1. Dictionaries < > = 6 2 9 1 4 8 12/3/2018 8:58 AM Dictionaries

2. Outline and Reading Dictionary ADT (§8.1) Log file (§8.2)
Binary search
Lookup table (§8.5)
Binary search tree (§9.1)
Search
Insertion
Deletion
Performance
12/3/2018 8:58 AM
Dictionaries

3. Dictionary ADT
The dictionary ADT models a searchable collection of key-element items
The main operations of a dictionary are searching, inserting, and deleting items
Multiple items with the same key are allowed
Applications:
address book
credit card authorization
mapping host names (e.g., cs16.net) to internet addresses (e.g., )
Dictionary ADT methods:
findElement(k): if the dictionary has an item with key k, returns its element, else, returns the special element NO_SUCH_KEY
insertItem(k, o): inserts item (k, o) into the dictionary
removeElement(k): if the dictionary has an item with key k, removes it from the dictionary and returns its element, else returns the special element NO_SUCH_KEY
size(), isEmpty()
keys(), Elements()
12/3/2018 8:58 AM
Dictionaries

4. Log File
A log file is a dictionary implemented by means of an unsorted sequence
We store the items of the dictionary in a sequence (based on a doubly-linked lists or a circular array), in arbitrary order
Performance:
insertItem takes O(1) time since we can insert the new item at the beginning or at the end of the sequence
findElement and removeElement take O(n) time since in the worst case (the item is not found) we traverse the entire sequence to look for an item with the given key
The log file is effective only for dictionaries of small size or for dictionaries on which insertions are the most common operations, while searches and removals are rarely performed (e.g., historical record of logins to a workstation)
12/3/2018 8:58 AM
Dictionaries

5. Binary Search
Binary search performs operation findElement(k) on a dictionary implemented by means of an array-based sequence, sorted by key
similar to the high-low game
at each step, the number of candidate items is halved
terminates after a logarithmic number of steps
Example: findElement(7) 1 3 4 5 7 8 9 11 14 16 18 19 l m h 1 3 4 5 7 8 9 11 14 16 18 19 l m h 1 3 4 5 7 8 9 11 14 16 18 19 l m h 1 3 4 5 7 8 9 11 14 16 18 19
l=m =h
12/3/2018 8:58 AM
Dictionaries

6. Lookup Table
A lookup table is a dictionary implemented by means of a sorted sequence
We store the items of the dictionary in an array-based sequence, sorted by key
We use an external comparator for the keys
Performance:
findElement takes O(log n) time, using binary search
insertItem takes O(n) time since in the worst case we have to shift n/2 items to make room for the new item
removeElement take O(n) time since in the worst case we have to shift n/2 items to compact the items after the removal
The lookup table is effective only for dictionaries of small size or for dictionaries on which searches are the most common operations, while insertions and removals are rarely performed (e.g., credit card authorizations)
12/3/2018 8:58 AM
Dictionaries

7. Binary Search Tree
A binary search tree is a binary tree storing keys (or key-element pairs) at its internal nodes and satisfying the following property:
Let u, v, and w be three nodes such that u is in the left subtree of v and w is in the right subtree of v. We have key(u)  key(v)  key(w)
External nodes do not store items
An inorder traversal of a binary search trees visits the keys in increasing order 6 9 2 4 1 8
12/3/2018 8:58 AM
Dictionaries

8. Dictionaries
12/3/2018 8:58 AM
Search
To search for a key k, we trace a downward path starting at the root
The next node visited depends on the outcome of the comparison of k with the key of the current node
If we reach a leaf, the key is not found and we return NO_SUCH_KEY
Example: findElement(4)
Algorithm findElement(k, v)
if T.isExternal (v)
return NO_SUCH_KEY
if k < key(v)
return findElement(k, T.leftChild(v))
else if k = key(v)
return element(v)
else { k > key(v) }
return findElement(k, T.rightChild(v))
< 6 2
> 9 1 4 = 8
12/3/2018 8:58 AM
Dictionaries

9. Insertion
< 6
To perform operation insertItem(k, o), we search for key k
Assume k is not already in the tree, and let w be the leaf reached by the search
We insert k at node w and expand w into an internal node
Example: insert 5 2 9
> 1 4 8
> w 6 2 9 1 4 8 w 5
12/3/2018 8:58 AM
Dictionaries

10. Deletion To perform operation removeElement(k), we search for key k6
To perform operation removeElement(k), we search for key k
Assume key k is in the tree, and let v be the node storing k
If node v has a leaf child w, we remove v and w from the tree with operation removeAboveExternal(w)
Example: remove 4
< 2 9
> v 1 4 8 w 5 6 2 9 1 5 8
12/3/2018 8:58 AM
Dictionaries

11. Deletion (cont.) 1 v
We consider the case where the key k to be removed is stored at a node v whose children are both internal
we find the internal node w that follows v in an inorder traversal
we copy key(w) into node v
we remove node w and its left child z (which must be a leaf) by means of operation removeAboveExternal(z)
Example: remove 3 3 2 8 6 9 w 5 z 1 v 5 2 8 6 9
12/3/2018 8:58 AM
Dictionaries

12. Performance
Consider a dictionary with n items implemented by means of a binary search tree of height h
the space used is O(n)
methods findElement , insertItem and removeElement take O(h) time
The height h is O(n) in the worst case and O(log n) in the best case
12/3/2018 8:58 AM
Dictionaries

13. Skip Lists S3 + - S2 + - S1 + - S0 + - 15 23 15 10 36 23 15
Dictionaries
12/3/2018 8:58 AM
Skip Lists S3 + - S2 + - 15 S1 + - 23 15 S0 + - 10 36 23 15
12/3/2018 8:58 AM
Dictionaries

14. Outline and Reading What is a skip list (§8.6) Operations
Search
Insertion
Deletion
Implementation
Analysis
Space usage
Search and update times
12/3/2018 8:58 AM
Dictionaries

15. What is a Skip List
A skip list for a set S of distinct (key, element) items is a series of lists S0, S1 , … , Sh such that
Each list Si contains the special keys + and -
List S0 contains the keys of S in nondecreasing order
Each list is a subsequence of the previous one, i.e., S0  S1  …  Sh
List Sh contains only the two special keys
We show how to use a skip list to implement the dictionary ADT S3 + - S2 - 31 + S1 - 23 31 34 64 + S0 56 64 78 + 31 34 44 - 12 23 26
12/3/2018 8:58 AM
Dictionaries

16. Search S3 S2 S1 S0 We search for a key x in a a skip list as follows:
We start at the first position of the top list
At the current position p, we compare x with y  key(after(p))
x = y: we return element(after(p))
x > y: we “scan forward”
x < y: we “drop down”
If we try to drop down past the bottom list, we return NO_SUCH_KEY
Example: search for 78 S3 + - S2 - 31 + S1 - 23 31 34 64 + S0 - 12 23 26 31 34 44 56 64 78 +
12/3/2018 8:58 AM
Dictionaries

17. Search: Algorithm SkipSearch(k):
Input: A search key k and the skip-list S
Output: Position p in S such that the item at p has the
largest key less than or equal to k
Let p be the top-most, - of S (S has at least two levels)
while below(p) != NULL do
p  below(p) // drop-down
while key(after(p))  k do
let p  after(p) //scan-forward
return p;
12/3/2018 8:58 AM
Dictionaries

18. Randomized Algorithms
Dictionaries
12/3/2018 8:58 AM
Randomized Algorithms
A randomized algorithm performs coin tosses (i.e., uses random bits) to control its execution
It contains statements of the type
b  random()
if b = 0
do A …
else { b = 1}
do B …
Its running time depends on the outcomes of the coin tosses
We analyze the expected running time of a randomized algorithm under the following assumptions
the coins are unbiased, and
the coin tosses are independent
The worst-case running time of a randomized algorithm is often large but has very low probability (e.g., it occurs when all the coin tosses give “heads”)
We use a randomized algorithm to insert items into a skip list
12/3/2018 8:58 AM
Dictionaries

19. Insertion
To insert an item (x, o) into a skip list, we use a randomized algorithm:
We repeatedly toss a coin until we get tails, and we denote with i the number of times the coin came up heads
If i  h, we add to the skip list new lists Sh+1, … , Si +1, each containing only the two special keys
We search for x in the skip list and find the positions p0, p1 , …, pi of the items with largest key less than x in each list S0, S1, … , Si
For j  0, …, i, we insert item (x, o) into list Sj after position pj
Example: insert key 15, with i = 2 S3 + - p2 S2 S2 - + + - 15 p1 S1 S1 - 23 + + - 23 15 p0 S0 S0 - 10 23 36 + + - 10 36 23 15
12/3/2018 8:58 AM
Dictionaries

20. Deletion
To remove an item with key x from a skip list, we proceed as follows:
We search for x in the skip list and find the positions p0, p1 , …, pi of the items with key x, where position pj is in list Sj
We remove positions p0, p1 , …, pi from the lists S0, S1, … , Si
We remove all but one list containing only the two special keys
Example: remove key 34 S3 - + p2 S2 S2 - 34 + - + p1 S1 S1 - 23 34 + - 23 + p0 S0 S0 - 12 23 34 45 + - 12 23 45 +
12/3/2018 8:58 AM
Dictionaries

21. Implementation x quad-node
We can implement a skip list with quad-nodes
A quad-node stores:
item
link to the node before
link to the node after
link to the node below
link to the node above
Also, we define special keys PLUS_INF and MINUS_INF, and we modify the key comparator to handle them
quad-node x
12/3/2018 8:58 AM
Dictionaries

22. insertAfterAbove(p, q, new)
new.after = p.after;
new.before = p;
new.below = q;
new.up = NULL;
p.after.before = new;
p.after = new;
If (q != NULL)
q.above = new; q P
new q
12/3/2018 8:58 AM
Dictionaries

23. SkipInsert: Algorithm SkipInsert(k, e): Input: item(k, e) Output: none
p  SkipSearch(k); // go to the bottom
q  insertAfterAbove(p, NULL, item(k, e));
while random() < 0.5 do
while above(p) = NULL do
p  before(p) // scan backward
p  above(p) // jump up to higher level
insertAfterAbove(p, q, item(k, e))
12/3/2018 8:58 AM
Dictionaries

24. Space Usage Consider a skip list with n items
By Fact 1, we insert an item in list Si with probability 1/2i
By Fact 2, the expected size of list Si is n/2i
The expected number of nodes used by the skip list is
The space used by a skip list depends on the random bits used by each invocation of the insertion algorithm
We use the following two basic probabilistic facts:
Fact 1: The probability of getting i consecutive heads when flipping a coin is 1/2i
Fact 2: If each of n items is present in a set with probability p, the expected size of the set is np
Thus, the expected space usage of a skip list with n items is O(n)
12/3/2018 8:58 AM
Dictionaries

25. Height
The running time of the search an insertion algorithms is affected by the height h of the skip list
We show that with high probability, a skip list with n items has height O(log n)
We use the following additional probabilistic fact:
Fact 3: If each of n events has probability p, the probability that at least one event occurs is at most np
Consider a skip list with n items
By Fact 1, we insert an item in list Si with probability 1/2i
By Fact 3, the probability that list Si has at least one item is at most n/2i
By picking i = 3log n, we have that the probability that S3log n has at least one item is at most n/23log n = n/n3 = 1/n2
Thus a skip list with n items has height at most 3log n with probability at least /n2
12/3/2018 8:58 AM
Dictionaries

26. Search and Update Times
The search time in a skip list is proportional to
the number of drop-down steps, plus
the number of scan-forward steps
The drop-down steps are bounded by the height of the skip list and thus are O(log n) with high probability
To analyze the scan-forward steps, we use yet another probabilistic fact:
Fact 4: The expected number of coin tosses required in order to get tails is 2
When we scan forward in a list, the destination key does not belong to a higher list
A scan-forward step is associated with a former coin toss that gave tails
By Fact 4, in each list the expected number of scan-forward steps is 2
Thus, the expected number of scan-forward steps is O(log n)
We conclude that a search in a skip list takes O(log n) expected time
The analysis of insertion and deletion gives similar results
12/3/2018 8:58 AM
Dictionaries

27. Summary
A skip list is a data structure for dictionaries that uses a randomized insertion algorithm
In a skip list with n items
The expected space used is O(n)
The expected search, insertion and deletion time is O(log n)
Using a more complex probabilistic analysis, one can show that these performance bounds also hold with high probability
Skip lists are fast and simple to implement in practice
12/3/2018 8:58 AM
Dictionaries