409. Longest Palindrome
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Problem Statement
Given a string s which consists of lowercase or uppercase letters, return the length of the longest palindrome that can be built with those letters.
Constraints
Letters are case sensitive, for example, "Aa" is not considered a palindrome.
1 <= s.length <= 2000
s consists of lowercase and/or uppercase English letters only.
Examples
Example 1
Input: s = "abccccdd"
Output: 7
Explanation: One longest palindrome that can be built is "dccaccd", whose length is 7.
Example 2
Input: s = "a"
Output: 1
Explanation: The longest palindrome that can be built is "a", whose length is 1.


Q- what you explain the problem statement ? 
You are given a string “s” you task to find the number of characters that can construct  the the longest palindrome string from it 
Q- what is the input/output format ? 

Input : s:string 
Output : string 

Q- we have a definition we need to specify ? 
A plandfrom string is a spring that you can read foreword and backward 

Q- do you practical solution ? 

To build the longest palindrom possible string 
We need to use all the characters of even “occurrences “ + the maximum number of char “with” odd occurrences this number most be odd 
For example 
Abccccdd

Here for example we have c → repeated 4 times (even pair) 
dd → repeated 2 times (even pair) 

A and b reperarte only one time (1) —> so w peak one of them 

Q- how to solve it using functional programming ? 

Step 1 : compute the hashmap from the “s” 
Step 2 : filter the hashmap by summing the values of char that has even number of occurrences 
Step 3 : peek the value of the element that has the biggest odd number of occurrence 

Q- exmple 

"abccccdd"

Step 1 : “abccccdd” 
Compute the hashmap {“a”:1,”b”:1,”c”:4,”d”:2} 
Step 2 : filter 
v1=6 
Step3 : peak
v2=1
Return 7

Q- use more mathemiactical notation to formalize my thinking process here ?
Certainly! Here's a properly formatted version of your content:

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**Problem Statement**  
We aim to determine the length of the longest palindrome that can be constructed from a given string \( s \).

### Definitions:
1. Let \( s \) be a string of length \( n \), where \( s = s_1s_2s_3 \dots s_n \).
2. Let \( C \) be the set of unique characters in \( s \).
3. For each character \( c \in C \), let \( f_c \) denote the frequency (count) of \( c \) in \( s \).

### Objective:
Determine the maximum possible length \( L \) of a palindrome that can be constructed using the characters from \( s \).

### Constraints:
- We can rearrange the characters of \( s \) in any order.
- Each character \( c \) can be used up to \( f_c \) times.
- Letters are case-sensitive (e.g., 'A' and 'a' are different characters).

### Mathematical Formulation:
To construct the longest possible palindrome:

1. **Use Even Counts Symmetrically**:
   - For each character \( c \), we can use it symmetrically by placing the same number of instances on both sides of the palindrome.
   - The maximum number of times we can use \( c \) symmetrically is the largest even number less than or equal to \( f_c \), which is \( 2 \times \left\lfloor \frac{f_c}{2} \right\rfloor \).

2. **Include One Odd Count in the Center (if any)**:
   - If there is at least one character with an odd frequency (\( f_c \mod 2 = 1 \)), we can place one instance of such a character at the center of the palindrome.

### Calculation of Palindrome Length \( L \):
The total length \( L \) of the longest palindrome can be calculated using the following formula:
\[
L = 2 \times \sum_{c \in C} \left\lfloor \frac{f_c}{2} \right\rfloor + \delta
\]
where:
- \( \left\lfloor x \right\rfloor \) denotes the floor function (greatest integer less than or equal to \( x \)).
- \( \delta = 
\begin{cases} 
1, & \text{if there exists at least one } c \text{ with } f_c \mod 2 = 1 \\
0, & \text{otherwise} 
\end{cases}
\)

### Explanation:
- \( 2 \times \sum_{c \in C} \left\lfloor \frac{f_c}{2} \right\rfloor \) accounts for the characters that can be used symmetrically on both sides of the palindrome.
- \( \delta \) accounts for a single character with an odd count that can be placed in the center of the palindrome.

### Alternate Expression:
Another way to express the calculation is:
\[
L = \sum_{c \in C} \left( f_c - (f_c \mod 2) \right) + \delta
\]
where \( f_c - (f_c \mod 2) \) computes the largest even number less than or equal to \( f_c \).

### Step-by-Step Solution:
1. **Compute Frequencies**:  
   Create a frequency map \( \{ f_c \} \) for all characters in \( s \).

2. **Calculate Symmetric Counts**:  
   For each \( c \), calculate \( e_c = f_c - (f_c \mod 2) \).  
   Sum all \( e_c \) to get \( E = \sum_{c \in C} e_c \).

3. **Determine Central Character**:  
   Check if there's any \( c \) with \( f_c \mod 2 = 1 \).  
   Set \( \delta = 1 \) if such a character exists; otherwise, \( \delta = 0 \).

4. **Compute Total Length**:  
   Calculate \( L = E + \delta \).

### Examples:
1. **Example 1**:
   - Input: \( s = "abccccdd" \)
   - Frequencies:
     - \( f_a = 1 \)
     - \( f_b = 1 \)
     - \( f_c = 4 \)
     - \( f_d = 2 \)
   - Symmetric Counts:
     - \( e_a = 1 - (1 \mod 2) = 1 - 1 = 0 \)
     - \( e_b = 1 - 1 = 0 \)
     - \( e_c = 4 - 0 = 4 \)
     - \( e_d = 2 - 0 = 2 \)
   - \( E = 0 + 0 + 4 + 2 = 6 \)
   - Central Character:
     - Characters 'a' and 'b' have odd counts.
     - \( \delta = 1 \)
   - Total Length:
     - \( L = 6 + 1 = 7 \)
   - Conclusion:
     - The longest palindrome has a length of 7.

2. **Example 2**:
   - Input: \( s = "a" \)
   - Frequencies:
     - \( f_a = 1 \)
   - Symmetric Counts:
     - \( e_a = 1 - 1 = 0 \)
     - \( E = 0 \)
   - Central Character:
     - 'a' has an odd count.
     - \( \delta = 1 \)
   - Total Length:
     - \( L = 0 + 1 = 1 \)
   - Conclusion:
     - The longest palindrome has a length of 1.

### Summary:
- The length of the longest palindrome is determined by utilizing all even counts of characters and possibly one character with an odd count for the center.
- Mathematically, this is expressed by summing the largest even counts of all characters and adding 1 if any odd counts exist.

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