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Introduction to Hashing in Computer Science

Learn about the properties of a good hash function, handling collisions, open addressing, chaining, time complexity, load factor, Cuckoo Hashing, and Bloom Filters.

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Introduction to Hashing in Computer Science

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  1. Hashing II CS2110 Spring 2018

  2. Hash Functions 0 1 1 4 3 • Properties of a good hash: • fast • collision-resistant • evenly distributed • hard to invert • Requirements: • deterministic • return a number in [0..n]

  3. Hash Table add(“CA”) Hash hunction mod 6 CA 5 b 0 1 2 3 4 5 CA Two ways of handling collisions: Chaining 2. Open Addressing MA NY

  4. HashSet and HashMap Set<V>{ boolean add(V value); boolean contains(V value); boolean remove(V value); } Map<K,V>{ V put(K key, V value); V get(K key); V remove(K key); }

  5. Remove put('a') put('b') put('c') put('d') get('d') remove('c') get('d') put('e') Open Addressing Chaining e a c d b 0 0 1 1 2 2 3 3 a e c b d

  6. Time Complexity (no resizing)

  7. Load Factor Load factor

  8. Expected Chain Length • For each bucket, probability that a single object is hashed to that bucket is 1/length of array • There are n objects in the hash table • Expected length of chain is n/length of array = 0 1 2 3 4 5 a b c d e

  9. Expected Time Complexity (no resizing)

  10. Expected Number of Probes • We always have to probe H(v) • With probability , first location is full, have to probe again • With probability , second location is also full, have to probe yet again • … • Expected #probes = 0 1 2 3 4 5

  11. Expected Time Complexity (no resizing) Assuming constant load factor We need to dynamically resize!

  12. Amortized Analysis • In an amortized analysis, the time required to perform a sequence of operations is averaged over all the operations • Can be used to calculate average cost of operation vs.

  13. Amortized Analysis of put • Assume dynamic resizing with load factor : • Most put operations take (expected) time • If , put takes time • Total time to perform n put operations is • Average time to perform 1 put operation is

  14. Expected Time Complexity (with dynamic resizing)

  15. Cuckoo Hashing

  16. Cuckoo Hashing • Alternative solution to collisions • Assume you have two hash functions H1 and H2 0 1 2 3 4 5 d c e b d c a b What if there are loops?

  17. Complexity of Cuckoo Hashing • Worst Case: • Expected Case:

  18. Bloom Filters • Assume we only want to implement a set • What if you had stored the value at "all" hash locations (instead of one)? 0 1 2 3 4 5 🔵 🔵 🔵 🔵

  19. Features of Bloom Filters • Worst-case put, get, and remove • Works well with higher load factors • But: false positives

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