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

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

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

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! 4/20/2017 Randomized Algorithms – Skip List

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. 4/20/2017 Randomized Algorithms – Skip List

Randomized Algorithms – Skip List Search cost What is the search cost? 4/20/2017 Randomized Algorithms – Skip List

Randomized Algorithms – Skip List Search cost – Cont. This is minimized when: 4/20/2017 Randomized Algorithms – Skip List

Discovering skip lists If we keep adding linked list layers we get: 4/20/2017 Randomized Algorithms – Skip List

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

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: 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

Randomized Algorithms - Treaps Skip List Example For example, for this leveling: This is the skip list corresponding to it: 4/20/2017 Randomized Algorithms - Treaps

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: 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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

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

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? 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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! 4/20/2017 Randomized Algorithms - Treaps

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 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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). 4/20/2017 Randomized Algorithms - Treaps

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. 4/20/2017 Randomized Algorithms - Treaps

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

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. 4/20/2017 Randomized Algorithms - Treaps

.- 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. 4/20/2017 Randomized Algorithms - Treaps