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Edit Comment Close Premium member Presentation Transcript ADVANCED DATA STRUCTURES : ADVANCED DATA STRUCTURES B-Trees B-Trees 1 SOORAJ KUMAR 07-0113-CS Contents : Contents Idea Definition Construction Insertion Deletion Analysis of B-trees Time analysis B-Trees 2 Why do we use B-trees : Why do we use B-trees It was difficult to access a large amount of data from the secondary memory Many of the algorithms were introduced to make our search very fast, to access the required data from the secondary memory B-trees are more effective and faster B-trees are used in many of the database management system B-Trees 3 Factors that affect the time : Factors that affect the time Many factors can effect the speed of accessing data within the secondary memory The moving of arms on the platters Finding the track and sector on the secondary memory’s platters Read and write time B-Trees 4 Slide 5: B-Trees 5 Definition of a B-tree : B-Trees 6 Definition 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 tree 2. all leaves are on the same level 3. all non-leaf nodes except the root have at least ?m / 2? children 4. the root is either a leaf node, or it has from two to m children 5. a leaf node contains no more than m – 1 keys The number m should always be odd An example B-Tree : B-Trees 7 An example B-Tree 51 62 42 6 12 26 55 60 70 64 90 45 1 2 4 7 8 13 15 18 25 27 29 46 48 53 A B-tree of order 5 containing 26 items Note that all the leaves are at the same level Constructing a B-tree : B-Trees 8 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 root Constructing a B-tree 1 2 8 12 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 root Constructing a B-tree (contd.) : B-Trees 9 Constructing a B-tree (contd.) 1 2 8 12 25 Constructing a B-tree (contd.) : B-Trees 10 Constructing 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 leaf 8 17 12 14 25 28 1 2 6 Constructing a B-tree (contd.) : B-Trees 11 Constructing 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 leaves 3 8 17 48 52 53 55 68 25 26 28 29 1 2 6 7 12 14 16 Constructing a B-tree (contd.) : B-Trees 12 Constructing a B-tree (contd.) 17 3 8 28 48 1 2 6 7 12 14 16 52 53 55 68 25 26 29 45 Inserting into a B-Tree : B-Trees 13 Inserting 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 higher Inserting into a B-Tree : B-Trees 14 Inserting into a B-Tree Removal from a B-tree : B-Trees 15 Removal 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 we can delete the key and promote the predecessor or successor key to the non-leaf deleted key’s position. Removal from a B-tree (2) : B-Trees 16 Removal 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 leave the parent with too few keys then we repeat the process up to the root itself, if required Type #1: Simple leaf deletion : B-Trees 17 Type #1: Simple leaf deletion Delete 2: Since there are enough keys in the node, just delete it Assuming a 5-way B-Tree, as before... Type #2: Simple non-leaf deletion : B-Trees 18 Type #2: Simple non-leaf deletion Delete 52 Borrow the predecessor or (in this case) successor 56 Type #4: Too few keys in node and its siblings : B-Trees 19 Type #4: Too few keys in node and its siblings Delete 72 Too few keys! Type #4: Too few keys in node and its siblings : B-Trees 20 Type #4: Too few keys in node and its siblings Type #3: Enough siblings : B-Trees 21 Type #3: Enough siblings Delete 22 Type #3: Enough siblings : B-Trees 22 Type #3: Enough siblings 12 29 7 9 15 31 Algorithm for deletion : Algorithm for deletion B-Trees 23 Continued… : Continued… B-Trees 24 Analysis of B-Trees : B-Trees 25 Analysis of B-Trees The maximum number of items in a B-tree of order m and height h: root m – 1 level 1 m(m – 1) level 2 m2(m – 1) . . . level h mh(m – 1) So, the total number of items is (1 + m + m2 + m3 + … + mh)(m – 1) = [(mh+1 – 1)/ (m – 1)] (m – 1) = mh+1 – 1 When m = 5 and h = 2 this gives 53 – 1 = 124 B-tree insert runtime : • O(k) runtime per node • Path has height h = O(logk n) • CPU-time: O(k logk n) • O(k) runtime per node • Path has height h = O(logk n) • CPU-time: O(k logk n) B-Trees 26 B-tree search runtime B-tree insert runtime Time Complexity of Operations in B-tree : Time Complexity of Operations in B-tree Search/Insert/Delete all take up to the number of items in a path from the root to a leaf. The total number of operations is no more than the height of the tree The height of a tree is no more than log(n) where n is the number of items in a B-tree. B-Trees 27 Reasons for using B-Trees : B-Trees 28 Reasons for using B-Trees When searching tables held on disc, the cost of each disc transfer is high but doesn't depend much on the amount of data transferred, especially if consecutive items are transferred If we use a B-tree of order 101, say, we can transfer each node in one disc read operation A B-tree of order 101 and height 3 can hold 1014 – 1 items (approximately 100 million) and any item can be accessed with 3 disc reads (assuming we hold the root in memory) If we take m = 3, we get a 2-3 tree, in which non-leaf nodes have two or three children (i.e., one or two keys) B-Trees are always balanced (since the leaves are all at the same level), so 2-3 trees make a good type of balanced tree Slide 29: B-Trees 29 Thanks You do not have the permission to view this presentation. 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B-Trees Presentation aSGuest18521 Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 4586 Category: Education License: All Rights Reserved Like it (8) Dislike it (0) Added: May 13, 2009 This Presentation is Public Favorites: 2 Presentation Description No description available. Comments Posting comment... By: shilpimca (7 month(s) ago) u plz send this ppt to my id: shilpi.22aug.mca@gmail.com thx..... Saving..... Post Reply Close Saving..... Edit Comment Close By: shyambtech (8 month(s) ago) hi,please can u send this(B_TREE) presentation(shyambtech@yahoo.co.in) Saving..... Post Reply Close Saving..... 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Edit Comment Close Premium member Presentation Transcript ADVANCED DATA STRUCTURES : ADVANCED DATA STRUCTURES B-Trees B-Trees 1 SOORAJ KUMAR 07-0113-CS Contents : Contents Idea Definition Construction Insertion Deletion Analysis of B-trees Time analysis B-Trees 2 Why do we use B-trees : Why do we use B-trees It was difficult to access a large amount of data from the secondary memory Many of the algorithms were introduced to make our search very fast, to access the required data from the secondary memory B-trees are more effective and faster B-trees are used in many of the database management system B-Trees 3 Factors that affect the time : Factors that affect the time Many factors can effect the speed of accessing data within the secondary memory The moving of arms on the platters Finding the track and sector on the secondary memory’s platters Read and write time B-Trees 4 Slide 5: B-Trees 5 Definition of a B-tree : B-Trees 6 Definition 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 tree 2. all leaves are on the same level 3. all non-leaf nodes except the root have at least ?m / 2? children 4. the root is either a leaf node, or it has from two to m children 5. a leaf node contains no more than m – 1 keys The number m should always be odd An example B-Tree : B-Trees 7 An example B-Tree 51 62 42 6 12 26 55 60 70 64 90 45 1 2 4 7 8 13 15 18 25 27 29 46 48 53 A B-tree of order 5 containing 26 items Note that all the leaves are at the same level Constructing a B-tree : B-Trees 8 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 root Constructing a B-tree 1 2 8 12 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 root Constructing a B-tree (contd.) : B-Trees 9 Constructing a B-tree (contd.) 1 2 8 12 25 Constructing a B-tree (contd.) : B-Trees 10 Constructing 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 leaf 8 17 12 14 25 28 1 2 6 Constructing a B-tree (contd.) : B-Trees 11 Constructing 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 leaves 3 8 17 48 52 53 55 68 25 26 28 29 1 2 6 7 12 14 16 Constructing a B-tree (contd.) : B-Trees 12 Constructing a B-tree (contd.) 17 3 8 28 48 1 2 6 7 12 14 16 52 53 55 68 25 26 29 45 Inserting into a B-Tree : B-Trees 13 Inserting 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 higher Inserting into a B-Tree : B-Trees 14 Inserting into a B-Tree Removal from a B-tree : B-Trees 15 Removal 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 we can delete the key and promote the predecessor or successor key to the non-leaf deleted key’s position. Removal from a B-tree (2) : B-Trees 16 Removal 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 leave the parent with too few keys then we repeat the process up to the root itself, if required Type #1: Simple leaf deletion : B-Trees 17 Type #1: Simple leaf deletion Delete 2: Since there are enough keys in the node, just delete it Assuming a 5-way B-Tree, as before... Type #2: Simple non-leaf deletion : B-Trees 18 Type #2: Simple non-leaf deletion Delete 52 Borrow the predecessor or (in this case) successor 56 Type #4: Too few keys in node and its siblings : B-Trees 19 Type #4: Too few keys in node and its siblings Delete 72 Too few keys! Type #4: Too few keys in node and its siblings : B-Trees 20 Type #4: Too few keys in node and its siblings Type #3: Enough siblings : B-Trees 21 Type #3: Enough siblings Delete 22 Type #3: Enough siblings : B-Trees 22 Type #3: Enough siblings 12 29 7 9 15 31 Algorithm for deletion : Algorithm for deletion B-Trees 23 Continued… : Continued… B-Trees 24 Analysis of B-Trees : B-Trees 25 Analysis of B-Trees The maximum number of items in a B-tree of order m and height h: root m – 1 level 1 m(m – 1) level 2 m2(m – 1) . . . level h mh(m – 1) So, the total number of items is (1 + m + m2 + m3 + … + mh)(m – 1) = [(mh+1 – 1)/ (m – 1)] (m – 1) = mh+1 – 1 When m = 5 and h = 2 this gives 53 – 1 = 124 B-tree insert runtime : • O(k) runtime per node • Path has height h = O(logk n) • CPU-time: O(k logk n) • O(k) runtime per node • Path has height h = O(logk n) • CPU-time: O(k logk n) B-Trees 26 B-tree search runtime B-tree insert runtime Time Complexity of Operations in B-tree : Time Complexity of Operations in B-tree Search/Insert/Delete all take up to the number of items in a path from the root to a leaf. The total number of operations is no more than the height of the tree The height of a tree is no more than log(n) where n is the number of items in a B-tree. B-Trees 27 Reasons for using B-Trees : B-Trees 28 Reasons for using B-Trees When searching tables held on disc, the cost of each disc transfer is high but doesn't depend much on the amount of data transferred, especially if consecutive items are transferred If we use a B-tree of order 101, say, we can transfer each node in one disc read operation A B-tree of order 101 and height 3 can hold 1014 – 1 items (approximately 100 million) and any item can be accessed with 3 disc reads (assuming we hold the root in memory) If we take m = 3, we get a 2-3 tree, in which non-leaf nodes have two or three children (i.e., one or two keys) B-Trees are always balanced (since the leaves are all at the same level), so 2-3 trees make a good type of balanced tree Slide 29: B-Trees 29 Thanks