Data Structure Algorithms Articles - Page 491 of 25

In this section we will see some important properties of one binary tree data structure. Suppose we have a binary tree like this.Some properties are −The maximum number of nodes at level ‘l’ will be $2^{l-1}$ . Here level is the number of nodes on path from root to the node, including the root itself. We are considering the level of root is 1.Maximum number of nodes present in binary tree of height h is $2^{h}-1$ . Here height is the max number of nodes on root to leaf path. Here we are considering height of a tree with one ... Read More

Here we will see how to represent a binary tree in computers memory. There are two different methods for representing. These are using array and using linked list.Suppose we have one tree like this −The array representation stores the tree data by scanning elements using level order fashion. So it stores nodes level by level. If some element is missing, it left blank spaces for it. The representation of the above tree is like below −12345678910111213141510516-81520-------23The index 1 is holding the root, it has two children 5 and 16, they are placed at location 2 and 3. Some children are ... Read More

Here we will see some basic operations of array data structure. These operations are −TraverseInsertionDeletionSearchUpdateThe traverse is scanning all elements of an array. The insert operation is adding some elements at given position in an array, delete is deleting element from an array and update the respective positions of other elements after deleting. The searching is to find some element that is present in an array, and update is updating the value of element at given position. Let us see one C++ example code to get better idea.Example Live Demo#include #include using namespace std; main(){    vector arr;    //insert elements ... Read More

Amortize AnalysisThis analysis is used when the occasional operation is very slow, but most of the operations which are executing very frequently are faster. In Data structures we need amortized analysis for Hash Tables, Disjoint Sets etc.In the Hash-table, the most of the time the searching time complexity is O(1), but sometimes it executes O(n) operations. When we want to search or insert an element in a hash table for most of the cases it is constant time taking the task, but when a collision occurs, it needs O(n) times operations for collision resolution.Aggregate MethodThe aggregate method is used to ... Read More

In this section we will see how to multiply two matrices. The matrix multiplication can only be performed, if it satisfies this condition. Suppose two matrices are A and B, and their dimensions are A (m x n) and B (p x q) the resultant matrix can be found if and only if n = p. Then the order of the resultant matrix C will be (m x q).AlgorithmmatrixMultiply(A, B): Assume dimension of A is (m x n), dimension of B is (p x q) Begin    if n is not same as p, then exit    otherwise define C ... Read More

The step count method is one of the method to analyze the algorithm. In this method, we count number of times one instruction is executing. From that we will try to find the complexity of the algorithm.Suppose we have one algorithm to perform sequential search. Suppose each instruction will take c1, c2, …. amount of time to execute, then we will try to find out the time complexity of this algorithmAlgorithmNumber of timesCostseqSearch(arr, n, key)i := 0while i < n, do   if arr[i] = key, then      break   end ifdonereturn i1n+1n0/11c1c2c3c4c5Now if we add the cost by multiplying the ... Read More

Fibonacci numbers are the numbers such that every number in the series after the first two is the sum of the two preceding ones. The series starts with 1, 1. Example −1, 1, 2, 3, 5, 8, 13, 21, 34, ….We can write a program to generate nth as follows −functionfibNaive(n) {    if (n

Dynamic programming breaks down the problem into smaller and yet smaller possible sub-problems. These sub-problems are not solved independently. Rather, results of these smaller sub-problems are remembered and used for similar or overlapping sub-problems. Dynamic programming is used where we have problems, which can be divided into similar sub-problems so that their results can be re-used. Mostly, these algorithms are used for optimization. Before solving the in-hand sub-problem, the dynamic algorithm will try to examine the results of the previously solved sub-problems. The solutions of sub-problems are combined in order to achieve the best solution. For a problem to be ... Read More

Like the binary search, it also separates the lists into sub-lists. This procedure divides the list into three parts using two intermediate mid values. As the lists are divided into more subdivisions, so it reduces the time to search a key value.The complexity of Ternary Search TechniqueTime Complexity: O(log3 n)Space Complexity: O(1)Input and OutputInput: A sorted list of data: 12 25 48 52 67 79 88 93 The search key 52 Output: Item found at location: 3AlgorithmternarySearch(array, start, end, key)Input − An sorted array, start and end location, and the search keyOutput − location of the key (if found), otherwise wrong ... Read More

Exponential search is also known as doubling or galloping search. This mechanism is used to find the range where the search key may present. If L and U are the upper and lower bound of the list, then L and U both are the power of 2. For the last section, the U is the last position of the list. For that reason, it is known as exponential.After finding the specific range, it uses the binary search technique to find the exact location of the search key.The complexity of Exponential Search TechniqueTime Complexity: O(1) for the best case. O(log2 i) ... Read More