B-TreesMotivation for B-TreesMotivation (cont.)Bottom line …Slide 5Definition of a B-treeAn example B-TreeConstructing a B-treeConstructing a B-tree (contd.)Slide 10Slide 11Slide 12Inserting into a B-TreeExercise in Inserting a B-TreeRemoval from a B-treeRemoval from a B-tree (2)Type #1: Simple leaf deletionType #2: Simple non-leaf deletionType #4: Too few keys in node and its siblingsSlide 20Type #3: Enough siblingsSlide 22Exercise in Removal from a B-TreeAnalysis of B-TreesReasons for using B-TreesComparing TreesB*-treesB-TreesCSCI 2720KraemerFall 2005CSCI 27202Motivation for B-TreesThus far we have assumed that we can store an entire data structure in main memoryWhat if we have so much data that it won’t fit?We will have to use disk storage but when this happens our time complexity failsThe problem is that Big-Oh analysis assumes that all operations take roughly equal timeThis is not the case when disk access is involvedCSCI 27203Motivation (cont.)Assume that a disk spins at 3600 RPMone revolution occurs in 1/60 of a second, 16.7msOn average what we want is half way round this disk takes 8msSounds good until:you realize that we get 120 disk accesses a second – the same time as 25 million instructionsCSCI 27204Bottom line …One disk access takes about the same time as 200,000 instructionsIt is worth executing many instructions to avoid a disk accessCSCI 27205Motivation (cont.)Assume that we use an AVL tree to store records on the 20 million owners of iPods We still end up with a very deep tree with lots of different disk accesses; log2 20,000,000 is about 24, so this takes about 0.2 seconds (if there is only one user of the program)We know we can’t improve on the log n for a binary treeBut, the solution is to use more branches and thus less height!As branching increases, depth decreasesCSCI 27206Definition of a B-treeA B-tree of order m is an m-way tree (i.e., a tree where each node may have up to m children) in which:1. the number of keys in each non-leaf node is one less than the number of its children and these keys partition the keys in the children in the fashion of a search tree2. all leaves are on the same level3. all non-leaf nodes except the root have at least m / 2 children4. the root is either a leaf node, or it has from two to m children5. a leaf node contains no more than m – 1 keysThe number m should always be oddCSCI 27207An example B-Tree51 62426 122655 607064 90451 2 4 7 8 13 15 18 2527 29 46 48 53A B-tree of order 5 containing 26 itemsNote that all the leaves are at the same levelNote that all the leaves are at the same levelCSCI 27208Suppose we start with an empty B-tree and keys arrive in the following order: 1 12 8 2 25 5 14 28 17 7 52 16 48 68 3 26 29 53 55 45We want to construct a B-tree of order 5The first four items go into the root:To put the fifth item in the root would violate condition 5Therefore, when 25 arrives, pick the middle key to make a new rootConstructing a B-tree1 2 8 12CSCI 27209Constructing a B-tree (contd.)1 2812 256, 14, 28 get added to the leaf nodes:1 2812 146 25 28CSCI 272010Constructing a B-tree (contd.)Adding 17 to the right leaf node would over-fill it, so we take the middle key, promote it (to the root) and split the leaf8 1712 14 25 281 2 67, 52, 16, 48 get added to the leaf nodes8 1712 14 25 281 2 6 16 48 527CSCI 272011Constructing a B-tree (contd.)Adding 68 causes us to split the right most leaf, promoting 48 to the root, and adding 3 causes us to split the left most leaf, promoting 3 to the root; 26, 29, 53, 55 then go into the leaves3 8 17 4852 53 55 6825 26 28 291 2 6 7 12 14 16Adding 45 causes a split of 25 26 28 29and promoting 28 to the root then causes the root to splitCSCI 272012Constructing a B-tree (contd.)173 8 28 481 2 6 7 12 14 16 52 53 55 6825 26 29 45CSCI 272013Inserting into a B-TreeAttempt to insert the new key into a leafIf this would result in that leaf becoming too big, split the leaf into two, promoting the middle key to the leaf’s parentIf this would result in the parent becoming too big, split the parent into two, promoting the middle keyThis strategy might have to be repeated all the way to the topIf necessary, the root is split in two and the middle key is promoted to a new root, making the tree one level higherCSCI 272014Exercise in Inserting a B-Tree Insert the following keys to a 5-way B-tree:3, 7, 9, 23, 45, 1, 5, 14, 25, 24, 13, 11, 8, 19, 4, 31, 35, 56Check your approach with a neighbour and discuss any differences.CSCI 272015Removal from a B-treeDuring insertion, the key always goes into a leaf. For deletion we wish to remove from a leaf. There are three possible ways we can do this:1 - If the key is already in a leaf node, and removing it doesn’t cause that leaf node to have too few keys, then simply remove the key to be deleted.2 - If the key is not in a leaf then it is guaranteed (by the nature of a B-tree) that its predecessor or successor will be in a leaf -- in this case can we delete the key and promote the predecessor or successor key to the non-leaf deleted key’s position.CSCI 272016Removal from a B-tree (2)If (1) or (2) lead to a leaf node containing less than the minimum number of keys then we have to look at the siblings immediately adjacent to the leaf in question: 3: if one of them has more than the min number of keys then we can promote one of its keys to the parent and take the parent key into our lacking leaf 4: if neither of them has more than the min’ number of keys then the lacking leaf and one of its neighbours can be combined with their shared parent (the opposite of promoting a key) and the new leaf will have the correct number of keys; if this step leaves the parent with too few keys then we repeat the process up to the root itself, if requiredCSCI 272017Type #1: Simple leaf deletion1212292952522277991515222256566969727231314343Delete 2: Since there are enoughkeys in the node, just delete itAssuming a 5-wayB-Tree, as before...Note when printed: this slide is animatedCSCI 272018Type #2: Simple non-leaf deletion12122929525277991515222256566969727231314343Delete 52Borrow the predecessoror (in this case) successor5656Note when printed: this slide is animatedCSCI 272019Type #4: Too few keys in node and its siblings1212292956567799151522226969727231314343Delete 72Too few keys!Join back togetherNote when