    '........',
    '........',
    '........',
    '........',
    '........',
    '........',
    '#.......',
]
dp = count_tilings(tiling_no_corners_opposite)
print(dp[8][0][0])

. , ?

def cover_if_possible(tiling, dp=None):
    if dp is None:
        dp = count_tilings(tiling)
    n = len(dp) - 1
    m = len(dp[0])
    if dp[n][0][0] == 0:
        return None

    result = [[-1 if tiling[i][j] == '#' else 0 for j in range(m)] for i in range(n)]
    num = 0

    k = n
    j = 0
    mask = 0

    while k > 0 or j > 0:
        #print(k, j, mask)
        prev_j = j - 1
        prev_k = k
        if prev_j == -1:
            prev_j += m
            prev_k -= 1

        #   ,       i, j, mask
        #     ,         
        #   
        for prev_mask in range(2 ** m):
            if prev_k < n - 1 and tiling[prev_k][prev_j] == '.' and tiling[prev_k + 1][prev_j] == '.' and \
                (prev_mask & (1 << prev_j)) == 0 and (prev_mask + (1 << prev_j)) == mask and dp[prev_k][prev_j][prev_mask] != 0:
                mask = prev_mask
                result[prev_k][prev_j] = num
                result[prev_k + 1][prev_j] = num
                #print(f'Vertical at ({prev_k}, {prev_j}) ({prev_k + 1}, {prev_j})')
                num += 1
                break
            elif prev_j < m - 1 and tiling[prev_k][prev_j] == '.' and tiling[prev_k][prev_j + 1] == '.' and (prev_mask & (3 << prev_j)) == 0 and \
                prev_mask + (2 << prev_j) == mask and dp[prev_k][prev_j][prev_mask] != 0:
                mask = prev_mask
                result[prev_k][prev_j] = num
                result[prev_k][prev_j + 1] = num
                #print(f'Horisontal at ({prev_k}, {prev_j}) ({prev_k}, {prev_j + 1})')
                num += 1
                break
            elif (((1 << prev_j) & prev_mask) != 0 or tiling[prev_k][prev_j] != '.') and \
                (prev_mask | (1 << prev_j)) - (1 << prev_j) == mask and dp[prev_k][prev_j][prev_mask] != 0:
                mask = prev_mask
                break

        j = prev_j
        k = prev_k

    return result

filling = cover_if_possible(tiling)
draw_filling(filling)

tiling_random = [
    '........',
    '#.#.....',
    '..#.....',
    '........',
    '........',
    '........',
    '........',
    '...#....'
]
filling_random = cover_if_possible(tiling_random)
draw_filling(filling_random)

def maxmimum_cover(tiling):
    n = len(tiling)
    m = len(tiling[0])
    if ((n + 1) * m * 2 ** m) <= 10000000:
        dp = [[[(n * m if k != 0 or j != 0 or mask != 0 else 0) for mask in range(2 ** m)] for j in range(m)] for k in range(n + 1)]
    for k in range(n):
        for j in range(m):
            for mask in range(2 ** m):
                next_k, next_j = k + ((j + 1) // m), (j + 1) % m
                #  
                if k < n - 1 and tiling[k][j] == '.' and tiling[k + 1][j] == '.' and (mask &  (1 << j)) == 0:
                    dp[next_k][next_j][mask + (1 << j)] = min(dp[next_k][next_j][mask + (1 << j)], dp[k][j][mask])
                #  
                if j < m - 1 and tiling[k][j] == '.' and tiling[k][j + 1] == '.' and (mask & (3 << j)) == 0:
                    dp[next_k][next_j][mask + (2 << j)] = min(dp[next_k][next_j][mask + (2 << j)], dp[k][j][mask])
                #  
                if ((1 << j) &  mask) != 0 or tiling[k][j] != '.':
                    dp[next_k][next_j][(mask | (1 << j)) - (1 << j)] = \
                        min(dp[next_k][next_j][(mask | (1 << j)) - (1 << j)], dp[k][j][mask])
                #   ,    
                else:
                    dp[next_k][next_j][(mask | (1 << j)) - (1 << j)] = \
                        min(dp[next_k][next_j][(mask | (1 << j)) - (1 << j)], dp[k][j][mask] + 1)
    return dp

def cover_maximum_possible(tiling, dp=None):
    if dp is None:
        dp = maxmimum_cover(tiling)
    n = len(dp) - 1
    m = len(dp[0])

    result = [[-1 if tiling[i][j] == '#' else -2 for j in range(m)] for i in range(n)]
    num = 0

    k = n
    j = 0
    mask = 0

    while k > 0 or j > 0:
        #print(k, j, mask)
        prev_j = j - 1
        prev_k = k
        if prev_j == -1:
            prev_j += m
            prev_k -= 1

        #   ,       i, j, mask
        #     ,         
        #   
        for prev_mask in range(2 ** m):
            #    ,        0,  
            # ,       
            if prev_k < n - 1 and tiling[prev_k][prev_j] == '.' and tiling[prev_k + 1][prev_j] == '.' and \
                    (prev_mask & (1 << prev_j)) == 0 and (prev_mask + (1 << prev_j)) == mask and \
                    dp[prev_k][prev_j][prev_mask] == dp[k][j][mask]:
                mask = prev_mask
                result[prev_k][prev_j] = num
                result[prev_k + 1][prev_j] = num
                num += 1
                break
            elif prev_j < m - 1 and tiling[prev_k][prev_j] == '.' and tiling[prev_k][prev_j + 1] == '.' and (prev_mask & (3 << prev_j)) == 0 and \
                    prev_mask + (2 << prev_j) == mask and dp[prev_k][prev_j][prev_mask] == dp[k][j][mask]:
                mask = prev_mask
                result[prev_k][prev_j] = num
                result[prev_k][prev_j + 1] = num
                num += 1
                break
            elif (((1 << prev_j) & prev_mask) != 0 or tiling[prev_k][prev_j] != '.') and \
                    (prev_mask | (1 << prev_j)) - (1 << prev_j) == mask and dp[prev_k][prev_j][prev_mask] == dp[k][j][mask]:
                mask = prev_mask
                break
            elif ((1 << prev_j) & prev_mask) == 0 and tiling[prev_k][prev_j] == '.' and \
                    (prev_mask | (1 << prev_j)) - (1 << prev_j) == mask and dp[prev_k][prev_j][prev_mask] + 1 == dp[k][j][mask]:
                mask = prev_mask
                break

        j = prev_j
        k = prev_k

    return result

tiling_custom=[
    '...####',
    '....###',
    '......#',
    '#.#....',
    '#......',
    '##.....',
    '###...#',
]
filling = cover_maximum_possible(tiling_custom)
draw_filling(filling)

! , . , . $n\times m\times 2^m$ , , $\mathcal{O}(nm2^m)$ , $\mathcal{O}(nm2^{\min(n, m)})$ , $n<m$ . $\mathcal{O}(nm2^m)$ , , $\mathcal{O}(nm)$ , .

- . , — , , . , — . , , — , — , — . , , , . , . , , .

def check_valid(i, j, n, m, tiling):
    return 0 <= i and i < n and 0 <= j and j < m and tiling[i][j] != '#'

def find_augmenting_path(x, y, n, m, visited, matched, tiling):
    if not check_valid(x, y, n, m, tiling):
        return False
    if (x, y) in visited:
        return False
    visited.add((x, y))

    for dx, dy in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
        if not check_valid(x + dx, y + dy, n, m, tiling):
            continue
        if (x + dx, y + dy) not in matched or find_augmenting_path(*matched[(x + dx , y + dy)], n, m, visited, matched, tiling):
            matched[(x + dx, y + dy)] = (x, y)
            return True
    return False

def convert_match(matched, tiling, n, m):
    result = [[-1 if tiling[i][j] == '#' else -2 for j in range(m)] for i in range(n)]
    num = 0
    for x, y in matched:
        _x, _y = matched[(x, y)]
        result[x][y] = num
        result[_x][_y] = num
        num += 1
    return result

def match_with_flow(tiling):
    result_slices = []
    n = len(tiling)
    m = len(tiling[0])

    matched = dict()
    #   
    rows = list(range(n))
    columns = list(range(m))
    random.shuffle(rows)
    random.shuffle(columns)
    result_slices.append(convert_match(matched, tiling, n, m))

    for i in rows:
        for j in columns:
            if (i + j) % 2 == 1:
                continue
            visited = set()
            if find_augmenting_path(i, j, n, m, visited, matched, tiling):
                result_slices.append(convert_match(matched, tiling, n, m))

    return result_slices

sequencial_match = match_with_flow(tiling_custom)

( ) " ". , , , . , , , , , .

UPD. - . , :

( )
, .

, .

! 5 .

. , . , . : , , , . . 4 , . 5 , , ( 5 , )

5 रंगों में पेंटिंग डोमिनोज़

def color_5(filling):
    result = [[i for i in row] for row in filling]
    #  
    domino_tiles = [[] for i in range(max(map(max, filling)) + 1)]
    domino_neighbours = [set() for i in range(max(map(max, filling)) + 1)]
    degree = [0 for i in range(max(map(max, filling)) + 1)]

    n = len(filling)
    m = len(filling[0])

    for i, row in enumerate(filling):
        for j, num in enumerate(row):
            if num >= 0:
                domino_tiles[num].append((i, j))

    for i, tiles in enumerate(domino_tiles):
        for x, y in tiles:
            for dx, dy in [(-1, 0), (1, 0), (0, -1), (0, 1), (-1, -1), (-1, 1), (1, -1), (1, 1)]:
                a, b = x + dx, y + dy
                if 0 <= a and a < n and 0 <= b and b < m and filling[a][b] >= 0 and filling[a][b] != i \
                        and filling[a][b] not in domino_neighbours[i]:
                    domino_neighbours[i].add(filling[a][b])
                    degree[i] += 1

    #       ,      5  
    # .     ,   ,     
    #           ,     
    active_degrees = [set() for i in range(max(degree) + 1)]
    for i, deg in enumerate(degree):
        active_degrees[deg].add(i)

    reversed_order = []

    for step in range(len(domino_tiles)):
        min_degree = min([i for i, dominoes in enumerate(active_degrees) if len(dominoes) > 0])
        domino = active_degrees[min_degree].pop()

        reversed_order.append(domino)
        for other in domino_neighbours[domino]:
            if other in active_degrees[degree[other]]:
                active_degrees[degree[other]].remove(other)
                degree[other] -= 1
                active_degrees[degree[other]].add(other)                

    #             ,
    #     5 ,      ,
    #    .     
    colors = [-1 for domino in domino_tiles]
    slices = [draw_filling(result)]
    for domino in reversed(reversed_order):
        used_colors = [colors[other] for other in domino_neighbours[domino] if colors[other] != -1]

        domino_color = len(used_colors)
        for i, color in enumerate(sorted(set(used_colors))):
            if i != color:
                domino_color = i
                break

        if domino_color < 5:
            colors[domino] = domino_color
            for x, y in domino_tiles[domino]:
                result[x][y] = domino_color

            slices.append(draw_filling(result))        
            continue

        #      ,           
        c = 0
        other = [other for other in domino_neighbours[domino] if colors[other] == c]
        visited = set([other])
        q = Queue()
        q.put(other)
        domino_was_reached = False
        while not q.empty():
            cur = q.get()
            for other in domino_neighbours[cur]:
                if other == domino:
                    domino_was_reached = True
                    break
                if color[other] == c or color[other] == c + 1 and other not in visited:
                    visited.add(other)
                    q.put(other)

        if not domino_was_reached:
            for other in visited:
                color[other] = color[other] ^ 1
                for x, y in domino_tiles[other]:
                    result[x][y] = color[other]
            color[domino] = c
            for x, y in domino_tiles[domino]:
                result[x][y] = c

            slices.append(draw_filling(result))
            continue

        #     2  3.
        c = 2
        other = [other for other in domino_neighbours[domino] if colors[other] == c]
        visited = set([other])
        q = Queue()
        q.put(other)
        domino_was_reached = False
        while not q.empty():
            cur = q.get()
            for other in domino_neighbours[cur]:
                if other == domino:
                    domino_was_reached = True
                    break
                if color[other] == c or color[other] == c + 1 and other not in visited:
                    visited.add(other)
                    q.put(other)
        for other in visited:
            color[other] = color[other] ^ 1
            for x, y in domino_tiles[other]:
                result[x][y] = color[other]
        color[domino] = c
        for x, y in domino_tiles[domino]:
            result[x][y] = c
        slices.append(draw_filling(result))

    return result, slices

यदि आप इस कोड का उपयोग करने जा रहे हैं, तो ध्यान दें कि प्रत्येक चरण वहां खींचा गया है - यह एनीमेशन के लिए आवश्यक था, जो एल्गोरिथ्म को बहुत धीमा कर देता है। यदि आपको केवल अंतिम पेंटिंग की आवश्यकता है, तो स्लाइस चर का उपयोग करने वाले सभी कोड को हटा दें।

ठीक है, अंत में, उन उदाहरणों में से एक का प्रयास करें जो थोड़ा पहले था

डोमिनोज़ फ़र्श

प्रोफ़ाइल द्वारा डायनामिक प्रोग्रामिंग

! 5 .

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