analyze time complex and space in depth as if in a tech interview given ps and code 
"""
# Question Review

>The problem “Locking the Tree of Space” involves designing and implementing three operations on an M-Ary tree:

- **`lock(x, uid)`:** Grants exclusive access to the subtree rooted at `x` for user `uid`. Succeeds only if no ancestors or descendants of `x` are locked by any user.
- **`unlock(x, uid)`:** Reverts the lock on `x` by user `uid`. Succeeds only if `x` was previously locked by `uid`.
- **`upgradeLock(x, uid)`:** Upgrades a lock from a descendant of `x` to `x` itself for user `uid`. Succeeds only if:
    - `x` is not currently locked.
    - All locked descendants of `x` are locked by `uid`.
    - `x` has at least one locked descendant (if none, return `false`).

>**Assumptions:**

1. **Unique User IDs:** Each user has a unique identifier (`uid`) for distinction. And all user IDs in queries are valid.
2. **Valid Node Names:** All node names in queries are valid and exist in the tree.
3. **Synchronization:** The system handles concurrent operations to prevent race conditions.
4. **Error Handling:** The system handles operation failures; the system does not crash and instead returns a meaningful error message.
5. **Memory:** No memory limitations exist for storing the tree.
6. **Upgrade Restrictions:** Descendants are unlocked when their ancestor is upgraded. Upgrade fails if `x` has any locked ancestors.

>**Points to Remember:**

- Tree is fully balanced (each node has 0 or `m` children).
- Queries are formatted as `Operation_Type NodeName Userld` 
(1 for lock, 2 for unlock, 3 for upgrade).
- Expected time complexities:
    - lock/unlock: `O(logm(N))`
    - upgradeLock: `O(numberOfLockedNodes * logm(N))`
- Output: `true` for successful operations, `false` otherwise.

>**Use Cases:**

The problem “Locking The Tree of Space” can be applied in several real-world scenarios for JusPay, especially considering its role as a fintech company that deals with online payments. Here are some potential applications:

1. **Transaction Processing:** Ensure smooth concurrent processing of multiple transactions by locking relevant nodes (representing transaction stages) in the tree. This prevents data inconsistencies and accidental modifications.
2. **User Session Management:** Manage active user sessions and prevent resource conflicts by locking corresponding nodes in the tree. This ensures exclusive access to services and resources until the session ends.
3. **Resource Allocation:** Optimize resource allocation in cloud environments by locking nodes representing resources (CPU, memory, storage) when assigned to tasks. This prevents overbooking and ensures task completion without resource conflicts.
4. **Database Operations:** Maintain data integrity during concurrent database operations (updates, deletes) by locking relevant table nodes (Users, Orders, Items) in the tree. This serializes operations and prevents data corruption.
5. **Payment Gateway Integration:** Facilitate smooth payment processing through JusPay's integrated gateways by locking the chosen gateway node in the tree. This prevents concurrent access and ensures successful transaction completion on the chosen platform.
    
    >**a. Selection and Locking:**    
    >- When Customer A reaches the payment gateway selection step, the corresponding node in the tree (representing all available gateways) is locked.
    >- This prevents other customers from choosing any gateway until A makes their selection.
    >- Once A selects Visa, the specific Visa node within the tree is locked, granting A exclusive access to its processing resources.
    
    >**b. Concurrent Access Prevention:**    
    >- While A's transaction progresses through Visa's processing steps (e.g., authorization, settlement), the Visa node remains locked.
    >- This prevents other customers from choosing Visa during this period, ensuring A's transaction receives undivided attention from the gateway.
    
    >**c. Controlled Resource Allocation:**  
    >- Locking the Visa node also allocates necessary resources like processing power and network bandwidth specifically for A's transaction.
    >- This prevents resource overload from concurrent transactions on the same gateway, ensuring smooth and efficient processing for A.
    
    >**d. Unlock and Release:**
    >- Upon successful completion of A's transaction, both the general gateway node and the Visa node are unlocked.
    >- This releases the allocated resources and makes them available for other customers seeking to use Visa or any other gateway.
    
    >**Additional Benefits:**
    >- This locking mechanism minimizes the risk of conflicts or errors that might arise from concurrent transactions on the same gateway.
    >- It also improves overall system performance by preventing resource overload and ensuring efficient utilization of gateway resources.
"""
"""
from collections import defaultdict
import sys

class Node:
    def __init__(self, name):
        self.name = name
        self.user_id = None                 # ID of the user who locked the node
        self.parent = None                  # Parent node
        self.children = []                  # List of child nodes
        self.is_locked = False              # Lock status of the node
        self.locked_descendant_count = 0    # Number of locked descendants
        self.in_time = 0                    # Entry time during DFS traversal
        self.out_time = 0                   # Exit time during DFS traversal

class Tree:
    def __init__(self, n, m, name_list, queries):
        self.node_list = []
        self.node_map = {}  # Map from node name to Node object

        # Create nodes and map their names
        for name in name_list:
            node = Node(name)
            self.node_list.append(node)
            self.node_map[name] = node

        # Build the tree
        for i in range(n):
            parent = self.node_list[i]
            first_child_index = m * i + 1
            for j in range(m):
                child_index = first_child_index + j
                if child_index < n:
                    child = self.node_list[child_index]
                    parent.children.append(child)
                    child.parent = parent
                else:
                    break

        self.root = self.node_list[0]
        self.user_locked_nodes = defaultdict(set)
        self.assign_time(self.root, 0)  # Preprocess for ancestry checks

        # Process queries
        self.processed_queries = []
        for query in queries:
            self.processed_queries.append(query.strip().split())

    def assign_time(self, node, time):
        # Assign entry time (in_time)
        node.in_time = time
        time += 1

        for child in node.children:
            self.assign_time(child, time)

        # Assign exit time (out_time)
        node.out_time = time
        time += 1

    def lock(self, node, user_id):
        # If the node is already locked, return False
        if node.is_locked:
            return False

        # If any ancestors are locked, cannot lock this node
        if self.is_ancestor_locked(node):
            return False

        # If any descendants are locked, cannot lock this node
        if node.locked_descendant_count > 0:
            return False

        # Lock the node
        node.is_locked = True
        node.user_id = user_id
        self.user_locked_nodes[user_id].add(node)

        # Update the locked_descendant_count for ancestors
        parent = node.parent
        while parent:
            parent.locked_descendant_count += 1
            parent = parent.parent

        return True

    def unlock(self, node, user_id):
        # The node must be locked and the user_id must match
        if not node.is_locked or node.user_id != user_id:
            return False

        # Unlock the node
        node.is_locked = False
        node.user_id = None
        self.user_locked_nodes[user_id].remove(node)

        # Update the locked_descendant_count for ancestors
        parent = node.parent
        while parent:
            parent.locked_descendant_count -= 1
            parent = parent.parent

        return True

    def upgrade(self, node, user_id):
        # The node must not be locked
        if node.is_locked:
            return False

        # If any ancestors are locked, cannot upgrade this node
        if self.is_ancestor_locked(node):
            return False

        # The node must have at least one locked descendant
        if node.locked_descendant_count == 0:
            return False

        # Collect locked descendants locked by user_id
        locked_descendants = []
        success = self.collect_locked_descendants(node, user_id, locked_descendants)
        if not success:
            return False

        # Unlock all locked descendants
        for descendant in locked_descendants:
            descendant.is_locked = False
            descendant.user_id = None
            self.user_locked_nodes[user_id].remove(descendant)

            # Update the locked_descendant_count for ancestors
            parent = descendant.parent
            while parent:
                parent.locked_descendant_count -= 1
                parent = parent.parent

        # Lock the current node
        node.is_locked = True
        node.user_id = user_id
        self.user_locked_nodes[user_id].add(node)

        # Update the locked_descendant_count for ancestors
        parent = node.parent
        while parent:
            parent.locked_descendant_count += 1
            parent = parent.parent

        return True

    def collect_locked_descendants(self, node, user_id, locked_descendants):
        for child in node.children:
            if child.is_locked:
                if child.user_id == user_id:
                    locked_descendants.append(child)
                else:
                    return False  # Found a locked node by another user
            elif child.locked_descendant_count > 0:
                if not self.collect_locked_descendants(child, user_id, locked_descendants):
                    return False
        return True

    def is_ancestor_locked(self, node):
        current = node.parent
        while current:
            if current.is_locked:
                return True
            current = current.parent
        return False

    def process_queries(self):
        results = []
        for query in self.processed_queries:
            operation_type, node_name, user_id = query
            node = self.node_map[node_name]
            user_id = int(user_id)
            if operation_type == "1":
                result = self.lock(node, user_id)
            elif operation_type == "2":
                result = self.unlock(node, user_id)
            elif operation_type == "3":
                result = self.upgrade(node, user_id)
            else:
                result = False  # Invalid operation
            results.append('true' if result else 'false')
        return results

def main():
    sys.setrecursionlimit(1 << 25)  # Increase recursion limit if needed
    n = int(sys.stdin.readline())
    m = int(sys.stdin.readline())
    q = int(sys.stdin.readline())

    node_list = []
    for _ in range(n):
        node_list.append(sys.stdin.readline().strip())

    queries = []
    for _ in range(q):
        queries.append(sys.stdin.readline().strip())

    tree = Tree(n, m, node_list, queries)
    results = tree.process_queries()
    for res in results:
        print(res)
main()