library LinkedList {

    /*
    * @notice A permanent NULL node (0x0) in the circular double linked list.
    * NULL.next is the head, and NULL.previous is the tail.
    */
    address public constant NULL = address(0);

    /**
    * @notice A node points to the node before it, and the node after it. If
    * node.previous = NULL, then the node is the head of the list. If
    * node.next = NULL, then the node is the tail of the list.
    */
    struct Node {
        bool inList;
        address previous;
        address next;
    }

    /**
    * @notice LinkedList uses a mapping from address to nodes. Each address
    * uniquely identifies a node, and in this way they are used like pointers.
    */
    struct List {
        mapping (address => Node) list;
    }

    /**
    * @notice Insert a new node before an existing node.
    *
    * @param self The list being used.
    * @param target The existing node in the list.
    * @param newNode The next node to insert before the target.
    */
    function insertBefore(List storage self, address target, address newNode) internal {
        require(!isInList(self, newNode), "already in list");
        require(isInList(self, target) || target == NULL, "not in list");

        // It is expected that this value is sometimes NULL.
        address prev = self.list[target].previous;

        self.list[newNode].next = target;
        self.list[newNode].previous = prev;
        self.list[target].previous = newNode;
        self.list[prev].next = newNode;

        self.list[newNode].inList = true;
    }

    /**
    * @notice Insert a new node after an existing node.
    *
    * @param self The list being used.
    * @param target The existing node in the list.
    * @param newNode The next node to insert after the target.
    */
    function insertAfter(List storage self, address target, address newNode) internal {
        require(!isInList(self, newNode), "already in list");
        require(isInList(self, target) || target == NULL, "not in list");

        // It is expected that this value is sometimes NULL.
        address n = self.list[target].next;

        self.list[newNode].previous = target;
        self.list[newNode].next = n;
        self.list[target].next = newNode;
        self.list[n].previous = newNode;

        self.list[newNode].inList = true;
    }

    /**
    * @notice Remove a node from the list, and fix the previous and next
    * pointers that are pointing to the removed node. Removing anode that is not
    * in the list will do nothing.
    *
    * @param self The list being using.
    * @param node The node in the list to be removed.
    */
    function remove(List storage self, address node) internal {
        require(isInList(self, node), "not in list");
        if (node == NULL) {
            return;
        }
        address p = self.list[node].previous;
        address n = self.list[node].next;

        self.list[p].next = n;
        self.list[n].previous = p;

        // Deleting the node should set this value to false, but we set it here for
        // explicitness.
        self.list[node].inList = false;
        delete self.list[node];
    }

    /**
    * @notice Insert a node at the beginning of the list.
    *
    * @param self The list being used.
    * @param node The node to insert at the beginning of the list.
    */
    function prepend(List storage self, address node) internal {
        // isInList(node) is checked in insertBefore

        insertBefore(self, begin(self), node);
    }

    /**
    * @notice Insert a node at the end of the list.
    *
    * @param self The list being used.
    * @param node The node to insert at the end of the list.
    */
    function append(List storage self, address node) internal {
        // isInList(node) is checked in insertBefore

        insertAfter(self, end(self), node);
    }

    function swap(List storage self, address left, address right) internal {
        // isInList(left) and isInList(right) are checked in remove

        address previousRight = self.list[right].previous;
        remove(self, right);
        insertAfter(self, left, right);
        remove(self, left);
        insertAfter(self, previousRight, left);
    }

    function isInList(List storage self, address node) internal view returns (bool) {
        return self.list[node].inList;
    }

    /**
    * @notice Get the node at the beginning of a double linked list.
    *
    * @param self The list being used.
    *
    * @return A address identifying the node at the beginning of the double
    * linked list.
    */
    function begin(List storage self) internal view returns (address) {
        return self.list[NULL].next;
    }

    /**
    * @notice Get the node at the end of a double linked list.
    *
    * @param self The list being used.
    *
    * @return A address identifying the node at the end of the double linked
    * list.
    */
    function end(List storage self) internal view returns (address) {
        return self.list[NULL].previous;
    }

    function next(List storage self, address node) internal view returns (address) {
        require(isInList(self, node), "not in list");
        return self.list[node].next;
    }

    function previous(List storage self, address node) internal view returns (address) {
        require(isInList(self, node), "not in list");
        return self.list[node].previous;
    }

}
/**
 * @notice LinkedList is a library for a circular double linked list.
 */
function insertAfter(List storage self, address target, address newNode) internal {
        require(!isInList(self, newNode), "already in list");
        require(isInList(self, target) || target == NULL, "not in list");

        // It is expected that this value is sometimes NULL.
        address n = self.list[target].next;

        self.list[newNode].previous = target;
        self.list[newNode].next = n;
        self.list[target].next = newNode;
        self.list[n].previous = newNode;

        self.list[newNode].inList = true;
    }
/**
 * @notice Insert a new node after an existing node.
 *
 * @param self The list being used.
 * @param target The existing node in the list.
 * @param newNode The next node to insert after the target.
 */