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12 Hash-Table Data Structures

12 Hash-Table Data Structures. Hash-table principles. Closed-bucket and open-bucket hash tables. Searching, insertion, deletion. Hash-table design. Implementations of sets and maps using hash tables. © 2001, D.A. Watt and D.F. Brown. Hash-table principles (1).

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12 Hash-Table Data Structures

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  1. 12Hash-Table Data Structures • Hash-table principles. • Closed-bucket and open-bucket hash tables. • Searching, insertion, deletion. • Hash-table design. • Implementations of sets and maps using hash tables. © 2001, D.A. Watt and D.F. Brown

  2. Hash-table principles (1) • If a map’s keys are small integers, we can represent the map by a key-indexed array. Search, insertion, and deletion then have time complexity O(1). • Can we approach this performance with keys of other types? Yes! • Hashing: translate each key to a small integer, and use that integer to index an array. • A hash table is an array of mbuckets, together with a hash functionhash(k) that translates each key k to a bucket index (in the range 0…m–1).

  3. hash table(array of buckets) 0 1 2 3 4 5 m–2 m–1 hashing (translating keys to bucket indices) Hash-table principles (2) • Illustration: collision

  4. Hash-table principles (3) • Each key k has a home bucket in the hash table, namely the bucket with index hash(k). • To insert a new entry with key k into the hash table, assign that entry to k’s home bucket. • To search for an entry with key k in the hash table, look in k’s home bucket. • To delete an entry with key k from the hash table, look in k’s home bucket.

  5. Hash-table principles (4) • The hash function must be consistent: k1 = k2 implies hash(k1) = hash(k2). • In general, the hash function is many-to-one. • Therefore different keys may share the same home bucket: k1k2 but hash(k1) = hash(k2). This is called a collision. • Always prefer a hash function that makes collisions relatively infrequent.

  6. Example 1: a hash function for words • Suppose that the keys are English words. • Possible hash function: m = 26hash(w) = (initial letter of w) – ‘A’ • All words with initial letter ‘A’ share bucket 0;…all words with initial letter ‘Z’ share bucket 25. • This is a convenient choice for illustrative purposes. • This is a poor choice for practical purposes: collisions are likely to be frequent in some buckets.

  7. Hashing in Java (1) • Instance method in class Object: publicint hashCode ();// Translate this object to an integer, such that x.equals(y)// implies x.hashcode()==y.hashcode(). • Note that hashCode is consistent. We can use it to implement a hash function for a hash table with m buckets: int hash (Object k) {return Math.abs(k.hashCode()) % m;} Math.abs returns a nonnegative integer. Modulo-m arithmetic then gives an integer in the range 0…m–1.

  8. Hashing in Java (2) • Each subclass of Object should override hashCode. • Examples:

  9. Closed- vs open-bucket hash tables • Closed-bucket hash table: • Each bucket may be occupied by several entries. • Buckets are completely separate. • Open-bucket hash table: • Each bucket may be occupied by at most one entry. • Whenever there is a collision, displace the new entry to another bucket.

  10. Closed-bucket hash tables (1) • Closed-bucket hash table (CBHT): • Each bucket may be occupied by several entries. • Buckets are completely separate. • Simplest implementation: each bucket is an SLL. • In the following illustrations, keys are names of chemical elements. Assume: m = 26hash(e) = (initial letter of e) – ‘A’

  11. 0 Ar 18 1 Br 35 2 Cl 17 3 4 5 F 9 6 7 I 53 8 9 Kr 36 10 11 12 13 Ne 10 23 Xe 54 24 25 Closed-bucket hash tables (2) • Illustration (with no collisions): is represented by

  12. 0 Be 4 Ba 56 1 Ca 20 Cs 55 2 He 2 H 1 7 8 9 K 19 10 Li 3 11 Mg 12 12 13 Na 11 Rb 37 17 18 Sr 38 25 Closed-bucket hash tables (3) • Illustration (with collisions): is represented by

  13. Closed-bucket hash tables (4) • Java class implementing CBHTs: publicclass CBHT { private BucketNode[] buckets; public CBHT (int m) { buckets = new BucketNode[m]; } … // CBHT methods (see below) privateint hash (Object key) {return Math.abs(key.hashCode()) % buckets.length; }

  14. Closed-bucket hash tables (5) • Java class (continued): //////// Inner class for CBHT nodes //////// privatestaticclass BucketNode { private Object key, value;private BucketNode succ; private BucketNode (Object key, Object val, BucketNode succ) {… } }}

  15. CBHT search (1) • CBHT search algorithm: To find which if any node of a CBHT contains an entry whose key is equal to target-key: 1. Set b to hash(target-key).2. Find which if any node of the SLL of bucket b contains an entry whose key is equal to target-key, and terminate with that node as answer.

  16. CBHT search (2) • Implementation (in class CBHT): public BucketNode search (Object targetKey) {int b = hash(targetKey);for (BucketNode curr = buckets[b]; curr != null; curr = curr.succ) {if (targetKey.equals(curr.key))return curr; }returnnull;}

  17. CBHT insertion (1) • CBHT insertion algorithm: To insert the entry (key, val) into a CBHT: 1. Set b to hash(key).2. Insert the entry (key, val) into the SLL of bucket b, replacing any existing entry whose key is key.3. Terminate.

  18. CBHT insertion (2) • Implementation (in class CBHT): publicvoid insert (Object key, Object val) {int b = hash(key);for (BucketNode curr = buckets[b]; curr != null; curr = curr.succ) {if (key.equals(curr.key)) { curr.value = val; return; } } buckets[b] = new BucketNode(key, val, buckets[b]);}

  19. CBHT deletion (1) • CBHT deletion algorithm: To delete the entry (if any) whose key is equal to key from a CBHT: 1. Set b to hash(key).2. Delete the entry (if any) whose key is equal to key from the SLL of bucket b.3. Terminate.

  20. CBHT deletion (2) • Implementation (in class CBHT): publicvoid delete (Object key) {int b = hash(key);for (BucketNode pred = null, curr = buckets[b]; curr != null; pred = curr, curr = curr.succ) {if (key.equals(curr.key)) {if (pred == null) buckets[b] = curr.succ;else pred.succ = curr.succ;return; } }}

  21. CBHTs: analysis • Analysis of the CBHT search/insertion/deletion algorithms (counting comparisons): Let the number of entries be n. • In the best case, no bucket contains more than (say) 2 entries: Max. no. of comparisons = 2 Best-case time complexity is O(1). • In the worst case, one bucket contains all n entries: Max. no. of comparisons = n Worst-case time complexity is O(n).

  22. CBHTs: design • CBHT design consists of: • choosing the number of buckets m • choosing the hash function hash. • Design aims: • collisions are infrequent • entries are distributed evenly among the buckets, such that few buckets contain more than about 2 entries.

  23. CBHTs: choosing the no. of buckets • The load factor of a hash table is the average number of entries per bucket, n/m. • If n is (roughly) predictable, choose m such that the load factor is likely to be between 0.5 and 0.75. • A low load factor wastes space. • A high load factor tends to cause some buckets to have many entries. • Choose m to be a prime number. • Typically the hash function performs modulo-m arithmetic. If m is prime, the entries are more likely to be distributed evenly over the buckets, regardless of any pattern in the keys.

  24. CBHTs: choosing the hash function • The hash function should be efficient (performing few arithmetic operations). • The hash function should distribute the entries evenly among the buckets, regardless of any patterns in the keys. • Possible trade-off: • Speed up the hash function by using only part of the key. • But beware of any patterns in that part of the key.

  25. Example 2: hash table for English words (1) • Suppose that a hash table will contain about 1000 common English words. • Known patterns in the keys: • Letters vary in frequency: • A, E, I, N, S, T are common • Q, X, Z are uncommon. • Word lengths vary in frequency: • word lengths 4–8 are common • other word lengths are less common.

  26. Example 2 (2) • hash(w) can depend on any of w’s letters and/or length. • Consider m = 20, hash(w) = length of w – 1. – Far too few buckets. Load factor = 1000/20 = 50. – Very uneven distribution. • Consider m = 26, hash(w) = initial letter of w – ‘A’. – Far too few buckets. – Very uneven distribution.

  27. Example 2 (3) • Consider m = 520, hash(w) = 26  (length of w – 1) + (initial letter of w – ‘A’). – Too few buckets. Load factor = 1000/520  1.9. – Very uneven distribution. Since few words have length 0–2, buckets 0–51 will be sparsely populated. Since initial letter Z is uncommon, buckets 25, 51, 77, 103, … will be sparsely populated. And so on. • Consider m = 1499, hash(w) = (weighted sum of letters of w) modulo m i.e., (c1 1st letter of w + c2 2nd letter of w + …) modulo m + Good number of buckets. Load factor  0.67. + Reasonably even distribution.

  28. Open-bucket hash tables (1) • Open-bucket hash table (OBHT): • Each bucket may be occupied by at most one entry. • Whenever there is a collision, displace the new entry to another bucket. • Each bucket has three possible states: • never-occupied (has never contained an entry) • occupied (currently contains an entry) • formerly-occupied (previously contained an entry, which has been deleted and not yet replaced).

  29. Open-bucket hash tables (2) • In the following illustrations, keys are names of chemical elements. Assume: m = 26hash(e) = (initial letter of e) – ‘A’ • On a collision, insert the new entry in the next unoccupied bucket (treating the array as cyclic).

  30. 0 Ar 18 1 Br 35 2 Cl 17 3 4 5 F 9 6 7 8 I 53 9 10 Kr 36 11 12 13 Ne 10 14 22 23 Xe 54 24 25 Open-bucket hash tables (3) • Illustration (with no collisions): occupied never-occupied is represented by

  31. 0 1 Be 4 2 Ca 20 cluster 3 Cs 55 4 Ba 56 5 6 7 H 1 8 He 2 9 10 K 19 11 Li 3 cluster 12 Mg 12 13 Na 11 17 Rb 37 18 Sr 38 25 Open-bucket hash tables (4) • Illustration (with collisions): is represented by

  32. Open-bucket hash tables (5) • Java class implementing OHBTs: publicclass OBHT { private BucketEntry[] buckets; public OBHT (int m) { buckets = new BucketEntry[m]; } … privateint hash (Object k) {return Math.abs(k.hashCode()) % buckets.length; } OBHT methods

  33. Open-bucket hash tables (6) • Java class (continued): //////// Inner class for OBHT entries //////// privatestaticclass BucketEntry { private Object key, value; private BucketEntry (Object k, Object v) {this.key = k; this.value = v; } privatestaticfinal BucketEntry FORMER =new BucketEntry(null, null); }}

  34. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 17 18 25 Example 3: populating an OBHT • Animation: Be 4 Ca 20 Cs 55 Ba 56 H 1 He 2 K 19 Li 3 Mg 12 Na 11 Rb 37 Sr 38

  35. OBHT search (1) • OBHT search algorithm: To find which if any bucket of an OBHT is occupied by an entry whose key is equal to target-key: 1. Set b to hash(target-key).2. Repeat: 2.1. If bucket b is never-occupied: 2.1.1. Terminate with answer none. 2.2. If bucket b is occupied by an entry whose key is equal to target-key: 2.2.1. Terminate with answer b. 2.3. If bucket b is formerly-occupied, or is occupied by an entry whose key is not equal to target-key: 2.3.1. Increment b modulo m.

  36. 0 1 Be 4 2 Ca 20 3 Cs 55 4 Ba 56 5 6 7 H 1 8 He 2 9 10 K 19 11 Li 3 12 Mg 12 13 Na 11 17 Rb 37 18 Sr 38 19 25 OBHT search (2) • Illustrations: Searching for Ba: Searching for He: Searching for Mg: Searching for Ra:

  37. OBHT search (3) • Implementation (in class OBHT): publicint search (Object targetKey) {int b = hash(targetKey);for (;;) { BucketEntry old = buckets[b];if (old == null)return NONE;elseif (old != BucketEntry.FORMER && targetKey.equals(old.key))return b;elseb = (b + 1) % buckets.length; }}

  38. OBHT insertion (1) • OBHT insertion algorithm: To insert the entry (key, val) into an OBHT: 1. Set b to hash(key).2. Repeat: 2.1. If bucket b is never-occupied: 2.1.1. If bucket b is the last never-occupied bucket, treat the OBHT as full. 2.1.2. Make bucket b occupied by (key, val). 2.1.3. Terminate. 2.2. If bucket b is formerly-occupied, or is occupied by an entry whose key is equal to key: 2.2.1. Make bucket b occupied by (key, val). 2.2.2. Terminate. 2.3. …

  39. OBHT insertion (2) • OBHT insertion algorithm (continued): 2.3. If bucket b is occupied by an entry whose key is not equal to key: 2.3.1. Increment b modulo m.

  40. 0 0 0 1 1 1 Be 4 Be 4 Be 4 2 2 2 Ca 20 Ca 20 Ca 20 3 3 3 Cs 55 Cs 55 Cs 55 4 4 4 Ba 56 Ba 56 Ba 56 5 5 5 Fr 87 Fr 87 6 6 6 B 5 7 7 7 H 1 H 1 H 1 8 8 8 He 2 He 2 He 2 9 9 9 10 10 10 K 19 K 19 K 19 11 11 11 Li 3 Li 3 Li 3 12 12 12 Mg 12 Mg 12 Mg 12 13 13 13 Na 11 Na 11 Na 11 17 17 17 Rb 37 Rb 37 Rb 37 18 18 18 Sr 38 Sr 38 Sr 38 25 25 25 OBHT insertion (3) • Illustrations: Inserting(B, 5): Inserting(Fr, 87):

  41. OBHT insertion (4) • Implementation (in class OBHT): privateint load = 0; // no. of occupied or formerly- // occupied buckets in this OBHT publicvoid insert (Object key, Object val) { BucketEntry newest =new BucketEntry(key, val);int b = hash(key);for (;;) { BucketEntry old = buckets[b];if (old == null) {if (++load == buckets.length) …; buckets[b] = newest;return;

  42. OBHT insertion (5) • Implementation (continued): } elseif (old == BucketEntry.FORMER || key.equals(old.key)) { buckets[b] = newest;return; } else b = (b + 1) % buckets.length; }}

  43. OBHT deletion (1) • OBHT deletion algorithm: To delete the entry (if any) whose key is equal to key from an OBHT: 1. Set b to hash(key).2. Repeat: 2.1. If bucket b is never-occupied: 2.1.1. Terminate. 2.2. If bucket b is occupied by an entry whose key is equal to key: 2.2.1. Make bucket b formerly-occupied. 2.2.2. Terminate. 2.3. If bucket b is formerly-occupied, or is occupied by an entry whose key is not equal to key: 2.3.1. Increment b modulo m.

  44. 0 0 0 1 1 1 Be 4 Be 4 Be 4 2 2 2 Ca 20 3 3 3 Cs 55 Cs 55 Cs 55 4 4 4 Ba 56 Ba 56 5 5 5 6 6 6 7 7 7 H 1 H 1 H 1 8 8 8 He 2 He 2 He 2 9 9 9 10 10 10 K 19 K 19 K 19 11 11 11 Li 3 Li 3 Li 3 12 12 12 Mg 12 Mg 12 Mg 12 13 13 13 Na 11 Na 11 Na 11 17 17 17 Rb 37 Rb 37 Rb 37 18 18 18 Sr 38 Sr 38 Sr 38 25 25 25 OBHT deletion (2) • Illustrations: Deleting Ba: Deleting Ca: formerly-occupied

  45. OBHT deletion (3) • Implementation (in class OBHT): publicvoid delete (Object key) {int b = hash(key);for (;;) { BucketEntry old = buckets[b];if (old == null)return;elseif (old != BucketEntry.FORMER && key.equals(old.key)) { buckets[b] = BucketEntry.FORMER;return; } else b = (b + 1) % buckets.length; }}

  46. OBHTs: analysis • Analysis of OBHT search/insertion/deletion algorithm (counting comparisons): Let the number of entries be n. • In the best case, no cluster contains more than (say) 4 entries: Max. no. of comparisons = 4 Best-case time complexity is O(1). • In the worst case, one cluster contains all n entries: Max. no. of comparisons = n Worst-case time complexity is O(n).

  47. OBHTs: design • OBHT design consists of: • choosing the number of buckets m • choosing the hash function hash • choosing the step length s (explained later). • Design aims: • collisions are infrequent • entries are distributed evenly over the hash table, such that few clusters contain more than about 4 entries.

  48. OBHTs: choosing the no. of buckets • Recall: The load factor of a hash table is the average number of entries per bucket, n/m. • If n is (roughly) predictable, choose m such that the load factor is likely to be between 0.5 and 0.75. • A low load factor wastes space. • A high load factor tends to result in long clusters. • Choose m to be a prime number.

  49. OBHTs: choosing the hash function • The hash function should be efficient. • The hash function should distribute the entries evenly over the buckets, with few long clusters. • In an OHBT with s = 1, a cluster will form when several entries fall into the same or adjacent buckets.

  50. OBHTs: choosing the step length • To resolve a collision, the search/insertion/deletion algorithm increments the bucket index and tries again. • The step length, s, is the amount by which the bucket index is incremented. So far we have assumed s = 1. • Alternatively, we can use a fixed s > 1. • Choose m to be prime, and choose s to be in the range 2…m–1. • This ensures that s and m have no common factors. • Otherwise, if (say) m = 10 and s = 2, a typical search path would be 6–8–0–2–4, never reaching the remaining buckets!

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