In this YouTube video tutorial, a Google engineer explains data structures and their importance in creating faster and more powerful algorithms. The tutorial covers Big O notation and how it's used to standardize how much time and space an algorithm requires, and concrete examples are provided for different time complexities. The tutorial also covers the implementation of static and dynamic arrays, including their advantages and disadvantages, and delves into creating and inserting nodes in singly and doubly linked lists. Finally, the tutorial provides an introduction to the stack data structure and briefly explains its primary operations.
00:00:00 In this section, the speaker introduces the concept of data structures as a way of organizing data efficiently and how they are essential in creating faster and more powerful algorithms. The speaker talks about the importance of understanding how and when to use the appropriate data structure for the task at hand and how data structures can make code cleaner and easier to understand. The concept of abstract data type is also explained along with examples of how an abstract data type provides only the interface and not the specific details on how the data structure should be implemented. Additionally, the video briefly touches on computational complexity in order to understand the performance of data structures.
00:05:00 In this section, the concept of Big O notation is introduced as a way of standardizing how much time and space is required for an algorithm to run based on the worst possible arrangement of input. Big O only cares about what happens when inputs become really large and removes constant values added to big O notation. The concept of a function f is also introduced, and it's stated Big O of f of n is just n cubed, which is the biggest, most dominant term in that function.
00:10:00 In this section, the Google Engineer provides concrete examples of how Big O notation is used. The examples are categorized based on their time complexities—constant time, linear time, quadratic time, logarithmic time. He also provides a step-by-step breakdown of how a logarithmic time complexity is achieved using the example of binary search. Additionally, he demonstrates how to calculate the time complexity of a more complicated algorithm and explains the rule for determining an algorithm's complexity.
00:15:00 In this section, the speaker discusses the complexity analysis of a nested loop with an outer loop and an inner loop. The inner loop has a constant amount of work and the outer loop has a variable amount of work. The big O of the function is O(n^4) as n^3 is the dominant term. The speaker then introduces static arrays, which are a fixed-length container containing indexable elements that are contiguous chunks of memory. Static arrays are used everywhere, from temporarily storing objects to storing information from an input or output stream. The speaker outlines the basic structure of an array, common operations that can be performed on them, and their complexity analysis.
00:20:00 In this section, the instructor discusses the use of arrays in programming, including as a workaround for languages that only allow one return value and in dynamic programming. He explains that arrays have constant access time due to their indexable property, but searching may take up to linear time in the worst-case scenario. Insertion, appending, and deletion from a static array are not feasible, but with a dynamic array, resizing the internal array for appending results in a rare but constant time operation. Finally, he notes that elements can be iterated over using a for-each loop, and that array indexing in computer science starts at zero, which can be confusing for some beginners.
00:25:00 In this section, the video discusses the concept of indexing in arrays, where a square bracket denotes indexing, and how dynamic arrays can grow and shrink as needed, allowing for similar get set operations as static arrays. To implement a dynamic array, a static array is used, and when capacity is exceeded, the array size is doubled, and all elements are copied into the new static array. The video also shows the source code for the array class, which supports generics of type T and has instance variables for the internal static array, length, and capacity.
00:30:00 In this section of the tutorial, the instructor goes through the implementation of several methods for a dynamic array, including size, inset, clear, add, remove, index of, contains, and toString. The add method involves resizing the array by doubling its size when the capacity is reached, and the remove method uses two indices to copy all elements of the array except for the removal index. The instructor also demonstrates how to create an iterator for the dynamic array and talks about the benefits of using an iterator for iterating over the elements of an array. Overall, the tutorial provides a simple yet comprehensive introduction to implementing dynamic arrays.
00:35:00 In this section of the video, the instructor provides an introduction to singly and doubly linked lists, explaining that they are a sequential list of nodes that hold data and point to other nodes containing data. Linked lists are used in the implementation of lists, stacks, and queues, as well as circular lists, hash table separate chaining, and for adjacency lists and graphs. The instructor also covers some useful terminology for creating linked lists. Additionally, the advantages and disadvantages of singly and doubly linked lists are discussed, with singly linked lists being more memory efficient but lacking the ability to access previous elements, while doubly linked lists can be traversed backwards and allow for easy removal of a node.
00:40:00 In this section, the instructor explains the implementation details of creating and inserting nodes in a singly linked list and a doubly linked list. To insert a node in a singly linked list, a new pointer is created, and the traverser pointer is advanced to the desired position, following which the new node is created and linked to the other nodes. On the other hand, a doubly linked list has both next and previous pointers, and when inserting a new node, both pointers of the adjacent nodes and the new node need to be updated. Removing a node from a singly linked list involves using two pointers to advance to and remove the desired node, and then deallocating its memory later.
00:45:00 In this section, the speaker covers how to remove nodes from a doubly linked list, which is easier than removing nodes from a singly linked list since we don't have to manually maintain references to the last node. The speaker shows an implementation in Java and discusses the complexity of various operations in linked lists, such as searching and removing elements from the head or tail. While searching in a linked list is linear in the worst case scenario, inserting to the head or removing the head is constant time. Removing from the tail takes linear time in a singly linked list, but not in a doubly linked list because it has a reference to the previous node.
00:50:00 In this section of the video, the presenter explains the implementation of a doubly linked list with methods to clear the list, get size and check if empty, and add nodes to the beginning and end of the list. He also explains how to peek at the first or last element of the list, remove the first or last element, and remove an arbitrary node in the middle of the list. The presenter emphasizes the importance of deallocating memory properly, and sets the node class as private to prevent users from accessing it directly.
00:55:00 In this section of the video, the tutor explains how to remove a node of a particular index from a linked list, even though the nodes are not explicitly indexed. The Remove method supports removing an arbitrary value from the linked list and searching for null values. The tutor also explains the index of method for getting the index of an object within a linked list and the iterator method. Lastly, the tutor introduces the stack data structure and briefly provides an overview of its primary operations, push and pop. The tutor also highlights that the upcoming videos in the series will cover stack implementation, problems solved using stacks, and the time complexity associated with stack operations.
The "Data Structures Easy to Advanced Course" tutorial by a Google Engineer provides comprehensive coverage of various data structures and their functionalities. The tutorial guides the audience on the working principles of stacks and queues, their implementations using arrays and linked lists, and their importance in different applications, including graph traversal and managing server requests. The tutorial also explores priority queues and their implementation using heaps, clarifying the difference between priority queues and heaps as well as the types of heaps. The tutorial ends by providing a step-by-step demonstration of how to add and remove elements from a binary heap.
01:00:00 In this section, the video discusses how a stack data structure functions and its various uses in programming. The video provides a detailed example of how operations are added and removed from a stack and explains how stacks are used in text editors, compilers, and supporting recursion. Additionally, the video highlights how a stack can be used to perform a depth-first search on a graph and presents a cool example problem of how to determine whether a bracket sequence is valid or not using a stack. Lastly, the video includes a complexity analysis of stacks and how they function as a linked list.
01:05:00 In this section, the Google engineer demonstrates how to check if a bracket sequence is valid using a stack data structure. He goes through the algorithm step-by-step, pushing left brackets onto the stack and popping them off when encountering right brackets, and checks if they match. He also explains how the Tower of Hanoi game can be related to stacks, as each peg represents a stack and the discs represent elements that can only be moved under certain conditions. Finally, he discusses how stacks can be implemented using arrays or linked lists.
01:10:00 In this section, we learn about creating a stack using a singly linked list and popping elements off of the stack in a simple implementation of a stack data structure in the Java programming language. The stack is created by pointing the head to a null node which means that the stack is empty. New elements are inserted before the head, and popping elements is done by moving the head pointer to the next node and deallocating the last node. Memory leaks in data structures can cause problems, so it is important to watch out for them in all data structures, and to correct them when necessary.
01:15:00 In this section, the instructor discusses queues, a linear data structure used to model a real-world queue with two primary operations: enqueuing and dequeuing. The front and the back of the queue are used to insert and remove elements, respectively. The instructor also explains the various terminologies surrounding queues, with enqueuing sometimes referred to as adding, and dequeuing as polling or removing from the front of the queue. A classic example of a queue is a line at a movie theater or restaurant, and queues can be useful for keeping track of the x most recent elements in a sequence.
01:20:00 In this section, the video explains how queues are used to manage server requests and to conduct a breadth first search traversal on a graph. The concept of queuing requests is introduced, where an idle server can only handle a certain number of requests at a time and any excess requests are put into a queue. The video also covers the basics of breadth first search, in which nodes of a graph are visited in order of their distance from a starting node. The video concludes by explaining the implementation of n-queuing and dequeuing elements using a queue data structure.
01:25:00 In this section, the tutorial delves into queues and explains how the queue abstract data type can be implemented using either arrays or different kinds of linked lists - singly linked lists and doubly linked lists. The tutorial provides an example of how the singly linked list implementation of a queue can work and also discusses the implementation of various queue methods like peak, poll, and offer using Java programming language. The tutorial also shares a link to the source code of the queue implementation on github.com.
01:30:00 In this section of the tutorial, the Google Engineer explains the concept and implementation of data structures, specifically queues and priority queues. He discusses how a priority queue operates similar to a normal queue with the only difference being that each element has a certain priority, and elements with a higher priority come out of the queue first. He also emphasizes that priority queues only support comparable elements, meaning that the data we insert into the priority queue must be ordered. Furthermore, he provides an example to explain the operations of polling and adding elements into a priority queue. In the next sections of the tutorial, he will go into more detail about the common operations performed on priority queues, turning min priority queues into max priority queues, complexity analysis, and ways of implementing priority queues.
01:35:00 In this section, the Google Engineer explains how priority queues are implemented using heaps. A heap is a tree-based data structure that satisfies the heap invariant, which means that the value of the parent node is always greater than or equal to the value of the child node for all nodes, resulting in two types of heaps: Max heaps and min heaps. Heaps are important as they form the canonical underlying data structure for priority queues. The Google Engineer then shows examples of structures and asks the audience if they are heaps or not, demonstrating that all heaps must be trees and satisfy the heap invariant.
01:40:00 In this section, the importance of priority queues in algorithms such as Dijkstra's shortest path algorithm and Huffman encoding is discussed. Priority queues are useful in situations where there is a need to dynamically fetch the next best or worst element. The main focus is on priority queues implemented as binary heaps, and the operations, including adding an element, removing an element, checking containment, and changing min priority queues to max priority queues. Hash tables can be used to improve the removing time complexity to be logarithmic, while a linear scan is performed for all the elements for the naive method. The section concludes with a hack to convert min priority queues to max priority queues.
01:45:00 In this section, a Google engineer explains how to use negation as a simple way to implement reverse heaps in priority queues. He provides examples of both numerical and string manipulation using a min priority queue with the standard lexicographic comparator as well as a negated comparator. The engineer also introduces the concept of binary heaps and how to add elements to them.
01:50:00 In this section, the instructor explains the important terminology and concepts necessary for understanding how to add elements effectively to a priority queue using a binary heap. He clarifies that a priority queue is not a heap but an abstract data type that defines the behavior a priority queue should have, and that heaps are just the data structure that lets us actually implement that behavior. He also discusses the types of heaps, including binary heaps, and explains the complete binary tree property and how to represent a binary heap using an array construct.
01:55:00 In this section, the Google Engineer explains how to add nodes to a binary heap and maintain the heap invariant by using a technique to easily access a node's children and parent nodes. They illustrate with the example of inserting values into a min heap and demonstrate how to bubble up the nodes to satisfy the heap invariant. In the next section, the engineer covers removing elements from a binary heap and emphasizes the importance of removing the root value, which is the node of interest with the highest priority.
The Google Engineer presents a full tutorial on data structures, explaining how to remove nodes from a binary heap data structure, how to maintain the heap invariant in a priority queue, and how to swim and sink nodes in a binary heap data structure. The video also covers the union find data structure, which is used to track elements divided into disjoint sets and to merge two groups together, with an example about magnets to illustrate how the data structure works. Additionally, Kruskal's algorithm for finding a minimum spanning tree in a graph is explained, and the concept of path compression is introduced to make the union find data structure more efficient.
02:00:00 In this section, the Google engineer explains the process of removing nodes from a binary heap data structure using the pull method. Pulling is done in two cases, when removing the root node and when removing a specific node. When removing the root node, the node is swapped with the last node in the array, and the heap invariant is ensured through bubbling down. When removing a specific node, the node is searched for linearly, swapped with the last node in the heap, and then the heap invariant is ensured through bubbling up or down depending on the value of the swapped node. The engineer concludes that pulling takes logarithmic time for the root node and linear time for a specific node.
02:05:00 In this section of the tutorial, the Google Engineer explains how to remove nodes in a heap with improved time complexity by making use of a hash table. Instead of doing a linear scan to find the index of the node to remove, each node is mapped to the index it is found at. When we want to remove a particular node, we just look up its index to start doing it. The video also covers how to deal with the problem of multiple values in the heap and how to keep track of the positions of the values in the tree. The Engineer explains that as long as we satisfy the heap invariant, it does not matter which node we remove. The tutorial includes examples of adding, pulling, and removing elements with the new scheme proposed.
02:10:00 In this section, the speaker explains how to insert, remove, and maintain the heap invariant in a priority queue using a heap data structure. Then he dives into the source code and highlights some important instance variables and constructors necessary for implementing the priority queue. He explains that the elements allowed in the priority queue must implement the comparable interface and that logging and removals can be tracked using a map that maps an element to a set of integers representing the positions of the element in the heap.
02:15:00 In this section of the tutorial, the Google Engineer discusses different ways to create a priority queue, including creating an initially empty queue with a set capacity, creating a queue with a defined initial capacity, and constructing a queue in linear time using the heapify operation. The engineer also goes over some simple methods of a priority queue, including empty, clear, size, peek, poll, and contains. The add method is also discussed, with details on how to add elements to the queue and the importance of keeping track of the elements in a map. The swim function is also highlighted as a necessary step after adding an element to the end of the list.
02:20:00 In this section of the video, the Google engineer explains the methods for swimming and sinking nodes in a binary heap data structure. The swimming method involves swapping nodes with its parent and moving upwards until the node is in the correct position based on its value. The sinking method is similar but involves comparing the node with its child nodes and swapping with the smaller one until it reaches the correct position. The engineer also explains the swap, remove, and remove add methods implemented in the binary heap. The remove add method involves swapping the removed node with the last element in the heap and sinking or swimming the new node to the appropriate position.
02:25:00 In this section of the tutorial, the instructor explains how to remove a node from a heap data structure and ensure that the minimum heap's integrity is maintained using a method to sink or swim nodes depending on whether the swap affects the heap's order. The section also introduces the union find data structure and its two primary operations, find and union, which are used to track elements divided into disjoint sets and to merge two groups together, respectively.
02:30:00 In this section, the speaker uses the example of magnets to explain how the union find data structure works. The union find efficiently merges items or groups of items, assigning them an arbitrary color. The union find data structure is used in various algorithms, such as the minimum spanning tree algorithm and grid percolation. The union find has excellent complexity, and its construction is linear time, while its count components function can determine the number of components in constant time, making it a great data structure to have around.
02:35:00 In this section of the tutorial, the Google engineer explains Kruskal's algorithm for finding a minimum spanning tree in a graph. The algorithm involves sorting edges by ascending edge weight and then unifying nodes that do not belong to the same group, until all vertices are unified into one group. The union find data structure is used to efficiently merge groups and prevent cycles. The engineer provides an example graph and walks through the steps of the algorithm to illustrate how it works.
02:40:00 In this section, the tutorial explains the union find data structure and how it works internally. The first step is to create a mapping between the objects and the integers in the range of zero to N non-inclusive, where N is the number of elements. Then an array is constructed, and each index has an associated object, which is done through the mapping. The value in the array's position is initially the index it's at, and each node is a root node mapping to itself. As instructions are performed to unify objects into groups, the values in the array have changed to map to other objects. Smaller components are merged into larger ones, and the root nodes are used to unify groups.
02:45:00 In this section, the speaker talks about how to merge elements into different groups using the union find data structure. By finding and following the parent nodes of each component's root node, we can determine which component an element belongs to. To unify two components, we make one of the root nodes point to the become the parent of the other root. The number of components is equal to the number of root nodes remaining, and the number of root nodes only decreases as we unify components. The speaker also notes that the implementation of this structure currently does not have an amortized time complexity without the use of path compression, an optimization to be discussed in the next video.
02:50:00 In this section, the concept of path compression is introduced to show how it makes the union find data structure more efficient. Path compression involves compressing all nodes along the path to the root node, thus allowing for constant time lookup of the root node for any given component. The example of unifying groups with and without path compression is given to compare and contrast the two methods, demonstrating the efficiency gained through path compression.
02:55:00 In this section, the instructor discusses how path compression and union find work together to create an efficient data structure. He demonstrates how path compression dynamically compresses paths along the way until reaching a final state. The union find code is presented in the video, with explanations of the use of arrays for indexing and tracking parent-child relationships. Additionally, the code includes methods to check for the root node and compress the path to it. The instructor emphasizes the importance of watching other videos on union find to gain a full understanding of the topic.
This tutorial covers various data structures, starting with the union-find data structure and its methods, including find, connected, parent, size, and unify. The tutorial then moves on to trees, including definitions for trees, rooted trees, binary trees, and binary search trees. The video provides examples of inserting and removing nodes from binary search trees, as well as different traversal algorithms, including pre-order, in-order, post-order, and level order, and explains how to implement these constructs in Python and Java. Additionally, the video introduces hash tables and discusses the importance of hash functions and popular collision resolution methods.
03:00:00 In this section, the instructor introduces a union-find data structure and its methods, including find, connected, parent, size, and unify. He demonstrates how the root nodes of the structure contain the size of each connected component, and the unify method merges smaller groups into larger ones. Moving on to trees, the instructor gives a brief overview of trees as undirected graphs and introduces binary trees and binary search trees. He promises to cover how to insert and remove nodes in binary search trees, tree traversals, and how they can be applied to general trees in later tutorials.
03:05:00 In this section, the concept of trees is introduced, with multiple definitions provided. A tree can be defined as a non-directed graph that is connected and acyclic, as a graph with n nodes and n-1 edges, or as a graph where there is only one path between any two vertices. A rooted tree is introduced as well, where any node can become the root and child and parent nodes are defined. The parent of the root node is itself, and leaf nodes are defined as nodes with no children. The concept of a subtree is also introduced, denoted by triangles within a tree. Additionally, the definition of a binary tree is explained as a tree in which every node has at most two children.
03:10:00 In this section, the video covers binary search trees, which are binary trees that satisfy the binary search tree invariant. This means that the left subtree always has values smaller than the current node and the right subtree has larger values. The video presents various examples of binary search trees and challenges viewers to determine if they satisfy the invariant. Binary search trees are useful in many implementations of abstract data types and are used in balanced binary search trees and syntax trees. They also have a logarithmic time complexity in the average case for operations like insertion and search on random data.
03:15:00 In this section of the tutorial, the Google engineer discusses binary search trees and how to insert elements into them. Binary search trees have logarithmic behavior on average, which makes them easy to implement and efficient for most cases. However, in the worst case, a binary tree can become a linear data structure, which is not ideal. To insert an element in a binary search tree, the element must be comparable, and one of four cases may occur: recurse down the left subtree, recurse down the right subtree, handle duplicate values, or insert a new node. Finally, the engineer demonstrates inserting values into a binary search tree using animations.
03:20:00 In this section, the Google engineer explains the worst-case scenario when inserting values into a binary search tree, which causes it to become a linear structure. This behavior is undesirable because it slows down operations such as searching for nodes or deleting values. The engineer then explains how to remove elements from a binary search tree in two steps: first finding the node to remove, and then replacing it with its successor to maintain the binary search tree invariant. The video provides an example of how to search for values in a binary search tree to help understand the process.
03:25:00 In this section, the Google engineer explains the four cases of the Remove phase when dealing with a binary search tree. The first case is when the node to remove is a leaf node the second and third cases occur when there is only one subtree either to the left or to the right. The fourth case occurs when the node has both a left and right subtree, and the question is in which subtree will the successor of the node be? The answer is that the successor can be either the largest value in the left subtree or the smallest value in the right subtree, and there can be two possible successors.
03:30:00 In this section of the course, the instructor explains how to remove nodes from binary search trees using the concept of a successor node. The successor is either the smallest node in the right subtree or the largest node in the left subtree. The instructor demonstrates the removal of nodes through several examples, emphasizing the importance of rebalancing the tree after removal. The section concludes with an overview of different tree traversals, including pre-order, in-order, post-order, and level order.
03:35:00 In this section, the instructor explains the concepts of pre-order, in-order, and post-order traversals in binary trees. He explains that pre-order prints the current node's value followed by its left and right subtrees, and then provides an example of how pre-order traversal works in a binary tree, maintaining a call stack to keep track of which nodes are visited. Similarly, he explains how in-order traversal works, which involves traversing the left subtree, printing the value, and then traversing the right subtree. The instructor uses a binary search tree as an example and highlights the order in which the nodes are visited and printed during traversal.
03:40:00 In this section, the tutorial discusses different traversal algorithms for binary trees. First, the inorder traversal, which prints the values of the nodes in increasing order. Next, the post order traversal, which traverses the left subtree, then the right subtree, and only after both of those are traversed, prints the value of the node. The tutorial goes on to explain breadth first search and how it can be used for level order traversal, which prints the nodes one layer at a time. A queue is used to keep track of the nodes to be explored, and nodes are added to the queue as their parent nodes are visited.
03:45:00 In this section of the video, the instructor explains how to perform breadth-first search using a queue data structure. He demonstrates the process of exploring nodes and adding their children to the queue according to their order. He emphasizes the importance of using a queue instead of a stack when performing level-order traversal. The video then shifts focus to the implementation of a binary search tree in Java, with the instructor explaining the class structure, instance variables, and methods used to add elements to the tree. He also highlights the importance of choosing a comparable type when working with binary search trees.
03:50:00 In this section, the instructor explains the process of removing nodes from a binary search tree. They begin by discussing the public method for removing nodes and explaining that they will only remove the node if it exists within the tree, which is checked first. Then, they delve into the recursive method used to remove the node, which involves finding the node to be removed and then actually removing it based on whether it has left and/or right subtrees. The instructor explains three different cases for removal, which involve either a left or right subtree, or both a left and right subtree. They also provide helper methods for traversing the tree to find the successor node.
03:55:00 In this section, the Google engineer explains how to implement a binary search tree in Python, including inserting a new node, searching for a specific element, and calculating the height of the tree. He also shows how to implement binary tree traversals iteratively using a custom method called "traverse," which takes a tree traversal order as input and returns an iterator for that order. Next, he introduces hash tables and discusses the importance of hash functions, as well as popular collision resolution methods like separate chaining and open addressing.
The video tutorial covers various aspects of hash tables and their implementation. It discusses the importance of hash functions, which map keys to values, and how to handle hash collisions using techniques such as separate chaining and open addressing. The tutorial also covers methods for inserting, removing, and searching for entries in hash tables, as well as resizing and managing load factors. The speaker emphasizes the importance of choosing an appropriate probing function and table size to avoid infinite loops and performance issues. Practical examples are used throughout the tutorial to illustrate the concepts.
04:00:00 In this section, the instructor introduces the concept of hash tables and its implementation using hashing techniques. Hash tables are used to construct a mapping from keys to values, where each key is unique and is associated with a value. To map the keys to values, a hash function is used, which maps the keys to a whole number in some fixed range. The instructor shows how to define hash functions for arbitrary objects, such as strings, using the ASCII values of characters in the string. Hash tables are often used to track frequencies of values and construct mappings between key-value pairs, provided the keys are hashable. The section ends with a mini challenge to define a hash function for a database of people with three fields.
04:05:00 In this section, the instructor discusses hash functions and their properties. The hash function is defined arbitrarily and can have an infinite number of possibilities. The instructor emphasizes that hash functions must be deterministic to avoid screwing up the hash table. The uniformity of hash functions is also important to minimize hash collisions which occur when two objects hash to the same value. The instructor also illustrates how hash values can speed up object comparisons, and goes on to explain that sophisticated hash functions such as cryptographic hash functions and checksums are used for files instead of the hash functions used for hash tables.
04:10:00 In this section, the tutorial explains what makes a key of type 't' hashable and how a hash table works using a uniform hash function to index into it. It mentions that hash functions need to be deterministic and enforce immutable keys that are fixed and constant, such as immutable strings and integers. By using the hash function as a way to index into a hash table, we can achieve quick insertion, look-up, and removal times in constant time. The tutorial also provides an example of inserting key-value pairs into a table that eventually leads to a hash collision and explains how to handle collisions.
04:15:00 In this section, the speaker discusses hash collision resolution techniques, specifically separate chaining. Separate chaining is a way of dealing with hash collisions by maintaining an auxiliary data structure, usually a linked list, to hold all the different key-value pairs which hash to a specific value. The time complexity of a hash table can achieve constant time insertion, removal, and search on average, but with a terrible hash function that's not uniform, it could be linear time. The speaker also provides an example of how separate chaining works and how we can use it to handle collisions within our hash table by maintaining a linked list data structure for each index in the array.
04:20:00 In this section, the speaker explains how lookups work in a hash table with separate chaining, which uses linked lists to handle collisions. With a given key, the speaker demonstrates how to find the corresponding value by hashing the key and searching the linked list in the corresponding bucket of the hash table. The speaker also addresses some common questions about maintaining constant time complexity, removing key-value pairs, and using different data structures to handle buckets in the hash table. Finally, the speaker shares some source code for a hash table implementation using separate chaining.
04:25:00 In this section, the video introduces hash tables and how they are implemented in Java. The entry class is first discussed, with a focus on generics, hash codes, and equals method. The hash table itself is then explained including instance variables like maximum load factor, capacity, threshold and the table itself, which is an array of linked lists. Various methods such as size, empty, clear, contains key, and hash are also discussed along with their implementation details. Finally, the normalized index method is explained, which is used to convert a hash value into an index for lookup in the hash table.
04:30:00 In this section of the video, the speaker explains the methods involved in implementing a hash table such as insert, get, remove, seek entry, and resize table. The insert method adds a new entry to the hash table or updates an existing one, while get retrieves the value associated with a specific key. Remove removes the key value pair from the hash table, and seek entry method helps in finding the entry at a given bucket index. Lastly, resize table resizes the table by doubling its capacity and then creates a new table with the new capacity, moves the data from the old table to the new table, and removes the old data.
04:35:00 In this section, the instructor discusses the open addressing technique for resolving collisions in hash tables. This method stores the key-value pairs in the table itself instead of an auxiliary data structure. Therefore, it is crucial to manage the size of the hash table and the number of elements in it. The load factor, which is the ratio of items to the size of the table, needs to be kept below a certain threshold to prevent it from becoming exponentially bad. When inserting a new key, the hash function gives an original position for the key, but if there is a collision, a probing sequence is used to find the next open spot. The linear probing sequence is one of many to choose from.
04:40:00 In this section, the instructor discusses open addressing, a method of handling collisions in hash tables where the item is placed in a different location when the original hash index is already occupied. Various probing functions are introduced, such as linear probing, quadratic probing, double hashing, and pseudo random number generator probing function, each of which uses a different mathematical formula to find the next slot. However, the main difficulty with open addressing is the potential for producing cycles that are shorter than the size of the hash table, which can lead to an infinite loop. A practical example is given to illustrate this problem.
04:45:00 In this section, the speaker discusses the issue of cycles in probing functions used in hash tables and how to handle it. The speaker explains that techniques such as linear probing, quadratic probing, and double hashing are all subject to this issue of cycles but can be redefined to produce a cycle of length to avoid being stuck in an infinite loop while inserting an element. The constant "b" in linear probing is considered obsolete, and the speaker mentions that some linear functions may be unable to produce a full cycle of order "n." The solution to this problem is to choose probing functions that produce a cycle of exactly "n" by making sure that the constant "a" and the table size "n" are relatively prime to each other.
04:50:00 In this section, the instructor discusses hash tables and how they work, including probing functions, table size, load factor, and resizing. He uses an example with linear probing to demonstrate how to insert key value pairs into a hash table and avoid infinite loops caused by hash collisions. However, he points out that the choice of probing function and table size can significantly impact performance, and selecting a function whose greatest common denominator with the table size is not one can lead to cycles and cause problems.
04:55:00 In this section, we see an example of using a probing function to insert key-value pairs into a hash table without any collisions. If a collision does occur, the probing function will keep probing until an empty spot is found to avoid any cycle occurring. Once the number of elements exceeds the table's threshold, the table size is doubled while maintaining the GCD of N. The old elements are then placed in the new table using the new table size while keeping the same probing function. Finally, a new key-value pair is inserted, and if the spot is free, there are no issues.
This YouTube video titled "Data Structures Easy to Advanced Course - Full Tutorial from a Google Engineer" provides a comprehensive overview of hash tables, double hashing, and quadratic probing in collision resolution. The video covers the concepts of resizing and growing a hash table, handling collisions and deletion, and implementing hash tables using quadratic probing. The video also introduces the Fenwick tree, a data structure that supports range queries and point updates in logarithmic time with linear construction time. The video provides a step-by-step explanation of how to perform prefix sums and range queries using the Fenwick tree.
05:00:00 In this section, the instructor discusses quadratic probing in hash tables, which is used to address collisions in open addressing. This involves probing according to a quadratic formula using a selecting a random probing function. The instructor explains that not all quadratic functions are viable because they don't produce a cycle of order which results in getting stuck in an infinite loop, but most randomly selected quadratic functions will end up producing a cycle. The instructor provides three most popular ways to select a probing function, and focuses on the second one where p of x equals x squared plus x divided by two and the table size is a power of two.
05:05:00 In this section, the speaker discusses the process of inserting elements into hash tables using the double hashing and open addressing collision resolution technique. They explain the importance of the table size being a power of two and how to handle collisions using probing. The speaker also demonstrates the process of resizing the table when the load factor exceeds a certain threshold. They go on to discuss how to update existing elements and how to insert new ones into the table.
05:10:00 In this section, the instructor explains the concept of double hashing, which is a probing method used to handle collisions in hash tables. The double hashing scheme involves probing according to a constant multiple of another hash function, called the secondary hash function. The instructor warns about the issue of infinite cycles that can occur in the case of double hashing and provides a strategy to fix it, which involves picking a prime number for the table size and constructing a value called delta. He also discusses the systematic way of generating new hash functions for different data types using the same fundamental building block.
05:15:00 In this section, the speaker explains the concept of hash functions and how they can be combined to create a new hash function. They mention the use of Universal hash functions and provide an example using double hashing. They discuss the process of inserting key-value pairs into a hash table while also handling hash collisions and updates. The example uses a table size of seven and a max load factor of 0.75, and the hashing function uses integers and real numbers as building blocks.
05:20:00 In this section, the tutorial explains the process of resizing and growing a hash table with double hashing, which involves doubling the table size, finding the next prime number above that value, allocating a new table, and inserting the old elements into the new table. The tutorial provides an example where the original table has reached its maximum threshold after inserting five elements, and the table is resized to a new size of 17. The tutorial also explores the issues that arise when removing elements from a hash table using the open addressing scheme, and the importance of avoiding collisions during insertion and deletion operations.
05:25:00 In this section, the video explains how to handle hash collisions and deletion in a hash table. The naive method of deletion, which simply clears the contents of the bucket, is shown to be flawed because it impacts the ability to find elements in the hash table properly. Instead, the video recommends placing a unique marker called a tombstone in place of the deleted element. This marker can later be replaced with a new key-value pair or removed during hash table resizing. The video provides an example using quadratic probing to show how tombstones are used during hash table lookup.
05:30:00 is just a power of two. In this section, a Google engineer provides an overview of hash tables that use open two addresses as a collision resolution scheme. The engineer discusses lazy deletion or lazy relocation, which involves relocating key value pairs to avoid hitting a bunch of tombstones when probing. The engineer also provides a walk-through of the code for a hash table that uses quadratic probing, which involves storing key value pairs directly inside an array rather than having one array with a wrapper class for an entry. The engineer explores the constructor and default constants for the hash table, which allows users to initialize it without any parameters.
05:35:00 In this section of the course, the instructor explains the implementation of the hash table using quadratic probing. The method involves computing the threshold, normalizing the index and initializing the tables. The insert method info is provided where key-value pairs are inserted into the hash table or updated if the key already exists. The instructor also discusses key count, hash table's empty check, and put addition before moving onto the insertion method. The do-while loop for the insertion of a key is explained in detail.
05:40:00 In this section, the instructor explains the contains key, has key, get, and remove methods of a hash table implementation. The contains key and has key methods check whether a key exists in the hash table by calling the get method, which returns a true or false value for the contains flag. The get method finds the hash index and looks for the key. If the key exists, the contains flag is set to true, and the value is returned. The remove method is simpler, where a key is first found in the hash table, then decremented, and a tombstone is dumped in its place. The resize table method is called when new elements are being inserted to grow the table size, where a new table is allocated, and the current table is swapped with the new table to call the insert method.
05:45:00 In this section, the Fenwick tree, also known as the binary index tree, is introduced as a data structure to efficiently compute range queries on an array and perform point updates. The motivation behind the Fenwick tree is the inefficiency of linear queries when computing ranges in an array. By computing all prefix sums for the array, range queries can be calculated in constant time. However, updates to the array require recomputing all prefix sums, making the process inefficient. The Fenwick tree was created to solve this problem by supporting range queries and point updates in logarithmic time with linear construction time. The Fenwick tree works by having each cell responsible for a range of other cells based on the value of its least significant bit.
05:50:00 In this section of the video, the instructor discusses Fenwick trees and how they work for range queries. He shows a diagram with a one-based array and the binary representation of each number. The diagram shows the least significant bits responsible for themselves, and all other cells responsible for a range of 2, 4, 8, or 16 cells. To perform a range query, the instructor explains how to calculate the prefix sum up to a certain index by cascading downwards until reaching zero. He demonstrates how to find the prefix sum for a specific index and how to do an interval sum between two given indices.
05:55:00 In this section, we learn about calculating prefix sums using a cascading down effect and how to perform a range query algorithm using Fenwick trees. The prefix sum calculation involves starting at a given index and subtracting the least significant bit from the value until it reaches zero, and then adding all the values during each subtraction. The range query algorithm uses Fenwick trees where we take the difference between prefix sums for range queries. The algorithm requires bit manipulation and provides a neat little algorithm for faster computation. The video author recommends checking out the previous Fenwick tree range query video for more context on how the tree is set up and how operations are performed on it.
The video tutorial covers various topics, including the concept of a Fenwick tree for quick range queries and point updates, using suffix arrays and LCP array for finding unique substrings and longest common substrings, and solving the longest common substring problem using a sliding window technique. The tutorial also explains how to find the longest repeated substring efficiently using the LCP array and explores the properties and importance of balanced binary search trees, specifically AVL trees, and how they can be kept balanced by using tree rotations. The video provides detailed explanations, examples, and source code in Java available on GitHub.
06:00:00 In this section, the concept of point updates and Fenwick tree construction are explained by the Google Engineer. Point updates involve adding values to the tree at a specific index and finding the cells responsible for that range of responsibility. Meanwhile, the linear construction of the Fenwick tree involves updating the immediate cell responsible for a value by propagating the values throughout the tree in place, resulting in a fully functional Fenwick tree. The propagation process relies on the cascading idea of updating the parent responsible for a specific value.
06:05:00 In this section of the tutorial, the concept of a Fenwick tree is explained in detail. A Fenwick tree is a data structure that is used to quickly perform range queries and point updates on an array of values. The data structure utilizes a type of binary indexing in which each cell is responsible for propagating its value to its parent and so on. The construction algorithm for a Fenwick tree is also discussed, which involves turning an array of values into a Fenwick tree by cloning the original array and computing the parent cell for each element in the new tree structure. The source code for a Fenwick tree implementation in Java is shown and is available in a GitHub repository.
06:10:00 In this section, the instructor explains how to create a Fenwick tree class and its different constructors. He also explains the prefix sums, which allows to compute the prefix sum from one to i both inclusive. The video also covers how to add from a point update and an additional method to set the index equal to k. The instructor emphasizes the importance of using binary manipulation and the suffix array data structure, which is very useful in string processing.
06:15:00 In this section, the concept of suffixes and suffix arrays are introduced, with an example of constructing a suffix array for the word "camel". The suffix array is an array of sorted indices that allows for a compressed representation of the sorted suffixes without storing the suffixes themselves, making it a space efficient alternative to a suffix tree. The longest common prefix (LCP) array, which stores how many characters two sorted suffixes have in common with each other, is also introduced as an important piece of information associated with the suffix array, with an example of constructing the LCP array for a given string.
06:20:00 In this section, the video discusses the application of suffix arrays and LCP arrays in finding and counting unique substrings in a more efficient manner than the naive algorithm which requires a lot of space. By using the information stored inside the LCP array, it provides not just a quick but a space-efficient solution. The LCP array represents the shared number of characters between two adjacent suffixes of the original string. By examining the LCP values, one can determine which substrings are repeated and eliminate them to count all unique substrings effectively.
06:25:00 In this section, the speaker discusses the problem of finding the longest common substring shared between at least k out of n given strings. One approach to solving this problem is dynamic programming, but it can quickly become unwieldy. A better approach is to use a suffix array, which can find the solution in linear time. To do this, the speaker explains that we first concatenate all the strings into a larger string, adding unique sentinel values in between each string to avoid any intermingling of suffixes. Then, we construct the suffix array for this concatenated string, which allows us to find the longest common substring of K strings by looking for K strings with different colors that share the largest LCP value.
06:30:00 In this section, the video tutorial discusses how to find the longest common substring of k different colors within a string using a sliding window technique. The approach is to adjust the window size to contain k suffixes of different colors and to query the minimum LCP value within that range. The minimum range query can be solved using a linear solution or a minimum range query data structure such as a segment tree. The tutorial also recommends using a hash table to keep track of the colors in the window. The sliding window expands downwards to capture missing colors and shrinks when the criteria are met.
06:35:00 In this section of the video, the instructor demonstrates an example of solving the longest common substring problem with a suffix array. The window LCP and window LCS values help track the longest common prefix and longest common substring values for the current window, while the LCS length and LCS set track the best values so far. The example uses four strings and a minimum of two strings of the pool of four to share the longest common substring. The instructor demonstrates how to expand and shrink the window while searching for the longest common substring, and hints at a challenge for viewers to check out on the cast website and provides a link to an implementation of the algorithm on GitHub.
06:40:00 In this section, the video explains the process of solving the longest common substring problem using the sliding window technique. By expanding and shrinking the window size and using the longest common prefix array to identify the common substrings, the algorithm finds a solution with a linear time complexity. The video then introduces the longest repeated substring problem and explains how to use the longest common prefix array to efficiently find the longest repeated substring in a given string, compared to the naive approach that requires N squared time and lots of memory.
06:45:00 In this section, the Google engineer teaches about using a suffix array and an LCP array to find the longest repeated substring. The LCP value at an index gives the length of the longest common prefix between two adjacent suffixes. The maximum value in the LCP array gives the length of the longest repeated substring. In the case of ties, all the possible longest values must be found. Then the engineer explains the importance of balanced binary search trees and how they differ from traditional binary search trees in terms of being self-adjusting to maintain a logarithmic height relative to the number of nodes they hold, making insertion and deletion extremely fast. The key concept to achieve this is tree rotations, and they will be discussed further in the video.
06:50:00 In this section, the concept of tree invariant and tree rotations in balanced binary search trees is explained. A tree invariant is a rule imposed on a tree to be met at the end of every operation, which is ensured by applying a series of tree rotations. Tree rotations are legitimate transformations that move around nodes in a tree without breaking the binary search tree invariant if the ordering and placement of nodes are maintained. The process of right rotation is explained in detail, and the steps involved in updating pointers when nodes have references to both the child and parent nodes are discussed.
06:55:00 In this section, the video tutorial focuses on demonstrating how to insert nodes into an AVL tree using the tree rotation technique while also explaining the properties that keep AVL trees balanced. First, the video explains what an AVL tree is and its significance as the first type of balanced binary search tree. Then, the video introduces the balance factor, which is the difference between the height of the right subtree and the left subtree. The video emphasizes that the balance factor of every node should be either minus one, zero, or plus one to maintain the balance of the AVL tree. The tutorial then proceeds to discuss how to handle cases when this is not true, which can be resolved with tree rotations.
The YouTube video titled "Data Structures Easy to Advanced Course - Full Tutorial from a Google Engineer" covers a range of data structures and their implementations in detail. The video discusses AVL trees, explaining the insertion and removal methods, as well as how to augment the binary search tree removal method for AVL trees. The instructor also provides source code for a recursive AVL tree implementation in Java, explaining the private insert method and the update and balance methods. Later on, the video covers indexed priority queues in detail, explaining how to assign index values to keys, and providing pseudocode for useful operations like insertion, update, and removal. The video also details how to improve node removal from linear time complexity to logarithmic time complexity using node position lookups.
07:00:00 In this section, the instructor explains the four distinct cases of rotations needed to balance a left-heavy tree in an AVL tree. The instructor also provides pseudocode for inserting a node into an AVL tree and explains the update and balance methods. The update method calculates the height and balance factor of a node, while the balance method checks if the balance factor of a node violates the AVL tree property and determines the appropriate rotation to rebalance the tree. The left-right and right-left cases are also explained as they reduce to left-left and right-right cases after the first rotation.
07:05:00 In this section, the video explains how to remove elements from an avielle tree, which is very similar to removing elements from a regular binary search tree. The removal process can be broken down into two steps: finding the element and replacing it with a successor node to maintain the binary search tree invariant. Finding the node involves comparing the target element with nodes in the tree until a match is found or the search reaches the end of the tree. The replacement process depends on whether the node to remove is a leaf node or has only a left or right subtree, with the successor node being the immediate child in the latter two cases.
07:10:00 In this section, the instructor explains how to remove a node from a binary search tree and provides examples for the three cases of deletion. The first case is when the node to remove has no children, the second is when it has one child, and the last is when it has two children. The instructor also explains how to augment the binary search tree removal method for AVL trees by simply adding two lines of code to ensure that the tree remains balanced and that the balance factor and height values stay up to date. Finally, the instructor provides a link to their GitHub repository where viewers can find the source code for the AVL tree and a live demonstration of the AVL tree in action.
07:15:00 In this section, the instructor provides an explanation of the source code for a recursive AVL tree implementation in Java. The AVL tree accepts a generic type argument and stores the value inside the node, which must be comparable. The node subclass stores the left and right child pointers, and the height and balance factor of the node. The tree can be displayed on the terminal using the tree printer interface, and public methods such as size, is empty, and set contains are provided in the code. The insert method is also explained using base cases and comparative values to determine whether an insertion was successful or not.
07:20:00 In this section, the private insert method in the AVL tree is explained, which inserts new nodes either to the left or the right subtree, while updating the balance factor and height of the nodes accordingly. The update method updates the height and balance factor of a node, and the balance method calls the necessary rotation methods to maintain the balance of the tree. The left-left, left-right, right-right, and right-left cases are explained, and the order of the update methods after rotation is emphasized as crucial for the AVL tree to maintain its balance.
07:25:00 In this section, the instructor discusses the remove method of the binary search tree data structure. He explains that to remove an element, the method first checks if the element exists in the tree and subsequently calls the private remove method. The instructor outlines the four cases that can arise during removal and proposes a heuristic to determine which subtree to remove from when attempting to remove a node with both subtrees. Finally, he reminds viewers to call the update and rebalance method on the callback of the remove method to ensure the tree remains balanced despite node removals.
07:30:00 In this section, the Google engineer introduces the indexed priority queue, a data structure that supports quick updates and deletions of key-value pairs. It solves the problem of being able to quickly look up and dynamically change the values in a priority queue, which can be useful in various situations such as a hospital waiting room where patients need to be served by the highest priority first. The engineer provides an example of patients with different priorities and how the indexed priority queue can help in updating priorities on the fly.
07:35:00 In this section, the video discusses index priority queues which allow for the efficient dynamic updating of priorities for certain items. The first step in using an index priority queue is to assign index values to all keys, creating a bi-directional mapping. This mapping should be bi-directional and can be facilitated with a bi-directional hash table. The reason for using index values on keys is to enable indexing into arrays, which is often how priority queues are implemented.
07:40:00 In this section, the speaker introduces the concept of an index priority queue and explains the operations it should support, including deleting keys, getting the value associated with a key, checking if a key exists in the priority queue, getting the key index with the smallest value, getting the smallest value in the index, being able to insert and update key value pairs, and the specialized update operations increase and decrease the key. The time complexity for all these operations is either constant or logarithmic. The speaker also provides a refresher on the traditional priority queue data structure, including how binary heap works, how to insert a new value into the priority queue, and how to remove items from it.
07:45:00 this section: How do we implement an indexed priority queue with a binary heap? The first step is to assign each item a unique index value and an initial value for the index priority queue. Then, we use a min indexed priority queue to sort by the lowest value first, and we maintain a position map to tell us the index of a node in the heap for any given key index. Additionally, we maintain an inverse lookup table to find the key for a given node. This information enables us to efficiently access and manipulate the priority queue for a variety of applications, such as prioritizing patients in a hospital or customers in a restaurant.
07:50:00 In this section, the video discusses how to perform useful operations, such as inserting, updating, and removing key-value pairs in an index priority queue. The process for insertion is similar to a regular priority queue, but with the added step of updating the position and inverse maps. The video provides pseudocode for the insertion process, showing how the node is moved up through the heap until the heap invariant is satisfied. The swapping process involves updating the position and inverse maps, but not the values array, which remains indexed by the key index rather than the node index. The video also briefly touches on polling and removing key-value pairs, which involve similar steps as insertion but in reverse.
07:55:00 In this section, the video explains how removing elements from an indexed priority queue is improved from linear time complexity to logarithmic time complexity, making use of node position lookups that are now constant time. The video gives a step-by-step example of how to remove nodes, including swapping nodes, storing key value pairs before removal, cleaning up removed nodes, and restoring the heap invariant by moving the swapped node up or down. Additionally, pseudocode for removing key-value pairs is provided, showcasing a short five-line implementation and three-line cleanup process. Finally, the video describes the sync method and how it works, as well as key-value pair updates, which are similar to remove but also take logarithmic time.
This section of the video tutorial on data structures covers a Java implementation of a min indexed binary heap, which uses arrays to track indices and values for each node, and includes methods for insertion, deletion, updating, and adjusting key values. The tutorial provides detailed explanations of the code, including helper methods and bounds checking, and the source code is available on Github.
08:00:00 In this section, the video covers the source code for a min indexed binary heap, which requires passing in a comparable object to order key-value pairs. The heap uses arrays to track child and parent indices for each node, a position map and inverse map to track key indices, and a values array to store associated values. The section goes over methods including insert, delete, update, increase key, and decrease key, as well as helper methods for bounds checking and heap verification. The code is presented in Java and can be found on Github.