Data structure and algorithmic thinking with python pdf download

Data Structure and Algorithmic Thinking with Python. Data Structures Through C. Expert Data Structures with C. Data Structures Through C in Depth. Data Structures and Algorithms using C. A data structure is a way of storing data in a computer so that it can be used efficiently and it will allow the most efficient algorithm to be used.

The choice of the data structure begins with the choice of an abstract data type ADT. A well-designed data structure allows a variety of critical operations to be performed, using as few resources, both execution time and memory space, as possible. Better and easily comprehendable solutions can be easily searched in the internet.

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The algorithm is not the code; it is only the logic of a problem. These two things have become very important in every industry in recent times.

The data structure is the collection of values related to data. The knowledge of data structures and algorithms is not limited to the textbooks of class only.

Anyone interested in making a career in the field of data and technology must study data structure and algorithms. However, the right study materials can help you to get in-depth knowledge about these things. We have made a list of study materials that will aid in your study of algorithms and data structure.

It is not tough to learn about data structure and algorithms. However, there are several books available in the market for data structure and algorithms due to the growing popularity of these two subjects.

Students sometimes get confused to choose the right set of books. To solve this issue, we have made a list of reference books on data structure and algorithms by doing proper research. Before starting the course on data structure and algorithms, you must know the syllabus. Friends, if you need any E-Book PDF related to any topic or subjects and need any assistance and inquiry related to exams you can comment below. We will respond as soon as possible. Disclaimer: Sarkari Rush does not own books pdf, neither created nor scanned.

We just provide the link already available on the internet and in google drive. What else is between n and nlogn? How about nloglogn? That is, amortized analysis is a worst-case analysis, but for a sequence of operations rather than for individual operations. The motivation for amortized analysis is to better understand the running time of certain techniques, where standard worst case analysis provides an overly pessimistic bound.

Amortized analysis generally applies to a method that consists of a sequence of operations, where the vast majority of the operations are cheap, but some of the operations are expensive. If we can show that the expensive operations are particularly rare we can change them to the cheap operations, and only bound the cheap operations. The general approach is to assign an artificial cost to each operation in the sequence, such that the total of the artificial costs for the sequence of operations bounds the total of the real costs for the sequence.

This artificial cost is called the amortized cost of an operation. To analyze the running time, the amortized cost thus is a correct way of understanding the overall running time — but note that particular operations can still take longer so it is not a way of bounding the running time of any individual operation in the sequence. Example: Let us consider an array of elements from which we want to find the kth smallest element. We can solve this problem using sorting.

After sorting the given array, we just need to return the kth element from it. The cost of performing the sort assuming comparison based sorting algorithm is O nlogn. This clearly indicates that sorting once is reducing the complexity of subsequent operations.

Note: We can use the Subtraction and Conquer master theorem for this problem. Note that while the recurrence relation looks exponential, the solution to the recurrence relation here gives a different result. This is similar to Problem The complexity of the above function is O n2logn. The complexity of the above function is O nlog2n. Solution: Consider the comments in the function below. The complexity of the above function is O n.

Even though the inner loop is bounded by n, due to the break statement it is executing only once. Solution: By iteration: Note: We can use the Subtraction and Conquer master theorem for this problem. Note T n has two recurrence calls indicating a binary tree. Each step recursively calls the program for n reduced by 1 and 2, so the depth of the recurrence tree is O n. The number of leaves at depth n is 2n since this is a full binary tree, and each leaf takes at least O 1 computations for the constant factor.

Running time is clearly exponential in n and it is O 2n. Its running time is. First write the recurrence formula and then find its complexity. The recurrence for this code is. First write a recurrence formula, and show its solution using induction. Solution: Consider the comments in the function below: The if statement requires constant time [O 1 ]. With the for loop, we neglect the loop overhead and only count three times that the function is called recursively.

Now, the given function becomes: To make it simple we assume. This is same as that of Problem Solution: Consider the comments in below pseudo-code and call running time of function n as T n. T n can be defined as follows: Using the master theorem gives:. Solution: Consider the comments in the below function: Complexity of above program is: O nlogn. Solution: Consider the comments in the below function: The time complexity of this program is: O n2.

Note that, log n! Let us assume that the loop executes k times. After kth step the value of j is 2k. Taking logarithms on both sides gives. Since we are doing one more comparison for exiting from the loop, the answer is. Let T n denote the number of times the for loop is executed by the program on input n. Which of the following is true? Big O notation describes the tight upper bound and Big Omega notation describes the tight lower bound for an algorithm. How many recursive calls are made by this function?

Which one of the following is false? This indicates that tight lower bound and tight upper bound are the same. So option C is wrong. Solution: Start with 1 and multiply by 9 until reaching 9n.

Solution: Refer to the Divide and Conquer chapter. Solution: Let us solve this problem by method of guessing.

Solution: How much work do we do in each level of the recursion tree? In level 0, we take n2 time. At level 1, the two subproblems take time: At level 2 the four subproblems are of size and respectively. Let , the total runtime is then: That is, the first level provides a constant fraction of the total runtime. Time Complexity: O n2. Any function which calls itself is called recursive. A recursive method solves a problem by calling a copy of itself to work on a smaller problem.

This is called the recursion step. The recursion step can result in many more such recursive calls. It is important to ensure that the recursion terminates. Each time the function calls itself with a slightly simpler version of the original problem. The sequence of smaller problems must eventually converge on the base case. Recursion is a useful technique borrowed from mathematics. Recursive code is generally shorter and easier to write than iterative code.

Generally, loops are turned into recursive functions when they are compiled or interpreted. Recursion is most useful for tasks that can be defined in terms of similar subtasks. For example, sort, search, and traversal problems often have simple recursive solutions. At some point, the function encounters a subtask that it can perform without calling itself. This case, where the function does not recur, is called the base case. The former, where the function calls itself to perform a subtask, is referred to as the ecursive case.

We can write all recursive functions using the format: As an example consider the factorial function: n! The definition of recursive factorial looks like: This definition can easily be converted to recursive implementation. Here the problem is determining the value of n! In the recursive case, when n is greater than 1, the function calls itself to determine the value of n — l! In the base case, when n is 0 or 1, the function simply returns 1.

Once a method ends that is, returns some data , the copy of that returning method is removed from memory. The recursive solutions look simple but visualization and tracing takes time. For better understanding, let us consider the following example. The answer to this question depends on what we are trying to do. A recursive approach mirrors the problem that we are trying to solve.

A recursive approach makes it simpler to solve a problem that may not have the most obvious of answers. That means any problem that can be solved recursively can also be solved iteratively. By the time you complete reading the entire book, you will encounter many recursion problems. Solution: The Towers of Hanoi is a mathematical puzzle. It consists of three rods or pegs or towers , and a number of disks of different sizes which can slide onto any rod.

The puzzle starts with the disks on one rod in ascending order of size, the smallest at the top, thus making a conical shape. Once we solve Towers of Hanoi with three disks, we can solve it with any number of disks with the above algorithm. Solution: Time Complexity: O n. Space Complexity: O n for recursive stack space. Backtracking is an improvement of the brute force approach. It systematically searches for a solution to a problem among all available options.

In backtracking, we start with one possible option out of many available options and try to solve the problem if we are able to solve the problem with the selected move then we will print the solution else we will backtrack and select some other option and try to solve it. If none if the options work out we will claim that there is no solution for the problem. Backtracking is a form of recursion. The usual scenario is that you are faced with a number of options, and you must choose one of these.

This procedure is repeated over and over until you reach a final state. The tree is a way of representing some initial starting position the root node and a final goal state one of the leaves.

Backtracking allows us to deal with situations in which a raw brute-force approach would explode into an impossible number of options to consider. Backtracking is a sort of refined brute force. At each node, we eliminate choices that are obviously not possible and proceed to recursively check only those that have potential. In general, that will be at the most recent decision point.

Eventually, more and more of these decision points will have been fully explored, and we will have to backtrack further and further. If we backtrack all the way to our initial state and have explored all alternatives from there, we can conclude the particular problem is unsolvable.

In such a case, we will have done all the work of the exhaustive recursion and known that there is no viable solution possible. Assume A[ Assume function printf takes time O 1.

This means the algorithm for generating bit-strings is optimal. Solution: Let us assume we keep current k-ary string in an array A[ Call function k- string n, k : Let T n be the running time of k — string n.

Note: For more problems, refer to String Algorithms chapter. The filled cells that are connected form a region. Two cells are said to be connected if they are adjacent to each other horizontally, vertically or diagonally. There may be several regions in the matrix. How do you find the largest region in terms of number of cells in the matrix? Solution: The simplest idea is: for each location traverse in all 8 directions and in each of those directions keep track of maximum region found. Solution: At each level of the recurrence tree, the number of problems is double from the previous level, while the amount of work being done in each problem is half from the previous level.

Formally, the ith level has 2i problems, each requiring 2n—i work. Thus the ith level requires exactly 2n work. The depth of this tree is n, because at the ith level, the originating call will be T n — i.

Thus the total complexity for T n is T n2n. A linked list is a data structure used for storing collections of data. A linked list has the following properties.

It allocates memory as list grows. There are many other data structures that do the same thing as linked lists. Before discussing linked lists it is important to understand the difference between linked lists and arrays. Both linked lists and arrays are used to store collections of data, and since both are used for the same purpose, we need to differentiate their usage. That means in which cases arrays are suitable and in which cases linked lists are suitable.

The array elements can be accessed in constant time by using the index of the particular element as the subscript. To access an array element, the address of an element is computed as an offset from the base address of the array and one multiplication is needed to compute what is supposed to be added to the base address to get the memory address of the element. First the size of an element of that data type is calculated and then it is multiplied with the index of the element to get the value to be added to the base address.

This process takes one multiplication and one addition. Since these two operations take constant time, we can say the array access can be performed in constant time. This will create a position for us to insert the new element at the desired position.

If the position at which we want to add an element is at the beginning, then the shifting operation is more expensive. Dynamic Arrays Dynamic array also called as growable array, resizable array, dynamic table, or array list is a random access, variable-size list data structure that allows elements to be added or removed.

One simple way of implementing dynamic arrays is to initially start with some fixed size array. As soon as that array becomes full, create the new array double the size of the original array. Similarly, reduce the array size to half if the elements in the array are less than half.

Note: We will see the implementation for dynamic arrays in the Stacks, Queues and Hashing chapters. Advantages of Linked Lists Linked lists have both advantages and disadvantages. The advantage of linked lists is that they can be expanded in constant time.

To create an array, we must allocate memory for a certain number of elements. To add more elements to the array when full, we must create a new array and copy the old array into the new array. This can take a lot of time. We can prevent this by allocating lots of space initially but then we might allocate more than we need and waste memory.

With a linked list, we can start with space for just one allocated element and add on new elements easily without the need to do any copying and reallocating. Issues with Linked Lists Disadvantages There are a number of issues with linked lists. The main disadvantage of linked lists is access time to individual elements.

Array is random-access, which means it takes O 1 to access any element in the array. Linked lists take O n for access to an element in the list in the worst case. Another advantage of arrays in access time is spacial locality in memory. Arrays are defined as contiguous blocks of memory, and so any array element will be physically near its neighbors. This greatly benefits from modern CPU caching methods.

Although the dynamic allocation of storage is a great advantage, the overhead with storing and retrieving data can make a big difference. Sometimes linked lists are hard to manipulate. If the last item is deleted, the last but one must then have its pointer changed to hold a NULL reference. This requires that the list is traversed to find the last but one link, and its pointer set to a NULL reference.

Finally, linked lists waste memory in terms of extra reference points. This list consists of a number of nodes in which each node has a next pointer to the following element.

The link of the last node in the list is NULL, which indicates the end of the list. The ListLength function takes a linked list as input and counts the number of nodes in the list. The function given below can be used for printing the list data with extra print function. Time Complexity: O n , for scanning the list of size n. Space Complexity: O 1 , for creating a temporary variable. Inserting a Node in Singly Linked List at the Beginning In this case, a new node is inserted before the current head node.

Inserting a Node in Singly Linked List at the Ending In this case, we need to modify two next pointers last nodes next pointer and new nodes next pointer. Inserting a Node in Singly Linked List at the Middle Let us assume that we are given a position where we want to insert the new node. In this case also, we need to modify two next pointers. That means we traverse 2 nodes and insert the new node. For simplicity let us assume that the second node is called position node.

The new node points to the next node of the position where we want to add this node. Let us write the code for all three cases. We must update the first element pointer in the calling function, not just in the called function. For this reason we need to send a double pointer. The following code inserts a node in the singly linked list. Note: We can implement the three variations of the insert operation separately.

Time Complexity: O n , since, in the worst case, we may need to insert the node at the end of the list. Space Complexity: O 1 , for creating one temporary variable. This operation is a bit trickier than removing the first node, because the algorithm should find a node, which is previous to the tail.

By the time we reach the end of the list, we will have two pointers, one pointing to the tail node and the other pointing to the node before the tail node. Deleting an Intermediate Node in Singly Linked List In this case, the node to be removed is always located between two nodes.

Head and tail links are not updated in this case. Time Complexity: O n. In the worst case, we may need to delete the node at the end of the list. Space Complexity: O 1 , for one temporary variable. After freeing the current node, go to the next node with a temporary variable and repeat this process for all nodes. Time Complexity: O n , for scanning the complete list of size n. A node in a singly linked list cannot be removed unless we have the pointer to its predecessor.

Similar to a singly linked list, let us implement the operations of a doubly linked list. If you understand the singly linked list operations, then doubly linked list operations are obvious.

Inserting a Node in Doubly Linked List at the Middle As discussed in singly linked lists, traverse the list to the position node and insert the new node. Also, new node left pointer points to the position node. Now, let us write the code for all of these three cases. In the worst case, we may need to insert the node at the end of the list.

Then, dispose of the temporary node. Deleting the Last Node in Doubly Linked List This operation is a bit trickier than removing the first node, because the algorithm should find a node, which is previous to the tail first.