Searching

1
Searching

• Given distinct keys k1, k2, , kn and a
  collection of n records of the form
• (k1,I1), (k2,I2), , (kn, In)
• Search Problem - For key value K, locate the
  record (kj, Ij) in T such that kjK.
• Searching is a systematic method for locating the
  record(s) with key value kjK.
• A successful search is one in which a record with
  key kjK is found.
• An unsuccessful search is one in which no record
  with kjK is found (and does not exist).

2
Searching Ordered Arrays

• Binary Search - been there done that.
• Dictionary Search - interpolation search
• Determine how far from an endpoint your value is
  probably going to be.
• Pos(value-Alo)/(Ahi-Alow) (hi-lo)
• Look here rather than mid
• Assumes the data is evenly distributed.

3
Lists Ordered by Frequency

• Order lists by (expected) frequency of
  occurrence.
• Perform sequential search
• Cost for first record 1
• Cost for second record 2
• Search cost 1p1 2 p2 3p3 npn
• Worst case (n1)/2
• Best if a few items are accessed many times

4
Self Organizing Lists

• 80/20 rule 80 of the accesses are to 20 of the
  records
• expected search cost .122n
• Self organizing lists modify the order of records
  within the list basedon the actual pattern of
  record accesses.
• Self organizing lists use a rule called a
  heuristic for deciding how to reorder the list.

5
Self Organizing Heuristics

• Order by actual frequency - most frequently used
  first
• When a record is found, swap it with the first
  item
• When a record is found, move it to the front of
  the list
• When a record is found, swap it with the record
  ahead of it

6
Hashing

• The process of mapping a key value to a position
  in a table.
• A hash function maps key values to positions.
• A hash table is an array that holds the records.
• The hash table has M slots (0M-1)
• For any value K in the key range and some hash
  function h,
• h(k) I where 0 IltM, and key(TI)K

7
Hashing Situations

• Hashing is appropriate for unique keys.
• Good for both in-memory and disk based
  applications.
• Answers the question What record, if any, has
  key value K?
• Example Store the n records with keys in range
  0-(n-1).
• Store the record with key i in slot i.
• Uses the hash function h(k)k. (Identity
  function).

8
Collisions

• More reasonable example
• Store about 1000 records with keys in the range
  0-16,383.
• Impractical to keep a table of size 16,384.
• We need a hash function to map keys to a smaller
  range.
• Given a hash function h and different keys k1 and
  k2. Let ? be a position in the hash table.
• If h(k1 ) h(k2 ) ? then k1 and k2 have a
  collision at ? under h.

9
Collision Resolution

• To search for the record with key K
• Compute the table location h(K).
• Starting with slot h(K), locate the record
  containing key K using (if necessary) a collision
  resolution policy.
• Collisions are inevitable in most applications.
• Example In a group of 23 people the odds are
  good that at least one pair share a birthday.

10
Hash Functions

• Must return a value within the table range.
• Should evenly distribute the records to be stored
  among the table slots.
• Ideally, the function should distribute records
  with equal probability to all the positions. In
  reality, usually depends on the data.
• If we know nothing about the key distribution,
  evenly distribute the key range among the
  positions.
• If we know about the key distribution, use a
  distribution dependant hash function.

11
Example Hash Functions

• h(key)key 16 - uses only last 4 bits.
• H(key)key 1000 - uses last 4 digits.
• Use tablesize to make sure result is in the
  range.
• Mid-square method square the key and take the
  middle r bits for a table of size 2r
• Sum up ASCII characters and take results modulo
  tablesize (a folding technique).

12
Collision Handling Categories

• Open hashing - when there is a collision, put
  collided item outside the table.
• Closed hashing - when there is a collision, put
  collided item inside the table.

13
Open Hashing

• Look at each table element as the head of a
  linked list of items that has to that position.
• Can organize the linked lists in many ways
• ordered unsuccessful searches are quickly
  found.
• Ordered by frequency if a few are searched for
  frequently, then this is a good technique.
• If there are N records to be stored and the table
  is of size M then the average search length is
  O(N/M).
• Good for internal memory. Linked nodes may be in
  different blocks on disk and cause many disk
  accesses.

14
Closed Hashing - Linear Probe

• If the item you are looking for is not in the
  hash position, look in the next position.
• Do the same for insert until you find an empty
  location.
• When you reach the bottom, go to the beginning.
• Must have at least one empty slot or there will
  be an infinite loop.
• Tends to have clustering since the collision
  position is not uniformly distributed (i.e. if
  collide at position 4, go to position 5, then 6,
  independent of key).

15
Better Linear Probe

• Instead of going to the next slot, skip by some
  constant c.
• The tablesize M and c should be relatively prime.
• This assures the probing will cycle through all
  the table.
• Still has some clustering.

16
Quadratic Probe

• Instead of adding 1 to the key add i2
• i is the probe sequence, so add 1, 4, 9, 16,...
• Remember we also mod with table size.

17
Double Hashing

• After a collision, use a different hash function.
• Eliminates clustering to some degree.
• For example if h(k) causes a collision then use
• p(k,i) ih2(k)
• h2 is a different hash function
• generates a different probe sequence

18
Analysis of Closed Hashing

• load factor lfN/M
• N is the number of records
• M is the size of the table
• N/M is the percent full
• The larger the load factor the greater the
  probability of a collision
• Average search length is O(1/(1-lf))

19
Deletions

• If we delete a value it may stop the search
  prematurely (break the chain).
• Use a special mark to indicate something was
  deleted. When searching continue if see this
  mark rather than stopping as if it was empty.
• Once we have many deleted items we may wish to
  rehash everything remaining
• best if we rehash the most frequently accessed
  items first.