DAA - Optimal Cost Binary Search Trees

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In computer sciencebinary search trees BSTsometimes called ordered or sorted binary treesare a particular type of container: They allow fast lookup, addition and removal of items, and can be used to implement either dynamic sets of items, or lookup tables that allow finding an item by its key e.

Binary search trees keep their keys in sorted order, so that lookup and other operations can use the principle of binary search: On average, this means that each comparison allows the operations to skip about half of the tree, so that each lookup, insertion or deletion takes time proportional to the logarithm of the number of items stored in the tree.

This is much better than the linear time required to find items by key in an unsorted array, but slower than the corresponding operations on hash tables. Several variants of the binary search tree have been studied in computer science; this article deals primarily with the basic type, making references to more advanced types when appropriate.

A binary search tree is a rooted binary treewhose internal nodes each store a key and optionally, an associated value and each have two distinguished sub-trees, commonly denoted left and right. The tree additionally satisfies the binary search property, which states that the key in each node must be greater than or equal to any key stored in the left sub-tree, and less than or equal to any key stored in the right sub-tree.

Frequently, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their associated records.

The major advantage of optimal binary search tree algorithm using dynamic programming ppt search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient; they are also easy to code. Binary search trees are a fundamental data structure used to construct more abstract data structures such as setsmultisetsand associative arrays.

Binary search requires an order relation by which every element item can be compared with every other element in the sense of a total preorder. The part of the element which effectively takes place in the comparison is called its key. In the context of binary search trees a total preorder is realized most flexibly by means of a three-way comparison subroutine. Binary search trees support three main operations: Searching a binary search tree for a specific key can be programmed recursively or iteratively.

We begin by examining the root node. If the tree is nullthe key we are searching for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and we return the node.

If the key is less than that of the root, we search the left subtree. Similarly, if the key is greater than that of the root, we search the right subtree. This process is repeated until the key is found or the remaining subtree is null.

If the searched key is not found after a null subtree is reached, then the key is not present in the tree. This is easily expressed as a recursive algorithm implemented in Python:. If the order relation is only a total preorder a reasonable extension of the functionality is the following: A binary tree sort equipped with such a comparison function becomes stable.

Because in the worst case this algorithm must search from the root of the tree to the leaf farthest from the root, the search operation takes time proportional to the tree's height see tree terminology. On average, binary search trees with n nodes have O log n height. Insertion begins as a search would begin; if the key is not equal to that of the root, we search the left or right subtrees as before.

Eventually, we will reach an external node and add the new key-value pair here encoded as a record 'newNode' as its right or left child, depending on the node's key. In other words, we examine the root and recursively insert the new node to the left subtree if its key is less than that of the root, or the right subtree if its key is greater than or equal to the root.

The above destructive procedural variant modifies the tree in place. It uses only constant heap space and the iterative version uses constant stack space as wellbut the prior version of the tree is lost. Alternatively, as in the following Python example, we can reconstruct all ancestors of the inserted node; any reference to the original tree root remains valid, making the tree a persistent data structure:.

The part that is rebuilt uses O log n space in the average case and O n in the worst case. In either version, this operation requires time proportional to the height of the tree in the worst case, which is O log n time in the average case over all trees, but O n time in the worst case.

Another way to explain insertion is that in order to insert a new node in the tree, its key is first compared with that of the root. If its key is less than the root's, it is then compared with the key of the root's left child. If its key is greater, it is compared with the root's right child. This process continues, until the new node is compared with a leaf node, and then it is added as this node's right or left child, optimal binary search tree algorithm using dynamic programming ppt on its key: There are other ways of inserting nodes into a binary tree, but this is the only way of inserting nodes at the leaves and at the same time preserving the BST structure.

When removing a node from a binary search tree it is mandatory to maintain the in-order sequence of the nodes. There are many possibilities to do this. However, the following method which has been proposed by T. Hibbard in [2] guarantees that the heights of the subject subtrees are changed by at most one. There are three possible cases to consider:. Broadly speaking, nodes with children are harder to delete. As with all binary trees, a node's in-order successor is its right subtree's left-most child, and a node's in-order predecessor is the optimal binary search tree algorithm using dynamic programming ppt subtree's right-most child.

In either case, this node will have only one or no child at all. Delete it according to one of the two simpler cases above. Consistently using the in-order successor or the in-order predecessor for every instance of the two-child case can lead to an unbalanced tree, so some implementations select one or the other at different times. Although this operation does not always traverse the tree down to a leaf, this is always a possibility; thus in the worst case it requires time proportional to the height of the tree.

It does not require more even when the node has two children, since it still follows a single path and does not visit any node twice. Once the binary search tree has optimal binary search tree algorithm using dynamic programming ppt created, its elements can be retrieved in-order by recursively traversing the left subtree of the root node, accessing the node itself, then recursively traversing the right subtree of the node, continuing this pattern with each node in the tree as it's recursively accessed.

As with all binary trees, one may conduct a pre-order traversal or a post-order traversalbut neither are likely to be useful for binary search trees. An in-order traversal of a binary search tree will always result in a sorted list of node items numbers, strings or other comparable items. The code for in-order traversal in Python is given below. It will call callback some function the programmer wishes to call on the node's value, such as printing to the screen for every node in the tree.

Traversal requires O n time, since it must visit every node. This algorithm is also O nso it is asymptotically optimal. Traversal can also be implemented iteratively. For certain applications, e. This is, of course, implemented without the callback construct and takes O 1 on average and O log n in the worst case. Sometimes we already have a binary tree, and we need to determine whether it is a BST. This problem has a simple recursive solution. The BST property—every node on the right subtree has to be larger than the current node and every node on the left subtree has to be smaller than the current node—is the key to figuring out whether a tree is a BST or not.

The greedy algorithm —simply traverse the tree, at every node check whether the node contains a value larger than the value at the left child and optimal binary search tree algorithm using dynamic programming ppt than the value on the right child—does not work for all cases. Consider the following tree:. In the tree above, each node meets the condition that the node contains a value larger than its left child and smaller than its right child hold, optimal binary search tree algorithm using dynamic programming ppt yet it is not a BST: Instead of making a decision based solely on the values of a node and its children, we also need information flowing down from the parent as well.

In the case of the tree above, if we could remember about the node containing the value 20, we would see that the node with value 5 is violating the BST property contract. As pointed out in section Traversalan in-order traversal of a binary search tree returns the nodes sorted. A binary search tree can be used to implement a simple sorting algorithm.

Similar to heapsortwe insert all optimal binary search tree algorithm using dynamic programming ppt values we wish to sort into a new ordered data structure—in this case a binary search tree—and then traverse it in order. There are several schemes for overcoming this flaw with simple binary trees; the most common is the self-balancing binary search tree. If this same procedure is done using such a tree, the overall worst-case time is O n log nwhich is asymptotically optimal for a comparison sort.

In optimal binary search tree algorithm using dynamic programming ppt, the added overhead in time and space for a tree-based sort particularly for node allocation make it inferior to other asymptotically optimal sorts such as heapsort for static list sorting. On the other hand, it is one of the most efficient methods of incremental sortingadding items to a list over time while keeping the list sorted at all times.

Binary search trees can serve as priority queues: Insertion works as previously explained. Find-min walks the tree, following left pointers as far as it optimal binary search tree algorithm using dynamic programming ppt without hitting a leaf:.

Delete-min max can simply look up the minimum maximumthen delete it. This way, insertion and deletion both take logarithmic time, just as they do in a binary heapbut unlike a binary heap and most other priority queue implementations, a single tree can support all of find-minfind-maxdelete-min and delete-max at the same time, making binary search trees suitable as double-ended priority queues. There are many types of binary search trees.

AVL trees and red-black trees are both forms of self-balancing binary search trees. A splay tree is a binary search tree that automatically moves frequently accessed elements nearer to the root.

In a treap optimal binary search tree algorithm using dynamic programming ppt heapeach node also holds a randomly chosen priority and the parent node has higher priority than its children. Tango trees are trees optimized for fast searches. T-trees are binary search trees optimized to reduce storage space overhead, widely used for in-memory databases. A degenerate tree is a tree where for each parent node, there is only one associated child node. It is unbalanced and, in the worst case, performance degrades to that of a linked list.

If your add node function does not handle re-balancing, then you can easily construct a degenerate tree by feeding it with data that is already sorted. What this means is that in a performance measurement, the tree will essentially behave like a linked list data structure. Heger [4] presented a performance comparison of binary search trees. Treap was found to have the best average performance, while red-black tree was found to have the smallest amount of performance variations.

If we do not plan on modifying a search tree, and we know exactly how often optimal binary search tree algorithm using dynamic programming ppt item will be accessed, we can construct [5] an optimal binary search tree optimal binary search tree algorithm using dynamic programming ppt, which is a search tree where the average cost of looking up an item the optimal binary search tree algorithm using dynamic programming ppt search cost is minimized.

Even if we only have estimates of the search costs, such a system can considerably speed up lookups on average. For example, if you have a BST of English words used in a spell checkeryou might balance the tree based on word frequency in text corporaplacing words like the near the root and words like agerasia near the leaves.

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