proof – Huahua’s Tech Road

花花酱 LeetCode 646. Maximum Length of Pair Chain

By zxi on April 9, 2025

这题我选DP，因为不需要证明，直接干就行了。

方法1: DP

首先还是需要对pairs的right进行排序。一方面是为了方便chaining，另一方面是可以去重。

然后定义 dp[i] := 以pairs[i]作为结尾，最长的序列的长度。

状态转移：dp[i] = max(dp[j] + 1) where pairs[i].left > pairs[j].right and 0 < j < i.

解释：对于pairs[i]，找一个最优的pairs[j]，满足pairs[i].left > pairs[j].right，这样我就可以把pairs[i]追加到以pairs[j]结尾的最长序列后面了，长度+1。

边检条件：dp[0:n] = 1，每个pair以自己作为序列的最后元素，长度为1。

时间复杂度：O(n²)
空间复杂度：O(n)

class Solution {

public:

int findLongestChain(vector<vector<int>>& pairs) {

sort(begin(pairs), end(pairs), [](const auto& p1, const auto& p2){

return p1[1] < p2[1];

});

const int n = pairs.size();

vector<int> dp(n, 1);

for (int i = 0; i < n; ++i)

for (int j = 0; j < i; ++j)

if (pairs[i][0] > pairs[j][1])

dp[i] = max(dp[i], dp[j] + 1);

return *max_element(begin(dp), end(dp));

}

};

方法2: 贪心

和DP一样，对数据进行排序。

每当我看到 cur.left > prev.right，那么就直接把cur接在prev后面。我们需要证明这么做是最优的，而不是跳过它选后面的元素。

Case 0: cur已经是最后一个元素了，跳过就不是最优解了。

假设cur后面有next, 因为已经排序过了，我们可以得知 next.right >= cur.right。

Case 1: next.right == cur.right。这时候选cur和选next对后面的决策来说是一样的，但由于Case 0的存在，我必须选cur。

Case 2: next.right > cur.right。不管left的情况怎么样，由于right更小，这时候选择cur一定是优于next。

综上所述，看到有元素可以连接起来就贪心的选它就对了。

时间复杂度：O(nlogn)
空间复杂度：O(1)

class Solution {

public:

int findLongestChain(vector<vector<int>>& pairs) {

sort(begin(pairs), end(pairs), [](const auto& p1, const auto& p2){

return p1[1] < p2[1];

});

int right = INT_MIN;

int ans = 0;

for (const auto& p : pairs)

if (p[0] > right) {

right = p[1];

++ans;

}

return ans;

}

};

花花酱 LeetCode 2396. Strictly Palindromic Number

By zxi on September 7, 2022

An integer n is strictly palindromic if, for every base b between 2 and n - 2 (inclusive), the string representation of the integer n in base b is palindromic.

Given an integer n, return true if n is strictly palindromic and false otherwise.

A string is palindromic if it reads the same forward and backward.

Example 1:

Input: n = 9
Output: false
Explanation: In base 2: 9 = 1001 (base 2), which is palindromic.
In base 3: 9 = 100 (base 3), which is not palindromic.
Therefore, 9 is not strictly palindromic so we return false.
Note that in bases 4, 5, 6, and 7, n = 9 is also not palindromic.

Example 2:

Input: n = 4
Output: false
Explanation: We only consider base 2: 4 = 100 (base 2), which is not palindromic.
Therefore, we return false.

Constraints:

4 <= n <= 10⁵

Solution: Just return false

No such number.

Time complexity: O(1)
Space complexity: O(1)

C++

// Author: Huahua

class Solution {

public:

bool isStrictlyPalindromic(int n) {

return false;

}

};

花花酱 LeetCode 1903. Largest Odd Number in String

By zxi on August 15, 2021

You are given a string num, representing a large integer. Return the largest-valued odd integer (as a string) that is a non-empty substring of num, or an empty string "" if no odd integer exists.

A substring is a contiguous sequence of characters within a string.

Example 1:

Input: num = "52"
Output: "5"
Explanation: The only non-empty substrings are "5", "2", and "52". "5" is the only odd number.

Example 2:

Input: num = "4206"
Output: ""
Explanation: There are no odd numbers in "4206".

Example 3:

Input: num = "35427"
Output: "35427"
Explanation: "35427" is already an odd number.

Constraints:

1 <= num.length <= 10⁵
num only consists of digits and does not contain any leading zeros.

Solution: Find right most odd digit

We just need to find the right most digit that is odd, answer will be num[0:r].

Answer must start with num[0].
Proof:
Assume the largest number is num[i:r] i > 0, we can always extend to the left, e.g. num[i-1:r] which is also an odd number and it’s larger than num[i:r] which contradicts our assumption. Thus the largest odd number (if exists) must start with num[0].

Time complexity: O(n)
Space complexity: O(1)

C++

// Author: Huahua

class Solution {

public:

string largestOddNumber(string num) {

for (int i = num.length() - 1; i >= 0; --i)

if ((num[i] - '0') & 1) return num.substr(0, i + 1);

return "";

}

};

Posts tagged as “proof”

花花酱 LeetCode 646. Maximum Length of Pair Chain

花花酱 LeetCode 2396. Strictly Palindromic Number

Solution: Just return false

C++

花花酱 LeetCode 1903. Largest Odd Number in String

Solution: Find right most odd digit

C++