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5, 6, hash(Eis) = 7, load = 63% 3, 4, 5, 6, 7, 8, hash(Ast) = 9, load = 72% 9, hash(Zug) = 10, load = 81% 3, 4, 5, 6, 7, 8, 9, 10, hash(Hof) = 0, load = 90% 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, hash(Mal) = 1, load = 100% The size field contains the number of items hashed into the table. The used[] array flags which components are occupied in the table. The printHash() method prints the hash table index and the
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resulting load factor as a percent. When linear probing kicks in, each successive index number probe is printed. As seen in Example 8.4, the collision occurs with the insertion of "Tag" at line 14. This program shows that it had three collisions (at index numbers 3, 4, and 5) before finding a free hash location at index 6. After that insertion, the table is 54 percent full. Every item after that also collides. And of course, as the table fills up, the number of collisions becomes more frequent. The last item, "Mal", has 10 collisions. That means that thereafter, every time this item is accessed it will have to search every one of the 11 items before it is found; clearly an O(n) process. Notice the index wraparound on the insertion of "Mal": 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 1.
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Another collision resolution algorithm that usually performs better than linear probing is called quadratic probing. This algorithm jumps over items in its probing, with the result that the used components are more uniformly distributed with fewer large clusters. That improves performance because the resulting probe chains are shorter. EXAMPLE 8.7 Quadratic Probing
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This program is the same as the program in Example 8.6 except for the modified hash() function shown here.
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public class TestQuadraticProbing { private static final int MASK = 0x7FFFFFFF; // 2^32-1 private static final int CAPACITY = 11; private static int size = 0; private static boolean[] used = new boolean[CAPACITY]; public static void main(String[] args) { printHash("Rad"); printHash("Uhr"); printHash("Ohr"); printHash("Tor"); printHash("Hut"); printHash("Tag"); printHash("Eis"); printHash("Ast"); printHash("Zug"); printHash("Hof"); printHash("Mal"); } private static void printHash(String word) { System.out.printf("hash(%s) = %d, load = %d%%%n", word, hash(word), 100*size/CAPACITY); } private static int hash(Object object) { ++size;
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[CHAP. 8
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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; } }
The output is:
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%
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.
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
Here is how part of a definition for a HashTable class might look, using separate chaining:
public class HashTable { private static final int MASK = 0x7FFFFFFF; // 2^32-1
CHAP. 8]
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