Found 210 Articles for Analysis of Algorithms

In this section we will see what is the hashing by open addressing. The open addressing is another technique for collision resolution. Unlike chaining, it does not insert elements to some other data-structures. It inserts the data into the hash table itself. The size of the hash table should be larger than the number of keys.There are three different popular methods for open addressing techniques. These methods are −Linear ProbingQuadratic ProbingDouble HashingIn this technique, we use a hash function like other hash techniques. If the place is free, then insert the element into that location. Now if that place is ... Read More

In this section we will see what is the hashing with chaining. The Chaining is one collision resolution technique. We cannot avoid collision, but we can try to reduce the collision, and try to store multiple elements for same hash value.this technique suppose our hash function h(x) ranging from 0 to 6. So for more than 7 elements, there must be some elements, that will be places inside the same room. For that we will create a list to store them accordingly. In each time we will add at the beginning of the list to perform insertion in O(1) timeLet ... Read More

For any hash function we can say that if the table size m is much smaller than universe size u, then for any hash function h, there is some large subset of U that has the same hash value.To get rid of this problem, we need a set of hash functions, from which we can choose any one that works well for S. If most of the hash functions are better for S, we can choose random hash functionSuppose ℌ be a set of hash functions. We can say ℌ is universal if, for each x, y ∈ U, the ... Read More

Here we will discuss about the hashing with multiplication method. For this we use the hash function −ℎ(𝑥) = ⌊𝑚𝑥𝐴⌋ 𝑚𝑜𝑑 𝑚Here A is a real-valued constant. The advantage of this method is that the value of m is not so critical. We can take m as power of 2 also. Although any value of A gives the hash function, but some values of A are better than others.According to Knuth, we can use the golden ratio for A, So A will be$$A=\frac{\sqrt5-1}{2}=0.61803398$$Of course, no matter what value is chosen for A. The pigeonhole principle implies that if u ≥ ... Read More

Here we will discuss about the hashing with division. For this we use the hash function −ℎ(𝑥) = 𝑥 𝑚𝑜𝑑 𝑚To use this hash function we maintain an array A[0, … m – 1]. Where each element of the array is a pointer to the head of the linked list. The linked list Li is pointed to array element A[i] holds all elements x such that h(x) = i. This technique is known as hashing by chaining.In such hash table, if we want to increase an element x, this will take O(1) time. we compute the index i = h(x). ... Read More

Here we will discuss about the Hash tables with the integer keys. Here the key values 𝑥 comes from universe 𝑈 such that 𝑈 = {0, 1, … , 𝑢 – 2, 𝑢 – 1}. A hash function is ℎ. The domain of this hash function is 𝑈. The range is in the set {0, 1, … , 𝑚 – 1}, and 𝑚 ≤ 𝑢.A hash function h is said to be a perfect hash function for a set 𝑆 ⊆ 𝑈 if for every 𝑥 ∈ 𝑆, ℎ(𝑥) is unique. A perfect hash function ℎ for 𝑆 is minimal ... Read More

Here we will see how to delete elements from binary max heap data structures. Suppose the initial tree is like below −Deletion Algorithmdelete(heap, n) − Begin    if heap is empty, then exit    else       item := heap[1]       last := heap[n]       n := n – 1       for i := 1, j := 2, j = heap[j], then break          heap[i] := heap[j]       done    end if    heap[i] := last EndExampleSuppose we want to delete 30 from the final heap −

Here we will see how to insert and elements from binary max heap data structures. Suppose the initial tree is like below −Insertion Algorithminsert(heap, n, item) − Begin    if heap is full, then exit    else       n := n + 1       for i := n, i > 1, set i := i / 2 in each iteration, do          if item

Like Binomial heaps, Fibonacci heaps are collection of trees. They are loosely based on binomial heaps. Unlike trees with in binomial heaps are ordered trees within Fibonacci heaps are rooted but unordered.Each node x in Fibonacci heaps contains a pointer p[x] to its parent, and a pointer child[x] to any one of its children. The children of x are linked together in a circular doubly linked list known as child list of x. Each child y in a child list has pointers left[y] and right[y] to point left and right siblings of y respectively. If node y is only child ... Read More

A binomial Heap is a collection of Binomial Trees. A binomial tree Bk is an ordered tree defined recursively. A binomial Tree B0 is consists of a single node.A binomial tree Bk is consisting of two binomial tree Bk-1. That are linked together. The root of one is the left most child of the root of the otherSome binomial heaps are like below −Some properties of binomial trees are −Binomial tree with Bk has 2k nodes.Height of the tree is kThere are exactly $$\left(\begin{array}{c}k\ j\end{array}\right)$$ nodes at depth i for all i in range 0 to kBinomial HeapA binomial heap ... Read More