Are you prepared for questions like 'Can you describe the merge sort algorithm? What is its time complexity?' and similar? We've collected 40 interview questions for you to prepare for your next Algorithms interview.
Merge sort is a divide and conquer algorithm that splits the array into halves until each piece is just one element. Then, it merges those pieces back together in a sorted order. First, the array is divided recursively into two halves until the base case of a single element is reached. Then, these halves are merged together using a 'merge' function that combines two sorted arrays into one larger sorted array.
Its consistency in breaking down the array and merging them back in a sorted order renders it highly efficient in most cases. The time complexity of merge sort in both the average and worst-case scenarios is O(n log n), where 'n' is the number of elements in the array. This efficiency comes from the logarithmic number of levels needed to break down the array and the linear time required to merge each level.
A stack is a linear data structure that follows the Last In, First Out (LIFO) principle, meaning the last element added to the stack will be the first one to be removed. You can think of it like a stack of plates: you add to the top and also remove from the top.
In contrast, a queue follows the First In, First Out (FIFO) principle. The first element added to the queue will be the first one to be removed, similar to a line of people waiting to buy tickets. You add elements at the back and remove them from the front.
To detect a cycle in a linked list, one common approach is to use Floyd’s Cycle-Finding Algorithm, also known as the Tortoise and Hare algorithm. You use two pointers: a slow pointer that moves one step at a time and a fast pointer that moves two steps at a time. If there's no cycle, the fast pointer will eventually reach the end of the list. However, if there is a cycle, the fast pointer will eventually meet the slow pointer within the cycle, which confirms the presence of a cycle. This method is efficient with a time complexity of O(n) and doesn't require extra space, making it a great solution for this problem.
The Travelling Salesman Problem (TSP) involves finding the shortest possible route that allows a salesman to visit each city once and return to the original city. It's a classic optimization problem in algorithm theory. The difficulty stems from the fact that it’s an NP-hard problem, meaning that as the number of cities increases, the number of possible routes grows factorially, making it computationally infeasible to solve using brute-force for large datasets.
TSP's significance lies not just in theoretical aspects, but also in practical applications like route planning, manufacturing, and DNA sequencing. Solving or approximating TSP efficiently can lead to significant cost and time savings in various industries. Additionally, studying TSP has led to the development of various heuristic and approximation algorithms, which are useful for tackling other complex optimization problems.
A graph is a data structure consisting of a set of nodes (or vertices) and a set of edges that connect pairs of nodes. Graphs can be either directed, where edges have a direction, or undirected, where edges don't have a direction. Graphs can represent a wide range of real-world systems like social networks, web structures, and transport systems.
A tree is actually a specific type of graph—it's an acyclic, connected graph. This means a tree has no cycles (you can't start at a node, traverse some edges, and come back to the same node without retracing steps) and all its nodes are connected. Additionally, trees have a hierarchical structure with a single root node and any two nodes are connected by exactly one path. This makes trees a bit more restrictive and structured compared to general graphs.
Kruskal's algorithm is used to find the minimum spanning tree (MST) in a graph. It works by sorting all the edges of the graph in ascending order based on their weights. You start by creating a forest, which is a set of trees where each vertex in the graph is a separate tree. Then, you iterate over the sorted edges and add each edge to the forest, but only if it doesn't form a cycle with the existing edges in the forest. You continue this process until there are exactly (V-1) edges in the forest, where (V) is the number of vertices in the graph. This gives you the MST.
To efficiently check for cycles when adding edges, Kruskal's algorithm often uses a Union-Find data structure, which supports the quick union and find operations necessary for keeping track of which vertices belong to which components (or trees). This allows the algorithm to run in (O(E \log E)) time, where (E) is the number of edges in the graph, primarily due to the time complexity of sorting the edges.
A classic example of using a greedy algorithm is the problem of finding the minimum number of coins needed to make a certain amount of change. Imagine you have coins of different denominations, and you want to make change for a specific amount of money using the fewest coins possible. In this scenario, a greedy algorithm works by always picking the largest denomination coin that is less than or equal to the remaining amount to be paid. You keep subtracting the coin's value from the remaining amount until you reach zero.
This approach is efficient for this problem because it allows you to quickly zero in on the solution by making a series of optimal local choices. It's important to note, however, that a greedy algorithm guarantees an optimal solution for this specific problem only if the coin denominations are standard, like those used in many countries' currencies.
Prim's algorithm and Kruskal's algorithm are both used to find the Minimum Spanning Tree (MST) in a connected, weighted graph, but they approach the problem differently. Prim's algorithm grows the MST from a starting vertex by adding the lowest-weight edge that connects a vertex in the MST to a vertex outside of it. It works well with dense graphs because it can take advantage of a priority queue to efficiently select the minimum edge.
In contrast, Kruskal's algorithm builds the MST by sorting all the edges in the graph by weight and then adding them one by one to the MST, as long as they don't form a cycle. It uses a union-find (disjoint-set) data structure to efficiently keep track of which vertices are in the same connected component. Kruskal's is particularly effective for sparse graphs since it focuses on edges rather than vertices.
To sum up, Prim’s algorithm is vertex-oriented and performs better in dense graphs with many edges, while Kruskal's algorithm is edge-oriented and excels in sparse graphs with fewer edges.
AVL trees are a type of self-balancing binary search tree. Named after their inventors Adelson-Velsky and Landis, they maintain a balance factor for each node, which is the difference between the heights of the left and right subtrees. The tree ensures that the balance factor is always -1, 0, or 1, which keeps the tree height logarithmic relative to the number of nodes.
They are used to ensure that operations like insertion, deletion, and lookup remain efficient, typically O(log n) in time complexity. This is particularly useful in applications where the input size can grow large, and maintaining a balanced tree is crucial for keeping performance optimal.
Binary search is an efficient algorithm for finding an item from a sorted list of items. It works by repeatedly dividing in half the portion of the list that could contain the item, until you've narrowed down the possible locations to just one. You start by comparing the target value to the middle element of the list. If the target value is equal to the middle element, you've found the item. If the target value is less than the middle element, you repeat the search in the lower half, and if it's greater, you repeat the search in the upper half. This process continues until the target value is found or the search space is reduced to zero.
Arrays and linked lists are both data structures used to store collections of elements, but they differ in their structure and usage.
Arrays have a fixed size and elements are stored in contiguous memory locations, making access to elements via an index very fast. However, adding or removing elements can be costly since it requires shifting the subsequent elements.
Linked lists, on the other hand, consist of nodes where each node contains an element and a reference to the next node. This allows for efficient insertions and deletions because you only need to change the references, but accessing elements requires traversing the list from the beginning, which can be slower.
The average-case time complexity of quicksort is O(n log n). This occurs because the algorithm typically partitions the array in a balanced way, effectively halving the problem size with each recursive call. In the best case, you consistently divide the array into two nearly equal parts, which allows the divide-and-conquer strategy to work efficiently.
However, in the worst case, quicksort's time complexity is O(n^2). This happens when the pivot always ends up being the smallest or largest element in the array, leading to unbalanced partitions and thus, a deep recursion stack. This unbalanced partitioning occurs, for example, when the input array is already sorted in ascending or descending order and the pivot selection doesn't help in balancing. Nevertheless, with good pivot strategies, like choosing a random pivot or using the median-of-three rule, you can generally avoid the worst-case scenario.
Dynamic programming is essentially an optimization technique used to solve complex problems by breaking them down into simpler subproblems. It involves storing the results of subproblems to avoid redundant computations, often using a table to save these results and referring back to them when needed.
Consider the classic example of the Fibonacci series, where each number is the sum of the two preceding ones. Instead of recalculating Fibonacci values repeatedly, dynamic programming allows us to store the results of each Fibonacci calculation in a table. So, if you're calculating Fib(5)
, you first compute Fib(0)
to Fib(4)
and store these values. When you need Fib(5)
, you simply add the previously stored Fib(3)
and Fib(4)
. This turns an exponential time complexity problem into a linear one, drastically improving performance.
The Floyd-Warshall algorithm is an all-pairs shortest path algorithm that works for both directed and undirected weighted graphs. It systematically examines all paths between all pairs of vertices and updates the shortest known path. The algorithm uses a dynamic programming approach, with a matrix to store the shortest path distances between each pair of nodes.
It initializes the distances based on the direct edges between nodes and iteratively updates the matrix by considering each node as an intermediate point. Specifically, if the distance from node A to node B through an intermediate node C is shorter than the previously known distance from A to B, the algorithm updates the distance. This process repeats for all nodes as intermediates, ultimately providing the shortest paths between every pair of nodes.
Balancing a binary search tree often involves techniques to either maintain a balanced tree during insertion and deletion or to rebalance an unbalanced tree. The most common approach is using self-balancing binary search trees like AVL trees or Red-Black trees. AVL trees maintain balance by ensuring that the heights of the two child subtrees of any node differ by at most one. Red-Black trees enforce more relaxed balancing by coloring nodes and ensuring certain properties during insertions and deletions that keep the tree approximately balanced, providing O(log n) time complexity for operations. If you have an unbalanced tree already, you can convert it to a balanced one by performing a tree-to-doubly-linked list conversion followed by a balanced tree building from the sorted list.
A trie is a tree-like data structure that stores a dynamic set of strings. Essentially, each node of the trie represents a single character of the string. Strings are inserted into the trie character by character, with each character serving as a path from one node to the next. Starting from the root, each edge represents a character in the string, and traversing down from the root to a leaf node (or to a node that marks the end of a string) spells out the stored string.
To check if a string is in the trie, you'd start at the root and traverse the nodes following the characters of the string. If you can follow the characters all the way to a node that marks the end of a string, the string exists in the trie. This method makes searching for strings very efficient, O(l) where l is the length of the string. They are particularly useful for prefix-based searches, as you can retrieve all words with a given prefix by reaching the last node of the prefix and exploring all its descendant nodes.
The Bellman-Ford algorithm is used to find the shortest paths from a single source vertex to all other vertices in a weighted graph. It's particularly useful for graphs with negative weight edges because it can detect negative weight cycles, something Dijkstra's algorithm can't do. Although it's less efficient than Dijkstra’s algorithm in graphs with only positive weights, its ability to handle negative weights makes it a valuable tool in certain scenarios.
Backtracking is a method for solving problems incrementally, trying out possible solutions and abandoning them if they don't work. Essentially, you build a solution piece by piece and revert or "backtrack" when you find that the current piece doesn't lead to a valid solution.
Take the classic example of solving a maze. You start at the entrance and try to reach the exit by following a path. You move forward step-by-step, and if you hit a dead-end, you backtrack to the last decision point and try a different path. This process continues until you either find the exit or exhaust all possible paths. Another typical example is the N-Queens problem, where you place N queens on an NxN chessboard such that no two queens threaten each other. You place a queen on a row and proceed to position the next queen in subsequent rows, backtracking whenever a conflict is detected until a valid arrangement is found.
To implement a minimum heap, you can use an array or a dynamic array data structure since it provides efficient access to the elements and helps maintain the heap properties. Each element of the array represents a tree node, and for any element at index i
, its parent is at (i - 1) / 2
, its left child is at 2 * i + 1
, and its right child is at 2 * i + 2
.
Here’s a basic approach:
Insertion: When adding an element, place it at the end of the array, then "heapify up" by comparing it to its parent and swapping if necessary. Continue this until the heap property is restored.
Deletion (extracting the minimum): Replace the root (the minimum element) with the last element in the array, then remove the last element. "Heapify down" by comparing the new root to its children and swapping it with the smaller child if necessary. Continue until the heap property is restored.
Here's a simple code snippet in Python:
```python class MinHeap: def init(self): self.heap = []
def insert(self, val):
self.heap.append(val)
self._heapify_up(len(self.heap) - 1)
def extract_min(self):
if len(self.heap) == 1:
return self.heap.pop()
root = self.heap[0]
self.heap[0] = self.heap.pop()
self._heapify_down(0)
return root
def _heapify_up(self, index):
parent = (index - 1) // 2
if index > 0 and self.heap[index] < self.heap[parent]:
self.heap[index], self.heap[parent] = self.heap[parent], self.heap[index]
self._heapify_up(parent)
def _heapify_down(self, index):
smallest = index
left_child = 2 * index + 1
right_child = 2 * index + 2
if left_child < len(self.heap) and self.heap[left_child] < self.heap[smallest]:
smallest = left_child
if right_child < len(self.heap) and self.heap[right_child] < self.heap[smallest]:
smallest = right_child
if smallest != index:
self.heap[index], self.heap[smallest] = self.heap[smallest], self.heap[index]
self._heapify_down(smallest)
```
This gives you a basic functioning min-heap with insert and extract-min operations maintaining O(log n) complexity.
Depth-first search (DFS) and breadth-first search (BFS) are two fundamental algorithms for traversing or searching tree or graph data structures. The main difference between them lies in their approach to exploring nodes.
DFS dives as deep as possible along each branch before backtracking. It uses a stack, either explicitly with a data structure or implicitly via recursion, to keep track of the next node to visit. This approach is useful for scenarios where you need to explore all possible paths, like puzzle solving or finding connected components in a graph.
BFS, on the other hand, explores all neighbors of a node before moving on to the neighbors' neighbors. It uses a queue to keep track of nodes to explore. This method is ideal for finding the shortest path in unweighted graphs, like in shortest route problems or web crawlers.
Hash tables are data structures that allow for fast data retrieval. They use a key-value pair mechanism where each key is hashed to produce an index in an array, which is where the corresponding value is stored. The efficiency comes from the fact that searching, inserting, and deleting operations can average O(1) time complexity under good conditions.
To implement a hash table, you first need a hash function that will convert your keys into an integer index. This function should ideally distribute keys uniformly across the array to minimize collisions. Then, you need to handle collisions using techniques like chaining, where each array cell points to a linked list of entries, or open addressing, where you probe for the next free slot in a linear or quadratic manner.
Maintaining a good load factor, typically below 0.7, is crucial to ensure efficiency. This might require resizing the hash table, which involves creating a new larger array and rehashing all existing keys into this new array.
A heap is a specialized tree-based data structure that satisfies the heap property. In a max-heap, for example, every parent node has a value greater than or equal to the values of its children, making the largest value always at the root. Conversely, a min-heap has every parent node with a value less than or equal to its children, with the smallest value at the root. Heaps are typically implemented with arrays and are useful for algorithms like heapsort and for implementing priority queues.
One efficient way to find the kth largest element in an unsorted array is by using the Quickselect algorithm. Quickselect is a selection algorithm partially based on the QuickSort sorting algorithm. It works by first choosing a pivot element and partitioning the array into two sub-arrays: elements greater than the pivot and elements less than the pivot. Depending on the size of the sub-arrays relative to k, you can determine which sub-array contains the kth largest element and recursively apply Quickselect to that sub-array.
Alternatively, for a simpler approach especially when k is small and performance isn't a critical concern, you could use a min-heap. Insert the first k elements of the array into the heap. Then, for each remaining element, check if it's larger than the smallest element in the heap. If it is, remove the smallest element and insert the new element. By the end of the iteration, the root of the heap will be the kth largest element.
Dijkstra's algorithm is used to find the shortest path from a starting node to all other nodes in a weighted graph. The key principle behind it is the greedy approach, where it incrementally builds the shortest path by selecting the edge with the smallest weight that leads to an unvisited node. It starts at the source node, and in each step, it marks the nearest unvisited node as visited and updates the shortest paths to its neighboring nodes. This process continues until all nodes have been visited or the shortest paths to all nodes have been determined.
Memoization is a technique used to optimize programs by storing the results of expensive function calls and reusing them when the same inputs occur again. The main idea is to save time by avoiding the repeated computation of the same result. It's particularly useful in recursive algorithms where the same subproblems are solved multiple times. By caching previous results, you reduce the time complexity, often turning an exponential task into a polynomial one.
Reversing a linked list involves changing the direction of the pointers so that each node points to the previous node instead of the next one. You can achieve this iteratively or recursively. Let’s go through the iterative approach as it’s more intuitive and usually more efficient with respect to space complexity.
You keep three pointers: prev
, current
, and next
. Initially, set prev
to null
and current
to the head of the list. Then, iterate through the list, doing the following at each step: set next
to current.next
, reverse the current.next
to point to prev
, move prev
to current
, and current
to next
. Repeat this process until you’ve traversed the entire list. At the end, prev
will be the new head of the reversed list.
Here's a quick pseudocode to help visualize it:
prev = null
current = head
while current is not null:
next = current.next
current.next = prev
prev = current
current = next
head = prev
A binary search tree (BST) is a type of binary tree where each node has a value, and that value dictates the tree's structure. Specifically, for any given node, all values in the left subtree are less than the node's value, and all values in the right subtree are greater than the node's value. This property makes BSTs useful for operations like search, insert, and delete, which can be performed efficiently.
A BST allows these operations to be executed in O(log n) time on average, assuming the tree is balanced. This efficiency comes from the way it halves the search space at each step. However, if the tree becomes unbalanced and skews into a line (like a linked list), the time complexity degrades to O(n). Maintaining a balanced tree, with self-balancing variants like AVL trees or Red-Black trees, ensures optimal performance.
Recursive algorithms can make code simpler and more readable, especially for problems that have a natural recursive structure like tree traversal or the Fibonacci sequence. They break down complex problems into smaller, more manageable ones and can be easier to implement for certain tasks.
However, recursion can be less efficient due to the overhead of multiple function calls and the risk of stack overflow for deep recursions. Memory usage can also be higher compared to iterative solutions. Tail recursion can help mitigate some issues, but not all languages optimize for it.
You can swap two variables in-place using arithmetic operations or bitwise XOR.
Using arithmetic:
python
a = a + b
b = a - b
a = a - b
Using XOR:
python
a = a ^ b
b = a ^ b
a = a ^ b
The XOR method is generally preferred because it avoids potential issues with arithmetic overflow. Both methods will effectively swap the values of a
and b
without needing extra space for a temporary variable.
The Rabin-Karp string search algorithm uses hashing to find a pattern within a text efficiently. It works by first calculating a hash value for the pattern and then for each substring of the text with the same length as the pattern. Instead of checking the entire substring against the pattern each time, it first compares the hash values. If the hash values match, then it performs a direct comparison to account for any potential hash collisions. This can significantly speed up the search process, especially when multiple patterns need to be searched in a single text. The rolling hash technique is typically used to efficiently compute hash values of substrings.
The Boyer-Moore string search algorithm is efficient because it skips sections of the text to be searched rather than checking each character in sequence. It uses two main heuristics to accomplish this: the bad character rule and the good suffix rule.
The bad character rule helps decide how far to move the pattern if a mismatch occurs. If the mismatched character is in the pattern, the pattern is shifted so that this character in the text aligns with its last occurrence in the pattern. The good suffix rule deals with suffixes of the pattern that have matched properly; it shifts the pattern to align its next possible occurrence.
By combining these heuristics, Boyer-Moore often skips large portions of text, making it particularly efficient for long patterns within extensive texts. This makes it popular for use in various search applications, especially where performance is critical.
To find the shortest path in a weighted graph, Dijkstra's algorithm is frequently used. It works well for graphs with non-negative weights. You start from the source node, set the initial distance to zero, and all other distances to infinity. The algorithm repeatedly selects the node with the smallest known distance, examines its neighbors, and updates their distances based on the edge weights if a shorter path is found.
If the graph has negative weights, you might want to use the Bellman-Ford algorithm instead. It can handle graphs with negative weights and is designed to detect negative cycles. This algorithm repeatedly relaxes all edges, updating the distance to each node with the minimum distance found so far, for a number of iterations equal to the number of vertices minus one.
For graphs where all edges have the same weight, Breadth-First Search (BFS) can be used, treating the problem like one in an unweighted graph. If the graph is very large or has special structures, more advanced algorithms like A* might be more efficient.
Big O, Big Theta, and Big Omega are notations used to describe the time complexity of an algorithm, helping to understand its efficiency in terms of input size.
Big O, O(n), describes the upper bound of an algorithm's runtime. It tells us the worst-case scenario: how the runtime grows relative to the input size. For example, if an algorithm takes at most 3n^2 + 2n + 1 operations, we say its complexity is O(n^2) because the n^2 term grows the fastest and dominates for large inputs.
Big Theta, Θ(n), provides a tight bound on the runtime. It gives a more precise description by accounting for both the upper and lower bounds of the runtime. For instance, an algorithm with runtime 2n^2 + 3n + 4 is Θ(n^2), since n^2 is the dominant term, and the growth rate can be sandwiched between c1n^2 and c2n^2 for some constants c1 and c2.
Big Omega, Ω(n), represents the lower bound of an algorithm's runtime. This is best-case complexity. If an algorithm's runtime is at least n log n for an input size n, then we say it is Ω(n log n), indicating that, no matter what, the algorithm cannot perform better than n log n operations.
A* search algorithm is a popular and efficient search algorithm used in pathfinding and graph traversal. It is used to find the shortest path between a start node and a goal node. The algorithm uses a combination of a cost function (often denoted as g(n), which is the path cost from the start node to node n) and a heuristic function (denoted as h(n), which estimates the cost from node n to the goal node). The total cost function, f(n) = g(n) + h(n), guides the search process by balancing actual path cost and estimated future cost, leading to optimal and efficient pathfinding.
A* is widely used in various applications, such as in video games for character movement, in robotics for navigation and obstacle avoidance, and in geographic information systems for routing and map analysis. Its versatility and effectiveness stem from its ability to incorporate heuristics, allowing it to be tailored to specific types of problems and environments.
To find the longest common subsequence (LCS) between two sequences, you usually use dynamic programming. Basically, you create a 2D table, where the dimensions are the lengths of the two sequences you're comparing. You then iteratively fill this table based on whether the current characters of the sequences match or not.
If the characters match, you take the value from the diagonally previous cell and add one. If they don't match, you take the maximum value from either the cell directly above or to the left. After filling up the entire table, the value in the bottom-right cell will give you the length of the LCS, and tracing back from this cell will give you the actual subsequence.
This approach runs in O(mn) time complexity and O(mn) space complexity, where m and n are the lengths of the two sequences. If space is a concern, you can optimize the space complexity to O(min(m, n)) since you really only need the current and previous rows or columns to compute the LCS.
Heap sort involves two main steps: building a max heap and then repeatedly extracting the maximum element from the heap.
First, you build a max heap from the input array. This is done by rearranging the array elements so that they satisfy the heap property, where each parent node is greater than or equal to its children. You typically start from the last non-leaf node and move upwards, ensuring each subtree is a valid heap.
Once the max heap is built, the sorting process begins. You repeatedly swap the first element (the maximum) with the last element of the unsorted section of the array. After each swap, you reduce the heap size by one and heapify the root element to ensure the remaining elements still form a max heap. This process is repeated until the heap size is reduced to one. The array is now sorted.
To determine if a binary tree is a valid binary search tree (BST), you can perform an in-order traversal of the tree and check if the values are in ascending order. An in-order traversal visits the nodes in the left subtree, then the root node, and then the nodes in the right subtree. If the sequence of visited nodes forms a strictly increasing sequence, it's a valid BST.
Alternatively, you can use a recursive approach where you pass down the range of valid values for each node. For the root node, the range is (-∞, +∞). For each node, you check if it falls within the allowed range, then recursively check its left subtree with the updated range (-∞, node's value) and the right subtree with (node's value, +∞). If all nodes satisfy this property, the tree is a valid BST.
There are three main ways to traverse a binary tree: in-order, pre-order, and post-order.
In-order traversal processes the left subtree first, the node itself, and then the right subtree. It's particularly useful for binary search trees because it visits nodes in ascending order, making it great for tasks like printing values in sorted order.
Pre-order traversal processes the current node first, then the left subtree, and finally the right subtree. This method is useful for tasks like copying a tree or expressing the structure of the tree because it captures the hierarchy before diving into subtrees.
Post-order traversal processes the left subtree first, the right subtree second, and the node itself last. It's useful for tasks that require bottom-up processing, such as deleting nodes or evaluating expressions in expression trees.
The KMP algorithm is designed to efficiently search for a substring within a main string by preprocessing the pattern to avoid unnecessary comparisons. First, it builds a partial match table (also known as the "prefix" table) that keeps track of the longest prefix that is also a suffix for each sub-pattern within the pattern. This allows the algorithm to skip over already matched characters.
When performing the search, the KMP algorithm uses the partial match table to determine where to resume the search upon encountering a mismatch. Instead of restarting the search from the beginning of the pattern, the table tells you the next positions to continue, effectively reducing the time complexity to O(n + m), where n is the length of the text and m is the length of the pattern.
A balanced tree is a type of binary tree where the height difference between the left and right subtrees of any node is no more than one. This means the tree is structured in a way that both sides grow at roughly the same rate, ensuring that operations like insertion, deletion, and lookup can be performed efficiently.
Balanced trees are crucial because they maintain logarithmic time complexity for these operations, which is a significant improvement over linear time complexity. This efficiency is especially important for large datasets, as it helps keep performance predictable and manageable. Common examples include AVL trees and Red-Black trees, both of which implement specific rebalancing rules to maintain their balanced structure.
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