分段故障11链接列表

我的班级通过实现圆形单链接列表来对约瑟夫斯问题进行模拟。我的代码编译，但是当它运行时，我会得到一个分段错误：构建列表后的11。到目前为止，我的调试使我意识到，当程序进入主函数循环的最后一个循环时，发生错误。我认为这与我首先使用的方式有关，但我不确定。任何帮助都很棒，谢谢。我正在用C 编码，如果不是很明显。

#include <iostream>
#include <string>
using namespace std;

/*
 * Node structure used by the linked list.
 */
struct Node {
    // The name stored in this node.
    string name;
    // A pointer to the next node in the list.
    Node * next;
};
/* 
 * Build the linked list
 */
class LinkedList {
    private:
        //pointer to the head of the list
        Node *head;
    public:
        // constructor for LinkedList
        LinkedList(){
            head = NULL;
        }
        /*
        * Add the given value to the list, making it the head.
        */
        void insert(string name){

            // Remember where old head was
            Node *oldHead = head;
            // allocate a new node in memory
            head = new Node;
            // set new node's fields
            head->name = name;
            head->next = oldHead;
        }
        /*
        * Remove the item on the top of the stack, returning nothing.
        */
        void remove() {
            // Remember what the new head will be
            Node* newSecond = head->next->next;
            // Deallocate the head from memory
            delete head->next;
            // Set the head to the new head
            head->next = newSecond;
        }
        /*
         * Shifts the head forward one node.
         */
        void cycle(){
            head = head->next;
        }
        Node* getHead(){
            return head;
        }
         // This is the opposite of a constructor - a destructor! You (almost)
        // never need these in Java, but in C++ they are essential because 
        // you have to clean up memory by yourself. In particular, we need to
        // empty out the stack.
        ~LinkedList() {
            // While there's a head node still left, remove the head node.
            while (head != NULL) {
                remove();
            }
        }
};

int main(){
    //create the circular linked list
    LinkedList circle;
    int people;
    cin >> people;
    string soldier;
    Node* first = circle.getHead();
    //Insert all the names
    for(int i = 0; i < people; i++){
        cin >> soldier;
        circle.insert(soldier);
    }
    //begin the killing
    while(first->next != NULL){
        circle.cycle();
        cout << " killed " << endl;
        Node* temp = first->next->next;
        circle.remove();
        first->next = temp;
    }
}