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[pull] master from wisdompeak:master #321

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pull merged 3 commits into AlgorithmAndLeetCode:master from wisdompeak:master
Jun 10, 2025
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Original file line number Diff line number Diff line change
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class Solution {
public:
int countPartitions(vector<int>& nums, int k) {
int n = nums.size();
long M = 1e9+7;
nums.insert(nums.begin(), 0);

vector<long>dp(n+1);
vector<long>presum(n+1);
dp[0] = 1;
presum[0] = 1;

deque<int>dq1;
deque<int>dq2;
int left = 1;
for (int i=1; i<=n; i++) {
int x = nums[i];
while (!dq1.empty() && nums[dq1.back()]<x) {
dq1.pop_back();
}
dq1.push_back(i);
while (!dq2.empty() && nums[dq2.back()]>x) {
dq2.pop_back();
}
dq2.push_back(i);

while (left <= i && nums[dq1.front()] - nums[dq2.front()] > k) {
if (dq1.front()==left) dq1.pop_front();
if (dq2.front()==left) dq2.pop_front();
left++;
}
// Any valid parition that ends at left-1, left, left+1, ..., i-1 is good.

dp[i] = presum[i-1] - (left>=2?presum[left-2]:0);
presum[i] = presum[i-1] + dp[i];

dp[i] = (dp[i] + M) % M;
presum[i] = (presum[i] + M) % M;
}

return dp[n];
}
};
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### 3578.Count-Partitions-With-Max-Min-Difference-at-Most-K

我们很容易想到令dp[i]表示以i为最后一段区间的结尾,可以得到的切割方案。此时我们需要考虑最后一段的起点位置j可以在哪里。

题目条件中"区间最大值与区间最小值的差",很容易提示我们可以用deque来分别求区间的最大值和最小值。更具体地,当我们从nums[0]开始,不断加入元素,直至将nums[i]纳入区间时,我们可以得到此时区间的最大值a和最小值b:如果a-b>k,那么意味着区间太长,起点位置应该往右移动。这是因为区间越大,就越容易得到更大的a和更小的b,必然有更大的差值。区间越小,a-b的值就会越小,而且这是一个单调的过程。于是我们调整左端点j右移,每移动一次的过程中,我们查看一下nums[j]的退出是否会影响保存区间最大值和最小值的两个deque(因为我们知道当前最大值或最小值必然是deque的首元素)。直至我们将左端点移动到L的位置,意味着区间[L:i]恰好满足最大值与最小值之差小于k。

以上说明区间[L:i]是一个合法的切分,同理更小的区间[L+1:i],[L+2:i]...都是满足条件的切分。既然确定了最后一个区间的切法,那么就有`dp[i] = dp[L-1]+dp[L]+dp[L+1]...+dp[i-1]` 显然,我们会用一个DP的前缀和数组presum来辅助,即`dp[i] = presum[i-1]-presum[L-2]`.

记得得到上述的dp[i]之后,就可以同样更新presum[i]。

以1-index考虑的话,最终的结果就是dp[n]。
1 change: 1 addition & 0 deletions Readme.md
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Expand Up @@ -485,6 +485,7 @@
[2398.Maximum-Number-of-Robots-Within-Budget](https://github.com/wisdompeak/LeetCode/tree/master/Deque/2398.Maximum-Number-of-Robots-Within-Budget) (H-)
[2762.Continuous-Subarrays](https://github.com/wisdompeak/LeetCode/tree/master/Deque/2762.Continuous-Subarrays) (M+)
[2969.Minimum-Number-of-Coins-for-Fruits-II](https://github.com/wisdompeak/LeetCode/tree/master/Deque/2969.Minimum-Number-of-Coins-for-Fruits-II) (H-)
[3578.Count-Partitions-With-Max-Min-Difference-at-Most-K](https://github.com/wisdompeak/LeetCode/tree/master/Deque/3578.Count-Partitions-With-Max-Min-Difference-at-Most-K) (H-)

#### [Priority Queue](https://github.com/wisdompeak/LeetCode/tree/master/Priority_Queue)
[004.Median-of-Two-Sorted-Arrays](https://github.com/wisdompeak/LeetCode/tree/master/Priority_Queue/004.Median-of-Two-Sorted-Arrays) (H)
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