1. Sorted Maps
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2. Sorted Maps (Ordered Maps)
exact match to the key
Sorted Maps
Allows inexact match to the key
E.g. Flights departing between a time range
(MLB, JFK, 3Oct, 1:00) to (MLB, JFK, 3Oct, 3:00)
Kayak.com
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3. Main operations floorEntry(k) ceilingEntry(k) Submap(k1,k2)
Entry with the greatest key less than or equal to k
The key “at or before” k
ceilingEntry(k)
Entry with the least key value greater than or equal to k
The key “at or after” k
Submap(k1,k2)
Entries between k1 and k2
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4. Example Index Element Search Key is G F<G<K F is the floor of G1 2 3 4 5 6 A D F K M O Q
Index
Element
Search Key is G
F<G<K
F is the floor of G
K is the ceiling of G
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5. Sorted Search Tables Sorted array by the key values Exact match/search
Binary search
Inexact match/search
Modify binary search
To handle “not found” differently
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6. Binary search (review)
Recall low and high specify the valid range where key is
If the mid point is the key
Key is found
If low > high (no valid range)
Key is not found
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7. Binary Search
Binary search can perform nearest neighbor queries on an ordered map that is implemented with an array, sorted by key
similar to the high-low children’s game
at each step, the number of candidate items is halved
terminates after O(log n) steps
Example: find(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
© 2014 Goodrich, Tamassia, Goldwasser
Binary Search Trees

8. Binary Search (inexact)1 2 3 4 5 6 A D F K M O Q
Index
Element
Search Key is G
Low=0, high=6, mid=3; G<K, check first half
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9. Binary Search (inexact)1 2 3 4 5 6 A D F K M O Q
Index
Element
Search Key is G
Low=0, high=6, mid=3; G<K, check first half
Low=0, high=2, mid=1; G>D, check second half
© 2014 Goodrich, Tamassia, Goldwasser
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10. Binary Search (inexact)1 2 3 4 5 6 A D F K M O Q
Index
Element
Search Key is G
Low=0, high=6, mid=3; G<K, check first half
Low=0, high=2, mid=1; G>D, check second half
Low=2, high=2, mid=2; G>F, check second half
© 2014 Goodrich, Tamassia, Goldwasser
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11. Binary Search (inexact)1 2 3 4 5 6 A D F K M O Q
Index
Element
Search Key is G
Low=0, high=6, mid=3; G<K, check first half
Low=0, high=2, mid=1; G>D, check second half
Low=2, high=2, mid=2; G>F, check second half
Low=3, high=2; low>high, not found
F<G<K
Low=3 is the ceiling of G; high=2 is the floor of G
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12. Disadvantage of a sorted array
Fixed maximum size
Put/remove could lead to moving a large number of entries
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13. Worst-case Time Complexity
n entries in a sorted array
Operation
Time Complexity
get(k)
put(k,v)
remove(k)
ceilingEntry(k)
floorEntry(k)
submap(k1,k2)
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14. Worst-case Time Complexity
n entries in a sorted array
Operation
Time Complexity
get(k)
O(log n)
put(k,v)
O(n)
remove(k)
ceilingEntry(k)
floorEntry(k)
submap(k1,k2)
O(log n + s), s entries reported
O(n) if all entries are reported
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15. No Fixed Maximum Size Sorted Linked List
Instead of a sorted array
Can we still use binary search?
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16. Skip Lists S3 + - S2 + - S1 + - S0 + - 15 23 15 10 36 23 15
12/2/2018
Skip Lists S3 + - S2 + - 15 S1 + - 23 15 S0 + - 10 36 23 15
© 2014 Goodrich, Tamassia, Goldwasser
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17. Linked List and Search Sorted array Sorted linked list Binary search
Can’t use binary search
No direct access to a node
Only sequential access via the next pointer
Additional data structure
Skip lists for faster search/updates
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18. 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 S3 + - S2 - 31 + S1 - 23 31 34 64 + S0 56 64 78 + 31 34 44 - 12 23 26
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19. Search for Key k
start at the first position of the top list
At the current position p, nextKey  key(next(p))
Compare k with nextKey
k = nextKey: we return element(next(p))
k >= nextKey: we “scan forward” (keep looking, similar to a regular linked list)
k < nextKey: we “drop down” (can’t find on the current list, try a lower list)
If we try to drop down past the bottom list, we return null
Example: search for 78 S3 + -
scan forward S2 - 31 +
drop down S1 - 23 31 34 64 + S0 - 12 23 26 31 34 44 56 64 78 +
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20. How to find the floor and ceiling of 70?
Search for Key k
start at the first position of the top list
At the current position p, nextKey  key(next(p))
Compare k with nextKey
k = nextKey: we return element(next(p))
k >= nextKey: we “scan forward”
k < nextKey: we “drop down”
If we try to drop down past the bottom list, we return null
Example: search for 78 S3 + -
scan forward S2 - 31 +
drop down S1 - 23 31 34 64 + S0 - 12 23 26 31 34 44 56 64 78 +
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21. Expected Height of Skip Lists
Items in list Si+1
Randomly selected from Si based on a “coin toss”
Si+1 has about half of the entries in Si
For N entries
Expected number of skip lists (height) is log N
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22. Pseudorandom Numbers Library functions Simulate a coin toss
Generate pseudorandom numbers
Not truly random
Starting with the same “seed”
The same sequence of pseudorandom numbers can be generated
Simulate a coin toss
0 is tail, 1 is head
Even is tail, odd is head
© 2014 Goodrich, Tamassia, Goldwasser
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23. Randomized Algorithms
Based on pseudorandom numbers
Perform an arbitrary operation out of a fixed set of operations
For example,
operation a if odd
operation b if even
Insertion to skip lists
uses a randomized algorithm
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24. Insertion: entry (k,v) Repeatedly toss a coin until we get tails
i = 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
For each list Sj,
find the position pj that is right before k (floor)
insert entry (k, v) 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
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25. Deletion: key k
search for x in the skip list and find the positions p0, p1 , …, pi of the items with key k, where position pj is in list Sj
remove positions p0, p1 , …, pi from the lists S0, S1, … , Si
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 +
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26. Similar to Binary Search
Roughly check the middle, eliminate half
31 in S2 divides the list into two halves
23 in S1 divides the first half into two smaller halves …
Each list has about half of the entries in the previous list S3 + - S2 - 31 + S1 - 23 31 34 64 + S0 56 64 78 + 31 34 44 - 12 23 26
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27. Put/insert Why # of heads before tail Though sound random
Determines the height of an entry
Or # of lists the entry is in
Though sound random
Each list has about half of the entries in the previous list
Expected height of log N
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28. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i T
H,T 1
H,H,T 2
H,H,H,T 3 …
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29. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i T
1/2
H,T 1
1/4
H,H,T 2
1/8
H,H,H,T 3
1/16 …
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30. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i (list i) T
1/2 50 ?
H,T 1
1/4 25
H,H,T 2
1/8
~12
H,H,H,T 3
1/16 ~6 …
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31. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i T
1/2 50
100
H,T 1
1/4 25
H,H,T 2
1/8
~12
H,H,H,T 3
1/16 ~6 …
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32. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i T
1/2 50
100
H,T 1
1/4 25
H,H,T 2
1/8
~12
H,H,H,T 3
1/16 ~6 …
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33. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i T
1/2 50
100
H,T 1
1/4 25
H,H,T 2
1/8
~12
H,H,H,T 3
1/16 ~6 …
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34. Each list has about half of the entries of the previous list
Coin toss sequence
Height (h)
Probability of sequence
% of entries with height h
% of entries at level i T
1/2 50
100
H,T 1
1/4 25
H,H,T 2
1/8
~12
H,H,H,T 3
1/16 ~6 …
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35. Implementation x quad-node
We can implement a skip list with quad-nodes
A quad-node stores:
element
link to the node prev
link to the node next
link to the node below
link to the node above
special keys PLUS_INF and MINUS_INF
quad-node x
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36. Expected (not Worst-case) Complexity
N entries
Expected Complexity
Height
O(log N)
Space Complexity
O(N)
Time Complexity
Get/Find
Put/Insert
Remove/Delete
Ceiling, Floor
Submap
Skipping the proof (beyond the scope of this course)
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37. Skipping the rest
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38. Space Usage
Depends on the random “coin toss” used by each invocation of the insertion algorithm
Two basic probabilistic facts:
Fact 1:
The probability of getting i consecutive heads when flipping a coin is (½)i =1/2i
Fact 2:
If each of n entries is present in a set with probability p
the expected size of the set is np
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39. Space Usage Consider a skip list with n entries
By Fact 1, we insert an entry 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
Thus, the expected space usage of a skip list with n items is O(n)
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40. Height 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 entries
By Fact 1, we insert an entry 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, the probability that S3log n has at least one entry is at most n/23log n = n/n3 = 1/n2
Thus a skip list with n entries has height at most 3log n with probability at least /n2
Highly likely, the height is O(log n)
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41. Time Complexity of Search
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
O(log n) with high probability
To analyze the scan-forward steps, we use:
Fact 4: The expected number of coin tosses required to get tails is 2
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42. Time Complexity of Search
scan forward in a list (and 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 2
Expected overall number of scan-forward steps
O(log n)
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43. Time Complexity of Search
Drop-down steps
O(log n)
Scan-forward steps
Overall: O(log n) expected time
The analysis of insertion and deletion gives similar results
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44. Summary
A skip list is a data structure for maps that uses a randomized insertion algorithm
In a skip list with n entries
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
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