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Building and Analyzing an Order-4 B+-Tree with Hashing Concepts

This exercise outlines the construction of an order-4 B+-Tree for a given set of key values (2, 3, 5, 7, 11, 17, 9, 6, 29, 4), assuming insertion in ascending order. After constructing the tree, we will delete specific keys (2, 5, 17) and analyze the tree structure. The objectives include surveying hashing concepts, exploring collision reduction strategies, and assessing file performance. Key topics like the characteristics of hash functions, collision resolution techniques, and access patterns will also be investigated for a comprehensive understanding of hashing and data structures.

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Building and Analyzing an Order-4 B+-Tree with Hashing Concepts

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  1. CpSc 3220File and Database Processing Hashing

  2. Exercise – Build a B+-Tree • Construct an order-4 B+-tree for the following set of key values: (2, 3, 5, 7, 11, 17, 9, 6, 29, and 4) • Assume the tree is initially empty and values are added in ascending order. • Now delete keys 2, 5, and 17

  3. Objectives • Survey Hashing Concepts • Investigate Hashing Algorithms • Study Collision Reduction • Analyze Performance • Investigate File Deterioration • Look at Patterns of Access

  4. Schematic View of Hash File 0 0 1 Record for Key 0 hash Key 1 Record for Keyx . . 0

  5. Basic Hashing Concepts • A hash file contains a fixed number of record spaces • Each record space is of a fixed size • A hash function determines the address of a record space for a given key • A hash function may give same address for two different records • A single address for different keys is called a collision. • Different keys that give identical addresses are called synonyms. • A hash function that gives no collisions is called a perfect hash function.

  6. Objectives for a Hash File Package • Keep collisions ‘low’ • Spread out (distribute) records over address space • Use extra memory (increase address space) • Put more than one record per address • Handle collisions efficiently

  7. Outline for a Simple Hashing Algorithm • Put Key in numerical form • Fold and Add to reduce numerical form to ‘integer’ size • Divide by the size of the address space and use remainder as RRN address (offset) of Key

  8. Simple Hash Function(when Key is an alphanumeric string) int Hash (string key) { intsum = 0; intlen = strlen(key); if (len % 2 == 1) key = concat(key, ‘ ‘)// make len even for (int j = 0; j < len; j += 2) sum = (sum + 256 * (ord)key[j] + (ord)key[j+1]) % FILE_SIZE; return sum; }

  9. Hash Function Distribution • Uniform (Perfect) • Random • Worse than random We will look at random distributions

  10. Predicting Record Distribution If r records are distributed randomly into N spaces, the probability that a given address will have exactly x records assigned to it is p(x) = (r!/( (r-x)! x! ) )/(1-(1/N))r-x(1/N)x p(0) – probability that an address is not used p(1) – probability that no collision occurs p(2) – probability that 1 collision occurs etc. Difficult to compute for large values of r and N.

  11. Poisson’s Function For large values of r and N, p(x) can be approximately by this function p(x) = ( (r/N)x e-(r/N) ) / x! The value r/N is the ratio of the number of records to the number of address spaces. If only one record is placed in each space it is a measure of the percent of storage space that will be used (the packing density).

  12. From Page 484 of File Structures by Folk, Zoellick, and Riccardi

  13. Collision Resolution Using Progressive Overflow ( Linear Probing) 0 0 1 Record for Key0 1 Record for Key1 hash Key3 1 Record for Key2 . . 0 Hi = (hash(key) + i) mod TableSize

  14. ASL = (total # probes)/(# of Recs)

  15. Address Spaces Can Hold More Than One Record 2 Key a Key d 0 1 Key r 2 Key k Key b 0 2 Key x Key t 1 Key w Packing Density = r/(bN) Address Density = r/N

  16. Implementation Issues • Loading a Hash File • Deletions • Tombstones • Performance Effects

  17. Other Collision Resolution Techniques • Quadratic Hashing • H(i) = (hash(key) + i2) mod TS • Double Hashing • H(i) = (hash(key) + f(i)) mod TS where f(i) = i*hash2(key) • Note that hash2(key) must never be zero • Separate Overflow Area • Chained Overflow with Separate Overflow Area • Scatter Tables

  18. Patterns of Record Access • 20 percent of records account for 80 percent of activity • Most active records must be in home address or performance deteriorates

  19. Summary • Hashing provides O(1) direct access performance. • If hash function gives collisions ASL may increase. • Collisions can be reduced by: • Spreading out records (choosing a better hash fct) • Using extra memory • Using buckets • Poisson Distribution allows us to analyze hash file performance • Better overflow handling can reduce ASL • Record Deletion requires special handling • Consider record access patterns • Hashing does not provide efficient sequential access • Hashing requires that we fix file size in advance

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