1. Advanced Data Structure using C++

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LECTURE NOTES OF ADVANCED DATA STRUCTURE (MT-CSE 110) the node is split into two nodes, with half the key values and pointer going into one node while the other half going into another node. These twin nodes have the same parent, which is modified for the additional key value and pointer. In the worst case, the splitting procedure may need to be carried all the way up the root of the tree. In that case, the height of the tree increases by one level. This insertion procedure is implemented as a recursion since the splitting procedure is the same, irrespective of the level. A B Tree insertion example with each iteration. All insertions start at a leaf node. To insert a new element Search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps: 1. If the node contains fewer than the maximum legal number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered. 2. Otherwise the node it is full, so evenly split it into two nodes. 1. A single median is chosen from among the leaf's elements and the new element. Prepared By : Er. Harvinder Singh Assist Prof., CSE, H.C.T.M (Kaithal) Page ‐ 216 ‐
LECTURE NOTES OF ADVANCED DATA STRUCTURE (MT-CSE 110) 2. Values less than the median are put in the new left node and values greater than the median are put in the new right node, with the median acting as a separation value. 3. Insert the separation value in the node's parent, which may cause it to be split, and so on. If the node has no parent (i.e., the node was the root), create a new root above this node (increasing the height of the tree). If the splitting goes all the way up to the root, it creates a new root with a single separator value and two children, which is why the lower bound on the size of internal nodes does not apply to the root. The maximum number of elements per node is U‐1. When a node is split, one element moves to the parent, but one element is added. So, it must be possible to divide the maximum number U‐1 of elements into two legal nodes. If this number is odd, then U=2L and one of the new nodes contains (U‐2)/2 = L‐1 elements, and hence is a legal node, and the other contains one more element, and hence it is legal too. If U‐1 is even, then U=2L‐1, so there are 2L‐2 elements in the node. Half of this number is L‐1, which is the minimum number of elements allowed per node. An improved algorithm supports a single pass down the tree from the root to the node where the insertion will take place, splitting any full nodes encountered on the way. This prevents the need to recall the parent nodes into memory, which may be expensive if the nodes are on secondary storage. However, to use this improved algorithm, we must be able to send one element to the parent and split the remaining U‐2 elements into two legal nodes, without adding a new element. This requires U = 2L rather than U = 2L‐1, which accounts for why some textbooks impose this requirement in defining B‐trees. Deletion There are two popular strategies for deletion from a B‐Tree. • locate and delete the item, then restructure the tree to regain its invariants Prepared By : Er. Harvinder Singh Assist Prof., CSE, H.C.T.M (Kaithal) Page ‐ 217 ‐
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ADVANCED DATA STRUCTURE NOTES (MT
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1. Advanced Data Structure using C++

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