1. Randomized Algorithms - Treaps
Randomized Skip List
Skip List
Or, In Hebrew: רשימות דילוג
4/20/2017
Randomized Algorithms - Treaps

2. Randomized Algorithms - Treaps
Outline
Motivation for the skipping
Skip List definitions and description
Analyzing performance
The role of randomness
4/20/2017
Randomized Algorithms - Treaps

3. Randomized Algorithms - Treaps
Outline
Motivation for the skipping
Skip List definitions and description
Analyzing performance
The role of randomness
4/20/2017
Randomized Algorithms - Treaps

4. Randomized Algorithms – Skip List
Starting from scratch
Initial goal just search, no updates (insert, delete).
Simplest Data Structure?
linked list!
Search? O(n)!
Can we do it faster?
yes we can!
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Randomized Algorithms – Skip List

5. Developing the skip list
Let’s add an express lane.
Can quickly jump from express stop to next express stop, or from any stop to next normal stop.
To search, first search in the express layer until about to go to far, then go down and search in the local layer.
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Randomized Algorithms – Skip List

6. Randomized Algorithms – Skip List
Search cost
What is the search cost?
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Randomized Algorithms – Skip List

7. Randomized Algorithms – Skip List
Search cost – Cont.
This is minimized when:
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Randomized Algorithms – Skip List

8. Discovering skip lists
If we keep adding linked list layers we get:
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Randomized Algorithms – Skip List

9. Randomized Algorithms - Treaps
Outline
Motivation for the skipping
Skip List definitions and description
Analyzing performance
The role of randomness
4/20/2017
Randomized Algorithms - Treaps

10. Randomized Algorithms - Treaps
Initial definitions
Let S be a totally ordered set of n elements.
A leveling with r levels of S, is a sequence of nested subsets (called levels) : where and
Given a leveling for S, the level of any element x in s is defined as:
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Randomized Algorithms - Treaps

11. Randomized Algorithms - Treaps
Skip List Description
Given any leveling of the set S, we define the skip list corresponding to this structure like this:
The level is stored in a sorted link list.
Each node x in this linked list has a pile of nodes above it.
There are horizontal and vertical pointers between nodes.
For convenience, we assume that two special elements and belong to each of the levels.
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Randomized Algorithms - Treaps

12. Randomized Algorithms - Treaps
Skip List Example
For example, for this leveling:
This is the skip list corresponding to it:
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Randomized Algorithms - Treaps

13. Skip list as binary tree
An interval level I, is the set of elements of S, spanned by a specific horizontal pointer at level i.
For example, for the previous skip list, this is the interval at level 2:
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Randomized Algorithms - Treaps

14. Skip list as binary tree
The interval partition structure is more conveniently viewed as a tree, where each node corresponds to an interval.
If an interval J at level i+1 contains as a subsets an interval I at level i, then node J is the parent of node I in the tree.
For interval I at level i+1, C(I) denotes the number of children of Interval I at level i.
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Randomized Algorithms - Treaps

15. Randomized Algorithms - Treaps
Searching skip lists
Consider an element y, that is not necessarily an member of S, and assume we want to search for it in skip list s.
Let be the interval at level j that contains y.
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Randomized Algorithms - Treaps

16. Searching skip lists – Cont.
We can now view the nested sequence of intervals as a root-leaf path in the tree representation of the skip list.
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Randomized Algorithms - Treaps

17. Randomized Algorithms - Treaps
Random skip list
To complete the description of the skip list, we have to specify the choice of the leveling that underlies it.
The basic idea is to chose a random leveling, thereby defining a random skip list.
A random leveling is defined as follows:
given the choice of level
the level is defined by independently choosing to retain each element with probability ½.
4/20/2017
Randomized Algorithms - Treaps

18. Randomized Algorithms - Treaps
Outline
Motivation for the skipping
Skip List definitions and description
Analyzing performance
The role of randomness
4/20/2017
Randomized Algorithms - Treaps

19. Randomized Algorithms - Treaps
Search cost
What is the expected time to find an element in a random skip list?
We will show it is O(logn) with high probability.
What is the expected space of a random skip list?
O(n). Why?
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Randomized Algorithms - Treaps

20. Randomized Algorithms - Treaps
Random procedure
An alternative view of the random construction is as follows:
Let l(x) for every be independent random variable, with geometric distribution.
Let r be one more than the maximum of these random variables.
Place x in each of the levels, …
As with random priorities in treaps, a random level is chosen once for every element in it’s insertion.
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Randomized Algorithms - Treaps

21. The expected number of levels
Lemma: the number of levels r in a random leveling of a set S of size n is O(logn) with high probability.
Proof: look at the board!
Does it mean that the E[r]=O(logn)? Why?
Is it enough?
No!
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Randomized Algorithms - Treaps

22. Randomized Algorithms - Treaps
The search path length
The last result implies that the tree representing the skip list has height o(logn) with high probability.
Unfortunately, since the tree need not be binary, it does not immediately follows that the search time is similarly bounded.
The implementation of Find(x,s) corresponds to walking down the path
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Randomized Algorithms - Treaps

23. The search path length – Cont.
Walking down the path is as follows:
At level j, starting at the node , use a vertical pointer to descend to the leftmost child of the current interval;
then using the horizontal pointers, move rightward till the node
The cost of FIND(x,s) proportional to the number of levels as well as the number of intervals visited at each level.
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Randomized Algorithms - Treaps

24. The search path length is O(logn)
Lemma 2: Let y be any element and consider the search path followed by FIND(y, S) in a random skip list for the set S of size n, then:
A is the length of the search path. B.
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Randomized Algorithms - Treaps

25. Randomized Algorithms - Treaps
Proof of Lemma 2
Proof of A: the number of nodes visited at level j does not exceed the number of children of the interval therefore in each level, you walk through elements and in total
Proof of B: look at the board!
This result shows that the expected time of search is o(logn).
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Randomized Algorithms - Treaps

26. Insert and delete in skip list
Insert and delete operation can be done both in o(logn) expected time also.
To insert an element y:
a random level l(y) (may exceed r) should be chosen for y as described earlier.
Then a search operation should find the search path of y
Then update the correct intervals, and add pointers.
Delete operation is just the converse of insert.
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Randomized Algorithms - Treaps

27. Randomized Algorithms - Treaps
Outline
Motivation for the skipping
Skip List definitions and description
Analyzing performance
The role of randomness
4/20/2017
Randomized Algorithms - Treaps

28. Randomized Algorithms - Treaps
The role of randomness
What is the role of the randomness in skip lists?
The random leveling of S enables us to avoid complicated “balancing” operations.
This simplifies our algorithm and decreases the overhead.
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Randomized Algorithms - Treaps

29. .- cont The role of randomness
It also saves us the need to “remember” the state of the node or the system.
Unlike binary search trees, skip lists behavior is indifferent to the input distribution.
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Randomized Algorithms - Treaps