CS 261: Data Structures. Skip Lists

Transcription

1 CS 261: Data Structures Skip Lists 1

2 Complexity Lists and Arrays OPERATIONS ORDINARY LISTS AND ARRAYS SORTED ARRAYS SORTED LISTS Add O(1) O(n) O(n) Remove O(n) O(n) O(n) Contains O(n) O(log n) O(n) 2

3 Sorted Linked Lists How to improve complexity of a sorted linked list? We could use two sorted linked lists, with pointers between them List

4 Two Sorted Linked Lists Constructed from the same elements Establish pointers between equal links down down down

5 Motivation Regular Trains vs. Express Trains down down down

6 Two Sorted Linked Lists : stores all elements : stores only a subset of elements down down down

7 How to Search for an Element? We start from the 1 st element of Stay on the "express line" as long as you can Then, take the "local line" down down down

8 How to Choose Elements for? Goal: Maximize fast access to all elements picks uniformly a subset of elements (e.g. every 2 nd, or rd, or 4 th element) down down down

9 How to Choose Elements for? But how exactly to uniformly sample elements for? down down down

10 The Goal: Minimize Complexity of Search down down down

11 What is Complexity of Search? num. of elements in L 1 + L 2 L 1 num. of elements in a segment of down down down

12 What is Complexity of Search? L 1 + L 2 L 1 down down down

13 What is Complexity of Search? minimize x + n x

14 What is Complexity of Search? minimum of the function d x + n x dx =0 14

15 What is Complexity of Search? d x + n x dx =1 n x 2 =0 ) x = p n 15

16 What is Complexity of Search? p n = = p n L 1 + L 2 L 1 = n down down down

17 What is Complexity of Search? 2 p n down down down

18 What is Complexity of Search? two sorted lists one sorted list down down down

19 How Many Lists Should We Form? If 2 lists have lower complexity, maybe lists will have even lower complexity down down down down down List

20 What is Complexity for Lists? down down down down down List

21 What is Complexity for Lists? down = x = x 2 down down down down List

22 What is Complexity for Lists? down down down down down List

23 What is Complexity for Lists? = p n down down down down down List

24 What is Complexity for Lists? lists 2 lists single list down down down down down List

25 What is Complexity for k Lists? for k linked lists: k kp n down down down down down List k

26 How Many Lists? minimize k kp n down down down down down List k

27 How Many Lists? minimize k kn 1 k = minimize k 2

28 How Many Lists? minimize k kn 1 k = minimize k log k + 1k log n 28

29 How Many Lists? d log k + 1 k log n dk =0 = 1 k 1 log n =0 k2 k = log n 29

30 Complexity for k=log n Lists? kn 1 k = (log n) n 1 log n down up down up down up down up down up List

31 n =2 log n n 1 log n =(2 log n ) 1 log n =2 (log n) n 1 log n = 2 log n

32 Complexity for log n Lists O(log n) down up down up down up down up down up List

33 Skip Lists Have It All Pugh 1989 Fast addition O(log n) Fast search O(log n) Fast removal O(log n) Disadvantage: - Slightly more complicated

34 Contains Skip List Makes a zig-zag motion top-to-bottom Complexity: O(log n), i.e., proportional to the number of linked lists 4

35 Contains Skip List 1. Start at topmost sentinel 2. Loop as follows 1. Slide right, get a link right before 2. If next element is OK, return true. If no down element, return false 4. Move down 5

36 Remove Skip List Makes a zig-zag motion top-to-bottom Only decrement the size at the bottom level Complexity: O(log n), i.e., proportional to the number of linked lists 6

37 Remove Skip List 1. Start at topmost sentinel 2. Loop as follows 1. Slide right, get a link right before 2. If next element is OK, remove it. If no down element, reduce size 4. Move down

38 How to construct a skip list when we do not know the number of elements in advance? 8

39 Add Skip List Add the element to the bottom list must increment size To move up to existing lists: Flip a coin, and if heads add the element up To add a new list at the top Flip a coin, and if heads make a new top list 9

40 Add Example Insert the following:

41 Add Example: Inserted : 9 14 Coin toss: T HT ( H = move up) down

42 Add Example: Inserted : Coin toss: T HT HHH T List down down down

43 Complexity of Add Proportional to the height, not to the number of nodes in the list O(log n) 4

44 Skip List Sorting Algorithm Problem: Sort an array A Step 1. Copy elements from A into a skip list Step 2. Copy elements from the skip list to A 44

45 Skip List Sorting Algorithm Complexity: Step 1. Copy elements from A into a skip list O(??) Step 2. Copy elements from the skip list to A O(??) 45