B tree, Hamiltonian & Eulerian path presentation
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The document discusses B-Trees, Hamiltonian cycles, and Eulerian paths, focusing on their structures, methods of insertion and deletion, and time complexities. It explains how B-Trees help manage data that doesn't fit in main memory by allowing efficient disk access and outlines definitions and applications for Hamiltonian and Eulerian paths in graph theory. Additionally, it highlights algorithms for finding Hamiltonian paths and Eulerian paths and their significance in various fields like computer graphics and DNA sequencing.
![Deletion
If P is Tp then (p-1)KE
For min ceil[p/2-1] KE
Underflow : Tp = KE](https://image.slidesharecdn.com/mbi2020015sappresentation-210516190415/85/B-tree-Hamiltonian-Eulerian-path-presentation-10-320.jpg)
B tree, Hamiltonian & Eulerian path presentation
- 1. B -Tree, Hamiltonian cycle & Eulerian path IIIT Allahabad Sanjay kumar MBI2020015
- 2. Motivation For B tree ● So far we have assume 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 involved
- 3. Motivation Cont.. ● Assume that a disk spins at 3600 RPM ● In 1 minute it makes 3600 revolutions, hence one revolution occurs in 1/60 of a second ● Assume that we use an AVL tree to store all the covid’19 patient details in india (about 20 lakh records) ● 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 decreases
- 4. What is B - Tree and Properties? ● A B -tree of order m is an m-way tree where each node may have up to m children ● Root : - It can have 2 children or ‘m’ children, where ‘m’ is called order of the tree ● Internal nodes : - Internal node can store upto ‘m-1’ keys - They can have between ⎡m / 2⎤ and ‘m’ children <Ptr < K1, Pr1 > Ptr<K2, Pr2>- - - - - - -<Kq-1, Pr-1>Pq>
- 5. Properties: ● Leaf: - A leaf may store between ⎡(m -1)/ 2⎤ and m-1 keys - All leaves are at the same depth
- 6. Example of B -Tree
- 7. The height of a B -tree ● If n =>1,then for any n -key B-tree of height h and minimum degree t =>2 h <= log t (n+1)/2.
- 8. Insertion into B - tree
- 9. Insertion Algorithm ● Insertion: when a node becomes too full, split it into two nodes and promote the middle key into a parent key. Repeat recursively on the parent key ● Num of node after adding the key: Fewer than ‘m’ - 1 Equal to ‘m’ Overflow
- 10. Deletion If P is Tp then (p-1)KE For min ceil[p/2-1] KE Underflow : Tp = KE
- 11. Time complexity ● Search (logn) ● Insert (logn) ● Delete (logn) Why we use B -tree ● 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
- 12. Cont... ● 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
- 13. Hamiltonian Cycle In the mathematical field of graph theory, a Hamiltonian cycle (or traceable path) is a path in which an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle in a dodecahedron
- 14. The figure has 20 red dots. Can you draw a path that visits each vertex (Dots) exactly once? Solution: Icosian game developed by William Rowan Hamilton
- 15. Hamiltonian path problem Knight Tours Revisited ● Let us form a graph G = (V, E) as follows: ● V = the squares of a chessboard ● E = the set of edges (v, w) where v and w are squares on the chessboard and a knight can jump from v to w in a single move. ● Hence, a knight tour is just a Hamiltonian Cycle in this graph
- 16. Theorem and Application of Hamiltonian cycle Theorem: The Hamiltonian Cycle Problem is NP-Complete ● This result explains why knight tours were so difficult to find, there is no known quick method to find them Application of Hamiltonian cycle: ● Computer graphics ● Mapping genomes ● Operation research
- 17. Eulerian Path ● An Euler path is a path in a graph that uses every edge of a graph exactly once ● An Euler path starts and ends at different vertices
- 18. How to find given graph is eulerian or not? ● Given a drawing on a piece of paper, try to cross all the lines with a pen such that the pen doesn’t leave the paper and you mustn’t cross a line more than once. ● An undirected graph has Eulerian Path if following two conditions are true: -All vertices with non-zero degree are connected. We don’t care about vertices with zero degree because they don’t belong to Eulerian Path -If a graph has Eulerian path then no. of odd degree vertices is either ‘0’ or ‘2’(start !=end)
- 19. Fleury’s algorithm for printing eulerian path or circuit Step 1: Make Sure the graph has either ‘0’ or ‘2’ odd vertices Step 2: If there are ‘0’ odd vertices start anywhere. If there are ‘2’ odd vertices start at one of them Step 3: fallow edges one at a time. If you have bridge and a non bridge always choose non bridge Step 4: Stop when run out of the edges
- 20. Example: 1. We pick ‘2-0’. We remove this edge and move to vertex ‘0’ 2. Euler tour ‘2-0’ 3. Euler tour ‘2-0 0-1’ 4. Euler tour ‘2-0 0-1 1-2’ 5. Euler tour ‘2-0 0-1 1-2 2-3’
- 21. Application of Eulerian graph ● 1735, Seven bridge of Königsber Here eulerian rule is violets so problem in impossible to solve ● DNA fragment assembly Hybrid approach on the basis of overlap-layout-consensus technique implicitly solves the Hamiltonian path problem and has a high rate of misassembly
- 22. Fragment assembly without repeat masking can be done in linear time with greater accuracy.
- 23. . Thank you sir!