<p>There is an <code>m x n</code> cake that needs to be cut into <code>1 x 1</code> pieces.</p>

<p>You are given integers <code>m</code>, <code>n</code>, and two arrays:</p>

<ul>
	<li><code>horizontalCut</code> of size <code>m - 1</code>, where <code>horizontalCut[i]</code> represents the cost to cut along the horizontal line <code>i</code>.</li>
	<li><code>verticalCut</code> of size <code>n - 1</code>, where <code>verticalCut[j]</code> represents the cost to cut along the vertical line <code>j</code>.</li>
</ul>

<p>In one operation, you can choose any piece of cake that is not yet a <code>1 x 1</code> square and perform one of the following cuts:</p>

<ol>
	<li>Cut along a horizontal line <code>i</code> at a cost of <code>horizontalCut[i]</code>.</li>
	<li>Cut along a vertical line <code>j</code> at a cost of <code>verticalCut[j]</code>.</li>
</ol>

<p>After the cut, the piece of cake is divided into two distinct pieces.</p>

<p>The cost of a cut depends only on the initial cost of the line and does not change.</p>

<p>Return the <strong>minimum</strong> total cost to cut the entire cake into <code>1 x 1</code> pieces.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">m = 3, n = 2, horizontalCut = [1,3], verticalCut = [5]</span></p>

<p><strong>Output:</strong> <span class="example-io">13</span></p>

<p><strong>Explanation:</strong></p>

<p><img alt="" src="https://assets.leetcode.com/uploads/2024/06/04/ezgifcom-animated-gif-maker-1.gif" style="width: 280px; height: 320px;" /></p>

<ul>
	<li>Perform a cut on the vertical line 0 with cost 5, current total cost is 5.</li>
	<li>Perform a cut on the horizontal line 0 on <code>3 x 1</code> subgrid with cost 1.</li>
	<li>Perform a cut on the horizontal line 0 on <code>3 x 1</code> subgrid with cost 1.</li>
	<li>Perform a cut on the horizontal line 1 on <code>2 x 1</code> subgrid with cost 3.</li>
	<li>Perform a cut on the horizontal line 1 on <code>2 x 1</code> subgrid with cost 3.</li>
</ul>

<p>The total cost is <code>5 + 1 + 1 + 3 + 3 = 13</code>.</p>
</div>

<p><strong class="example">Example 2:</strong></p>

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">m = 2, n = 2, horizontalCut = [7], verticalCut = [4]</span></p>

<p><strong>Output:</strong> <span class="example-io">15</span></p>

<p><strong>Explanation:</strong></p>

<ul>
	<li>Perform a cut on the horizontal line 0 with cost 7.</li>
	<li>Perform a cut on the vertical line 0 on <code>1 x 2</code> subgrid with cost 4.</li>
	<li>Perform a cut on the vertical line 0 on <code>1 x 2</code> subgrid with cost 4.</li>
</ul>

<p>The total cost is <code>7 + 4 + 4 = 15</code>.</p>
</div>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= m, n &lt;= 20</code></li>
	<li><code>horizontalCut.length == m - 1</code></li>
	<li><code>verticalCut.length == n - 1</code></li>
	<li><code>1 &lt;= horizontalCut[i], verticalCut[i] &lt;= 10<sup>3</sup></code></li>
</ul>

import numpy as np
class Solution:
    def recursion(self,
        h_bottom: int, h_top: int,
        w_bottom: int, w_top: int,
        ) -> int:

        # 1x1 slice
        if h_bottom == h_top and w_bottom == w_top:
            return 0

        if (h_bottom, h_top, w_bottom, w_top) in self.coordinates_to_minimum_cost:
            return self.coordinates_to_minimum_cost[(h_bottom, h_top, w_bottom, w_top)]

        min_val = np.inf

        for h_cut in range(h_bottom, h_top): # 0,1
            ret1 = self.recursion(h_bottom, h_cut, w_bottom, w_top)
            ret2 = self.recursion(h_cut+1, h_top, w_bottom, w_top)
            min_val = min(min_val, ret1 + ret2 + self.horizontalCut[h_cut])

        for w_cut in range(w_bottom, w_top): # 0
            ret1 = self.recursion(h_bottom, h_top, w_bottom, w_cut)
            ret2 = self.recursion(h_bottom, h_top, w_cut+1, w_top)
            min_val = min(min_val, ret1 + ret2 + self.verticalCut[w_cut])

        self.coordinates_to_minimum_cost[(h_bottom, h_top, w_bottom, w_top)] = min_val

        return min_val


    def minimumCost(self, m: int, n: int, horizontalCut: List[int], verticalCut: List[int]) -> int:
        self.coordinates_to_minimum_cost = dict()
        self.horizontalCut = horizontalCut; self.verticalCut = verticalCut
        return self.recursion(0, m-1, 0, n-1)

'''
Q are m,n bigger than zero?, positive values in cake

Brute force appreoach
recursion(
    h_bottom: int, # 0
    h_top: int, # 2
    w_bottom: int, # 0
    w_top: int, # 1
)
# consider cake h_bottom, ..., h_top inclusive i.e. can cut h_bottom,.., h_top-1
# consider cake w_bottom, ..., w_top inclusive i.e. can cut w_bottom,.., w_top-1

return condition if h_bottom == h_top and w_bottom == w_top # 1x1

if (h_bottom, h_top, w_bottom, w_top) in self.coordinates_to_minimum_cost:
    return self.coordinates_to_minimum_cost[(h_bottom, h_top, w_bottom, w_top)]

min_val = np.inf

for h_cut in range(h_bottom, h_top): # 0,1
    ret1 = recursion(h_bottom, h_cut, w_bottom, w_top)
    ret2 = recursion(h_cut+1, h_top, w_bottom, w_top)
    min_val = min(min_val, ret1 + ret2)

for w_cut in range(w_bottom, w_top): # 0
    ret1 = recursion(h_bottom, h_top, w_bottom, w_cut)
    ret2 = recursion(h_bottom, h_top, w_cut+1, w_top)
    min_val = min(min_val, ret1 + ret2)

self.coordinates_to_minimum_cost[(h_bottom, h_top, w_bottom, w_top)] = min_val

return min_val

'''

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