给定一个整数 n 和一个位置数组 ‘position[]’ (1 <= position[i] <= 2n)，找出长度为 2n 的合适括号表达式的方式数，使得给定的位置有左括号.
例子 ：
Input : n = 3, position[] = [2}
Output : 3
Explanation :
The proper bracket sequences of length 6 and
opening bracket at position 2 are:
[ [ ] ] [ ]
[ [ [ ] ] ]
[ [ ] [ ] ]
Input : n = 2, position[] = {1, 3}
Output : 1
Explanation: The only possibility is:
[ ] [ ]方法：这个问题可以通过动态规划解决。 .
令DP i, j是填充前 i 个位置的有效方式的数量，这样“[”类型的括号比“]”类型的括号多 j。有效的方式意味着它是匹配括号表达式的前缀，并且强制执行“[”括号的位置已经满足。不难看出，DP 2N, 0是最终答案。
DP 的基本情况是， DP 0, 0 = 1。我们需要用'[‘括号填充第一个位置，这是唯一的方法。
如果该位置有一个可以用散列数组标记的左括号序列，则重复发生为：
if(j != 0) dpi, j = dpi-1, j-1
else dpi, j = 0; 如果该位置没有左括号序列，则重复发生如下：
if(j != 0) dpi, j = dpi - 1, j - 1 + dpi - 1, j + 1
else dpi, j = dpi - 1, j + 1答案将是DP 2n, 0
下面给出了上述方法的实现：
C++
// CPP code to find number of ways of
// arranging bracket with proper expressions
#include
using namespace std;
#define N 1000
// function to calculate the number
// of proper bracket sequence
long long arrangeBraces(int n, int pos[], int k)
{
// hash array to mark the
// positions of opening brackets
bool h[N];
// dp 2d array
int dp[N][N];
memset(h, 0, sizeof h);
memset(dp, 0, sizeof dp);
// mark positions in hash array
for (int i = 0; i < k; i++)
h[pos[i]] = 1;
// first position marked as 1
dp[0][0] = 1;
// iterate and formulate the recurrences
for (int i = 1; i <= 2 * n; i++) {
for (int j = 0; j <= 2 * n; j++) {
// if position has a opening bracket
if (h[i]) {
if (j != 0)
dp[i][j] = dp[i - 1][j - 1];
else
dp[i][j] = 0;
}
else {
if (j != 0)
dp[i][j] = dp[i - 1][j - 1] +
dp[i - 1][j + 1];
else
dp[i][j] = dp[i - 1][j + 1];
}
}
}
// return answer
return dp[2 * n][0];
}
// driver code
int main()
{
int n = 3;
// positions where opening braces
// will be placed
int pos[] = { 2 };
int k = sizeof(pos)/sizeof(pos[0]);
cout << arrangeBraces(n, pos, k);
return 0;
} Java
// Java code to find number of ways of
// arranging bracket with proper expressions
public class GFG {
static final int N = 1000;
// function to calculate the number
// of proper bracket sequence
static long arrangeBraces(int n, int pos[], int k) {
// hash array to mark the
// positions of opening brackets
boolean h[] = new boolean[N];
// dp 2d array
int dp[][] = new int[N][N];
// mark positions in hash array
for (int i = 0; i < k; i++) {
h[pos[i]] = true;
}
// first position marked as 1
dp[0][0] = 1;
// iterate and formulate the recurrences
for (int i = 1; i <= 2 * n; i++) {
for (int j = 0; j <= 2 * n; j++) {
// if position has a opening bracket
if (h[i]) {
if (j != 0) {
dp[i][j] = dp[i - 1][j - 1];
} else {
dp[i][j] = 0;
}
} else if (j != 0) {
dp[i][j] = dp[i - 1][j - 1]
+ dp[i - 1][j + 1];
} else {
dp[i][j] = dp[i - 1][j + 1];
}
}
}
// return answer
return dp[2 * n][0];
}
// Driver code
public static void main(String[] args) {
int n = 3;
// positions where opening braces
// will be placed
int pos[] = {2};
int k = pos.length;
System.out.print(arrangeBraces(n, pos, k));
}
}
// This code is contributed by 29AjayKumarPython3
# Python 3 code to find number of ways of
# arranging bracket with proper expressions
N = 1000
# function to calculate the number
# of proper bracket sequence
def arrangeBraces(n, pos, k):
# hash array to mark the
# positions of opening brackets
h = [False for i in range(N)]
# dp 2d array
dp = [[0 for i in range(N)]
for i in range(N)]
# mark positions in hash array
for i in range(k):
h[pos[i]] = 1
# first position marked as 1
dp[0][0] = 1
# iterate and formulate the recurrences
for i in range(1, 2 * n + 1):
for j in range(2 * n + 1):
# if position has a opening bracket
if (h[i]):
if (j != 0):
dp[i][j] = dp[i - 1][j - 1]
else:
dp[i][j] = 0
else:
if (j != 0):
dp[i][j] = (dp[i - 1][j - 1] +
dp[i - 1][j + 1])
else:
dp[i][j] = dp[i - 1][j + 1]
# return answer
return dp[2 * n][0]
# Driver Code
n = 3
# positions where opening braces
# will be placed
pos = [ 2 ,];
k = len(pos)
print(arrangeBraces(n, pos, k))
# This code is contributed
# by sahishelangiaC#
// C# code to find number of ways of
// arranging bracket with proper expressions
using System;
class GFG
{
static int N = 1000;
// function to calculate the number
// of proper bracket sequence
public static long arrangeBraces(int n, int[] pos, int k)
{
// hash array to mark the
// positions of opening brackets
bool[] h = new bool[N];
// dp 2d array
int[,] dp = new int[N,N];
for(int i = 0; i < N; i++)
h[i] = false;
for(int i = 0; i < N; i++)
for(int j = 0; j < N; j++)
dp[i,j] = 0;
// mark positions in hash array
for (int i = 0; i < k; i++)
h[pos[i]] = true;
// first position marked as 1
dp[0,0] = 1;
// iterate and formulate the recurrences
for (int i = 1; i <= 2 * n; i++) {
for (int j = 0; j <= 2 * n; j++) {
// if position has a opening bracket
if (h[i]) {
if (j != 0)
dp[i,j] = dp[i - 1,j - 1];
else
dp[i,j] = 0;
}
else {
if (j != 0)
dp[i,j] = dp[i - 1,j - 1] +
dp[i - 1,j + 1];
else
dp[i,j] = dp[i - 1,j + 1];
}
}
}
// return answer
return dp[2 * n,0];
}
// driver code
static void Main()
{
int n = 3;
// positions where opening braces
// will be placed
int[] pos = new int[]{ 2 };
int k = pos.Length;
Console.Write(arrangeBraces(n, pos, k));
}
//This code is contributed by DrRoot_
}Javascript
输出 ：
3时间复杂度： O(n^2)
辅助空间： O(n^2)
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