给定一个字符串，找到最长的子字符串，即回文。例如，如果给定的字符串为“ forgeeksskeegfor”，则输出应为“ geeksskeeg”。
先决条件：回文树|最长回文子串
回文树的结构：
回文树的实际结构接近有向图。它实际上是两个Tree的合并结构，它们共享一些公共节点(请参见下图以更好地理解)。树节点通过存储其索引来存储给定字符串的回文子字符串。
该树由两种类型的边组成：
1.插入边缘(加权边缘)
2.最大回文后缀(未加权)
插入边：
从节点u到v的权重为x的插入边意味着节点v是通过在u的字符串的开头和结尾处插入x来形成的。由于“ u”已经是回文，因此节点v处的结果字符串也将是回文。 x将是每个边缘的单个字符。因此，一个节点最多可以有26个插入边(考虑到小写字母字符串)。
最大回文后缀边缘：
顾名思义，对于一个节点，该边将指向其“最大回文后缀字符串”节点。我们不会将完整的字符串本身视为最大回文后缀，因为这没有任何意义(自循环)。为了简单起见，我们将其称为后缀edge(这是指除完整字符串之外的最大后缀)。很明显，每个节点只有1个后缀边缘，因为我们不会在树中存储重复的字符串。
我们将创建所有回文子串，然后返回我们得到的最后一个，因为这将是迄今为止最长的回文子串。
由于回文树按某个字符的到达顺序存储回文，因此最长的将始终位于树数组的最后一个索引处。
下面是上述方法的实现：
C++
// CPP code for Longest Palindromic substring
// using Palindromic Tree data structure
#include
using namespace std;
#define MAXN 1000
struct Node
{
// store start and end indexes of current
// Node inclusively
int start, end;
// stores length of substring
int length;
// stores insertion Node for all characters a-z
int insertionEdge[26];
// stores the Maximum Palindromic Suffix Node for
// the current Node
int suffixEdge;
};
// two special dummy Nodes as explained above
Node root1, root2;
// stores Node information for constant time access
Node tree[MAXN];
// Keeps track the current Node while insertion
int currNode;
string s;
int ptr;
// Function to insert edge in tree
void insert(int currIndex)
{
// Finding X, such that s[currIndex]
// + X + s[currIndex] is palindrome.
int temp = currNode;
while (true)
{
int currLength = tree[temp].length;
if (currIndex - currLength >= 1 &&
(s[currIndex] == s[currIndex -
currLength - 1]))
break;
temp = tree[temp].suffixEdge;
}
// Check if s[currIndex] + X +
// s[currIndex] is already Present in tree.
if (tree[temp].insertionEdge[s[currIndex] -
'a'] != 0)
{
currNode =
tree[temp].insertionEdge[s[currIndex]
- 'a'];
return;
}
// Else Create new node;
ptr++;
tree[temp].insertionEdge[s[currIndex]
- 'a'] = ptr;
tree[ptr].end = currIndex;
tree[ptr].length = tree[temp].length + 2;
tree[ptr].start = tree[ptr].end -
tree[ptr].length + 1;
// Setting suffix edge for newly Created Node.
currNode = ptr;
temp = tree[temp].suffixEdge;
// Longest Palindromic suffix for a
// string of length 1 is a Null string.
if (tree[currNode].length == 1) {
tree[currNode].suffixEdge = 2;
return;
}
// Else
while (true) {
int currLength = tree[temp].length;
if (currIndex - currLength >= 1 &&
(s[currIndex] ==
s[currIndex - currLength - 1]))
break;
temp = tree[temp].suffixEdge;
}
tree[currNode].suffixEdge =
tree[temp].insertionEdge[s[currIndex] - 'a'];
}
// Driver code
int main()
{
// Imaginary root's suffix edge points to
// itself, since for an imaginary string
// of length = -1 has an imaginary suffix
// string. Imaginary root.
root1.length = -1;
root1.suffixEdge = 1;
// NULL root's suffix edge points to
// Imaginary root, since for a string
// of length = 0 has an imaginary suffix string.
root2.length = 0;
root2.suffixEdge = 1;
tree[1] = root1;
tree[2] = root2;
ptr = 2;
currNode = 1;
s = "forgeeksskeegfor";
for (int i = 0; i < s.size(); i++)
insert(i);
// last will be the index of our last substring
int last = ptr;
for (int i = tree[last].start;
i <= tree[last].end; i++)
cout << s[i];
return 0;
} Java
// JAVA code for Longest Palindromic subString
// using Palindromic Tree data structure
class GFG
{
static final int MAXN = 1000;
static class Node
{
// store start and end indexes of current
// Node inclusively
int start, end;
// stores length of subString
int length;
// stores insertion Node for all characters a-z
int[] insertionEdge = new int[26];
// stores the Maximum Palindromic Suffix Node for
// the current Node
int suffixEdge;
};
// two special dummy Nodes as explained above
static Node root1, root2;
// stores Node information for constant time access
static Node[] tree = new Node[MAXN];
// Keeps track the current Node while insertion
static int currNode;
static char[] s;
static int ptr;
// Function to insert edge in tree
static void insert(int currIndex)
{
// Finding X, such that s[currIndex]
// + X + s[currIndex] is palindrome.
int temp = currNode;
while (true) {
int currLength = tree[temp].length;
if (currIndex - currLength >= 1 &&
(s[currIndex] == s[currIndex - currLength - 1]))
break;
temp = tree[temp].suffixEdge;
}
// Check if s[currIndex] + X +
// s[currIndex] is already Present in tree.
if (tree[temp].insertionEdge[s[currIndex] - 'a'] != 0)
{
currNode = tree[temp].insertionEdge[s[currIndex] - 'a'];
return;
}
// Else Create new node;
ptr++;
tree[temp].insertionEdge[s[currIndex] - 'a'] = ptr;
tree[ptr].end = currIndex;
tree[ptr].length = tree[temp].length + 2;
tree[ptr].start = tree[ptr].end - tree[ptr].length + 1;
// Setting suffix edge for newly Created Node.
currNode = ptr;
temp = tree[temp].suffixEdge;
// Longest Palindromic suffix for a
// String of length 1 is a Null String.
if (tree[currNode].length == 1)
{
tree[currNode].suffixEdge = 2;
return;
}
// Else
while (true)
{
int currLength = tree[temp].length;
if (currIndex - currLength >= 1 &&
(s[currIndex] == s[currIndex - currLength - 1]))
break;
temp = tree[temp].suffixEdge;
}
tree[currNode].suffixEdge =
tree[temp].insertionEdge[s[currIndex] - 'a'];
}
// Driver code
public static void main(String[] args)
{
// Imaginary root's suffix edge points to
// itself, since for an imaginary String
// of length = -1 has an imaginary suffix
// String. Imaginary root.
root1 = new Node();
root1.length = -1;
root1.suffixEdge = 1;
// null root's suffix edge points to
// Imaginary root, since for a String
// of length = 0 has an imaginary suffix String.
root2 = new Node();
root2.length = 0;
root2.suffixEdge = 1;
for (int i = 0; i < MAXN; i++)
tree[i] = new Node();
tree[1] = root1;
tree[2] = root2;
ptr = 2;
currNode = 1;
s = "forgeeksskeegfor".toCharArray();
for (int i = 0; i < s.length; i++)
insert(i);
// last will be the index of our last subString
int last = ptr;
for (int i = tree[last].start; i <= tree[last].end; i++)
System.out.print(s[i]);
}
}
// This code is contributed by Rajput-JiPython3
# Python3 code for Longest Palindromic
# substring using Palindromic Tree
# data structure
class Node:
def __init__(self, length = None,
suffixEdge = None):
# store start and end indexes
# of current Node inclusively
self.start = None
self.end = None
# Stores length of substring
self.length = length
# stores insertion Node for all
# characters a-z
self.insertionEdge = [0] * 26
# stores the Maximum Palindromic
# Suffix Node for the current Node
self.suffixEdge = suffixEdge
# Function to insert edge in tree
def insert(currIndex):
global currNode, ptr
# Finding X, such that s[currIndex]
# + X + s[currIndex] is palindrome.
temp = currNode
while True:
currLength = tree[temp].length
if (currIndex - currLength >= 1 and
(s[currIndex] == s[currIndex -
currLength - 1])):
break
temp = tree[temp].suffixEdge
# Check if s[currIndex] + X +
# s[currIndex] is already Present in tree.
if tree[temp].insertionEdge[ord(s[currIndex]) -
ord('a')] != 0:
currNode = tree[temp].insertionEdge[ord(s[currIndex]) -
ord('a')]
return
# Else Create new node
ptr += 1
tree[temp].insertionEdge[ord(s[currIndex]) -
ord('a')] = ptr
tree[ptr].end = currIndex
tree[ptr].length = tree[temp].length + 2
tree[ptr].start = (tree[ptr].end -
tree[ptr].length + 1)
# Setting suffix edge for newly Created Node.
currNode = ptr
temp = tree[temp].suffixEdge
# Longest Palindromic suffix for a
# string of length 1 is a Null string.
if tree[currNode].length == 1:
tree[currNode].suffixEdge = 2
return
# Else
while True:
currLength = tree[temp].length
if (currIndex - currLength >= 1 and
s[currIndex] == s[currIndex -
currLength - 1]):
break
temp = tree[temp].suffixEdge
tree[currNode].suffixEdge = \
tree[temp].insertionEdge[ord(s[currIndex]) - ord('a')]
# Driver code
if __name__ == "__main__":
MAXN = 1000
# Imaginary root's suffix edge points to
# itself, since for an imaginary string
# of length = -1 has an imaginary suffix
# string. Imaginary root.
root1 = Node(-1, 1)
# NULL root's suffix edge points to
# Imaginary root, since for a string of
# length = 0 has an imaginary suffix string.
root2 = Node(0, 1)
# Stores Node information for
# constant time access
tree = [Node() for i in range(MAXN)]
# Keeps track the Current Node
# while insertion
currNode, ptr = 1, 2
tree[1] = root1
tree[2] = root2
s = "forgeeksskeegfor"
for i in range(0, len(s)):
insert(i)
# last will be the index of our
# last substring
last = ptr
for i in range(tree[last].start,
tree[last].end + 1):
print(s[i], end = "")
# This code is contributed by Rituraj JainC#
// C# code for longest Palindromic subString
// using Palindromic Tree data structure
using System;
class GFG
{
static readonly int MAXN = 1000;
class Node
{
// store start and end indexes of current
// Node inclusively
public int start, end;
// stores length of subString
public int Length;
// stores insertion Node for all characters a-z
public int[] insertionEdge = new int[26];
// stores the Maximum Palindromic Suffix Node for
// the current Node
public int suffixEdge;
};
// two special dummy Nodes as explained above
static Node root1, root2;
// stores Node information for constant time access
static Node[] tree = new Node[MAXN];
// Keeps track the current Node while insertion
static int currNode;
static char[] s;
static int ptr;
// Function to insert edge in tree
static void insert(int currIndex)
{
// Finding X, such that s[currIndex]
// + X + s[currIndex] is palindrome.
int temp = currNode;
while (true)
{
int currLength = tree[temp].Length;
if (currIndex - currLength >= 1 &&
(s[currIndex] == s[currIndex - currLength - 1]))
break;
temp = tree[temp].suffixEdge;
}
// Check if s[currIndex] + X +
// s[currIndex] is already Present in tree.
if (tree[temp].insertionEdge[s[currIndex] - 'a'] != 0)
{
currNode = tree[temp].insertionEdge[s[currIndex] - 'a'];
return;
}
// Else Create new node;
ptr++;
tree[temp].insertionEdge[s[currIndex] - 'a'] = ptr;
tree[ptr].end = currIndex;
tree[ptr].Length = tree[temp].Length + 2;
tree[ptr].start = tree[ptr].end - tree[ptr].Length + 1;
// Setting suffix edge for newly Created Node.
currNode = ptr;
temp = tree[temp].suffixEdge;
// longest Palindromic suffix for a
// String of length 1 is a Null String.
if (tree[currNode].Length == 1)
{
tree[currNode].suffixEdge = 2;
return;
}
// Else
while (true)
{
int currLength = tree[temp].Length;
if (currIndex - currLength >= 1 &&
(s[currIndex] == s[currIndex - currLength - 1]))
break;
temp = tree[temp].suffixEdge;
}
tree[currNode].suffixEdge =
tree[temp].insertionEdge[s[currIndex] - 'a'];
}
// Driver code
public static void Main(String[] args)
{
// Imaginary root's suffix edge points to
// itself, since for an imaginary String
// of length = -1 has an imaginary suffix
// String. Imaginary root.
root1 = new Node();
root1.Length = -1;
root1.suffixEdge = 1;
// null root's suffix edge points to
// Imaginary root, since for a String
// of length = 0 has an imaginary suffix String.
root2 = new Node();
root2.Length = 0;
root2.suffixEdge = 1;
for (int i = 0; i < MAXN; i++)
tree[i] = new Node();
tree[1] = root1;
tree[2] = root2;
ptr = 2;
currNode = 1;
s = "forgeeksskeegfor".ToCharArray();
for (int i = 0; i < s.Length; i++)
insert(i);
// last will be the index of our last subString
int last = ptr;
for (int i = tree[last].start; i <= tree[last].end; i++)
Console.Write(s[i]);
}
}
// This code is contributed by Rajput-Ji输出：
geeksskeeg