collision resolution strategies

Collision Resolution Strategies

Open Address Methods ('search paths' in secondary table positions)
External (Separate) Chaining (a linked list on each table address)
Coalesced Chaining (a hybrid method)

Open Address Method: The general approach is to use a 'probe sequence' H₁(key) + H_m(key) (mod tablesize) for m=1,2,3,4, ...

H_m tells you which slot to try next if the previous slots were all in use.

Linear Rehashing: This is the open address method that uses H_m(key) = m. The probe sequence is H₁(key) + m. One advantage of this is simplicity.

Search under linear rehashing: assuming table slots are numbered 0 through tableSize-1, do this: while (the key is not found & the current location is not empty & we have not probed all the slots yet) set m = (m+1 % tableSize), and probe at location m.
Insertion under linear rehashing: first search, then if not found and table not full, place element in first location in probe sequence marked "empty" or "deleted."
Deletion under linear rehashing: first search, then if found mark location "deleted" (do not mark "empty" - subsequent searching must not terminate prematurely at a location where an element has been deleted.)

Proliferation of "deleted" cells can clutter the table and slow searches. One partial solution is to move up elements after deletions, or do such compaction from time to time.

External Chaining: Each table slot contains a linked list which can grow dynamically.

Advantages of External Chaining

Simple and efficient search, insertion, and deletion
more dynamic allocation than with open addressing
It is possible to have more records in the table than array slots. (load factors α > 1)

Coalesced Chaining: The table contains link fields and a cellar for overflow.



0
1
2       address region
3
4

----------------------------

5
6         cellar

H maps into the address region only (using division in this example). The cellar is for keys that need to be rehashed from the address region.

The rule is to place a colliding key into the empty place with the largest address (epla).

The example below illustrates what happens when the key sequence is 27, 29, 32, 34, 37, 47, and 53.

Note that the probe sequence for overflows from cell 3 'coalesces' in the end with the probe sequence for overflows from cell 2.



table        table         table 
address      contents      contents
             (data)          (link)

0              53              nil
1              47               0 <-- coalesced link
2              27               6
3              37               1
4              29               5

--------------------------------------------

5              34              nil
6              32               3

Empirical studies show that a cellar about 15% of the size of the main table works well. The search effort is not much more than with external chaining. Deletion can be done without resorting to marking records "deleted" but deletion is more complicated than in the case of external chaining, because lists can coalesce. There is a problem finding the predecessor of a search key. (Vitter wrote extensively about coalesced chaining.) Coalesced chaining tends to conserve memory better than open addressing or external chaining in cases where the hashed elements are small and load factors are high.