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Randomized Algorithms - Treaps 104/19/23
Randomized Skip List
Skip List Or, In Hebrew: רשימות דילוג
Randomized Algorithms - Treaps 204/19/23
Outline
Motivation for the
skipping
Skip List definitions and
description
Analyzing performance
The role of randomness
Randomized Algorithms - Treaps 304/19/23
Outline
Motivation for the
skipping
Skip List definitions and
description
Analyzing performance
The role of randomness
Randomized Algorithms – Skip List 404/19/23
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!
Randomized Algorithms – Skip List 504/19/23
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.
Randomized Algorithms – Skip List 604/19/23
Search cost What is the search cost?
Randomized Algorithms – Skip List 704/19/23
Search cost – Cont.
This is minimized when:
Randomized Algorithms – Skip List 804/19/23
Discovering skip lists If we keep adding linked list layers we get:
Randomized Algorithms - Treaps 904/19/23
Outline
Motivation for the
skipping
Skip List definitions and
description
Analyzing performance
The role of randomness
Randomized Algorithms - Treaps 1004/19/23
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:
LLLL rr 121...
Randomized Algorithms - Treaps 1104/19/23
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.
Randomized Algorithms - Treaps 1204/19/23
Skip List Example For example, for this leveling:
This is the skip list corresponding to it:
}35,15,2{
}45,35,23,15,2{
}47,45,35,32,31,23,20,15,11,5,2{
3
2
1
LLL
Randomized Algorithms - Treaps 1304/19/23
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:
Randomized Algorithms - Treaps 1404/19/23
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.
Randomized Algorithms - Treaps 1504/19/23
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.
Randomized Algorithms - Treaps 1604/19/23
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.
Randomized Algorithms - Treaps 1704/19/23
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.
Randomized Algorithms - Treaps 1804/19/23
Outline
Motivation for the
skipping
Skip List definitions and
description
Analyzing performance
The role of randomness
Randomized Algorithms - Treaps 1904/19/23
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?
Randomized Algorithms - Treaps 2004/19/23
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.
Randomized Algorithms - Treaps 2104/19/23
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!
Randomized Algorithms - Treaps 2204/19/23
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
Randomized Algorithms - Treaps 2304/19/23
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.
Randomized Algorithms - Treaps 2404/19/23
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.
)(log])))((1([
1
nOyCEr
jjI
r
jjyC I
1
)))((1(
Randomized Algorithms - Treaps 2504/19/23
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).
))((1 yC I j
r
jjyC I
1
)))((1(
Randomized Algorithms - Treaps 2604/19/23
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.
Randomized Algorithms - Treaps 2704/19/23
Outline
Motivation for the
skipping
Skip List definitions and
description
Analyzing performance
The role of randomness
Randomized Algorithms - Treaps 2804/19/23
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.
Randomized Algorithms - Treaps 2904/19/23
. -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.