Randomized Skip List

Randomized Skip List • Skip List • Or, In Hebrew: רשימות דילוג Randomized Algorithms - Treaps

Outline • Motivation for the skipping • Skip List definitions and description • Analyzing performance • The role of randomness Randomized Algorithms - Treaps

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

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

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

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

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

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: 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. Randomized Algorithms - Treaps

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

The search path length – Cont. • Walking down the pathis 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

The search path length is O(logn) • Lemma 2: Let y be any element and consider the search pathfollowed 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. 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). 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. 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. Randomized Algorithms - Treaps

.- contThe 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. Randomized Algorithms - Treaps

Randomized Skip List