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int h = (object.hashCode() & MASK) % CAPACITY; if (used[h]) { int h0 = h; int jump = 1; while (used[h]) { System.out.printf("%d, ", h); h = (h0 + jump*jump)%CAPACITY; // squared increment ++jump; } } used[h] = true; return h; } }
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hash(Rad) = 3, load = 9% hash(Uhr) = 4, load = 18% hash(Ohr) = 2, load = 27% hash(Tor) = 8, load = 36% hash(Hut) = 5, load = 45% 3, 4, hash(Tag) = 7, load = 54% 5, hash(Eis) = 6, load = 63% 3, 4, 7, hash(Ast) = 1, load = 72% hash(Zug) = 9, load = 81%
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The essential difference here is in the sequence of index numbers probed within the while loop when a collision occurs. Instead of searching linearly, it uses a squared increment. For example, when the insertion of "Ast" collides at index 3, linear probing continued probing at indexes 4, 5, 6, 7, 8, and 9 (in Example 8.6). But with quadratic probing, only indexes 3, 4, 7, and 1 ( = 12 mod 11) are probed, using successive jumps of 1, 4, and 9 (12, 22, and 32). Linear probing required 50 percent more probes.
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The price that the quadratic probing algorithm pays for its improved performance is that it is more likely to result in an infinite loop. That happens in Example 8.7 with the next insertion. The string "Hof" hashes initially to index 3. After eight collisions, the linear probing algorithm found a free cell at index 0 ( = 11 mod 11). But the probe sequence used on this item by the quadratic probing algorithm is the same as for "Ast": 3, 4, 7, 1, 8, 6, 6, 8 1, 7, 4, 3, 4, . . . . This is computed from the unmodulated quadratic sequence 3, 4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, . . . . This continues indefinitely, probing only the six indexes 3, 4, 7, 1, 8, and 6, all of which have already been used. So even though the table is only 81% full, the insertion fails. That can t happen with linear probing. SEPARATE CHAINING Instead of devising a more effective collision resolution algorithm, we can avoid collisions altogether by allowing more than one item per table component. This method is called separate chaining, because is uses linked lists ( chains ) to hold the multiple items. In this context, the table components are usually called buckets. EXAMPLE 8.8 Separate Chaining
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Here is how part of a definition for a HashTable class might look, using separate chaining:
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public class HashTable { private static final int MASK = 0x7FFFFFFF; // 2^32-1
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private private private private static static static static
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int capacity = 101; int size = 0; float load = 0.75F; LinkedList[] buckets;
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HashTable() { buckets = new LinkedList[capacity]; for (int i = 0; i < capacity; i++) { buckets[i] = new LinkedList(); } } HashTable(int capacity, float load) { this(); this.capacity = capacity; this.load = load; } Object put(Object key, Object value) { int h = hash(key); LinkedList bucket=buckets[h]; Object oldValue = null; for (ListIterator it = bucket.iterator(); it.hasNext(); ) { Map.Entry entry = it.next(); if (entry.getKey().equals(key)) { break; } } if (entry.getKey().equals(key)) { oldValue = entry.setValue(value); } else { bucket.add(new Entry(key,value)); } return oldValue; } // more methods... }
Note that put() serves two different purposes. If the table already has an entry with the given key, it only changes the value of that entry. Otherwise, it adds a new entry with that key/value pair.
The java.util.HashMap class uses separate chaining in a way that is similar to that shown in Example 8.8. APPLICATIONS Tables are widely used in systems programming. Moreover, they are the primary building blocks of relational databases. Here is an example in applications programming. EXAMPLE 8.9 A Concordance
A concordance is a list of words that appear in a text document along with the numbers of the lines on which the words appear. It is just like an index of a book except that it lists line numbers instead of page
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