1. Advanced Data Structure using C++

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LECTURE NOTES OF ADVANCED DATA STRUCTURE (MT-CSE 110) • The space required to store the tree is O(n) • Inserting a record requires O(logbn) operations in the worst case • Finding a record requires O(logbn) operations in the worst case • Removing a (previously located) record requires O(logbn) operations in the worst case • Performing a range query with k elements occurring within the range requires O(logbn + k) operations in the worst case. Implementation The leaves (the bottom‐most index blocks) of the B+ tree are often linked to one another in a linked list; this makes range queries simpler and more efficient (though the aforementioned upper bound can be achieved even without this addition). This does not substantially increase space consumption or maintenance on the tree. If a storage system has a block size of B bytes, and the keys to be stored have a size of k, arguably the most efficient B+ tree is one where b = (B / k) − 1. Although theoretically the one‐off is unnecessary, in practice there is often a little extra space taken up by the index blocks (for example, the linked list references in the leaf blocks). Having an index block which is slightly larger than the storage system's actual block represents a significant performance decrease; therefore erring on the side of caution is preferable. If nodes of the B+ tree are organised as arrays of elements, then it may take a considerable time to insert or delete an element as half of the array will need to be shifted on average. To overcome this problem, elements inside a node can be organized in a binary tree or a B+ tree instead of an array. B+ trees can also be used for data stored in RAM. In this case a reasonable choice for block size would be the size of processor's cache line. However some studies have proved that a block size few times larger than processor's cache line can deliver better performance if cache prefetching is used. Space efficiency of B+ trees can be improved by using some compression techniques. One possibility is to use delta encoding to compress keys stored into Prepared By : Er. Harvinder Singh Assist Prof., CSE, H.C.T.M (Kaithal) Page ‐ 222 ‐
LECTURE NOTES OF ADVANCED DATA STRUCTURE (MT-CSE 110) each block. For internal blocks, space saving can be achieved by either compressing keys or pointers. For string keys, space can be saved by using the following technique: Normally the ith entry of an internal block contains the first key of block i+1. Instead of storing the full key, we could store the shortest prefix of the first key of block i+1 that is strictly greater (in lexicographic order) than last key of block i. There is also a simple way to compress pointers: if we suppose that some consecutive blocks i, i+1...i+k are stored contiguously, then it will suffice to store only a pointer to the first block and the count of consecutive blocks. All the above compression techniques have some drawbacks. First, a full block must be decompressed to extract a single element. One technique to overcome this problem is to divide each block into sub‐blocks and compress them separately. In this case searching or inserting an element will only need to decompress or compress a sub‐block instead of a full block. Another drawback of compression techniques is that the number of stored elements may vary considerably from a block to another depending on how well the elements are compressed inside each block. Deletion Before presenting pseudocode we provide a basic flowchart and algorithm to illuminate its function. Figure 1 shows how the initial downwards recursive search is followed by an upwards unwinding of the recursion, during which the deletion, and potentially the rebalancing of the tree, takes place. The second phase corresponds to the shaded area of the figure. A set of immediate neighbors and anchors, defined below, is calculated during the search phase, for use during the tree rebalancing. The algorithm outline is as follows: 1. recurse to a leaf node from root to find deletable entry: for nodes in the search path, calculate immediate neighbors and their anchors. 2. if entry found at leaf node continue else stop 3. remove appropriate entry fiom current node Prepared By : Er. Harvinder Singh Assist Prof., CSE, H.C.T.M (Kaithal) Page ‐ 223 ‐
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1. Advanced Data Structure using C++

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