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Optimizing Dynamic Range Queries with Skip Lists for Efficient Data Management

This document explores the innovative use of skip lists as a powerful data structure for dynamic range queries. It demonstrates augmenting skip lists to achieve optimal running times of O(log n) with high probability for various operations, such as finding elements by index and performing dynamic range minimum queries. Learn how skip lists facilitate efficient order maintenance and achieve quick access to elements with essential examples and practical applications in data handling.

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Optimizing Dynamic Range Queries with Skip Lists for Efficient Data Management

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  1. Hop, Jump, and Leap Dhruv Matani and Gaurav Menghani

  2. Hop, Jump, and Leap* Dhruv Matani and Gaurav Menghani * In O(lg n) with high probability

  3. Hop, Jump, and Leap • Skip Lists are a simple and powerful data-structure. • Augmenting skip-lists to achieve decent running times for most problems. • By decent, we mean O(lg n) w.h.p

  4. Hop, Jump, and Leap In class, we saw:

  5. Hop, Jump, and Leap Today, we shall see:

  6. Dynamic Range Minimum Query using Skip Lists

  7. 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  8. Find element at Index 4

  9. Find the element at index 4 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  10. Find the element at index 4 7 1 8 5 19 2 6 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  11. Find the element at index 4 7 1 8 5 19 2 1 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  12. Find the element at index 4 7 1 8 5 19 2 6 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  13. Find the element at index 4 7 1 8 5 19 2 7 1 8 5 19 2 1 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  14. Find the element at index 4 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 3 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  15. Find the element at index 4 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 6 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  16. Find the element at index 4 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 3 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  17. Find the element at index 4 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 4 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  18. Find minimum in range [3, 7]

  19. Find min [3,7] 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  20. Find min [3,7] 7 1 8 5 19 2 [3,7] 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  21. Find min [3,7] [3,7] = [3,6], [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  22. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [3,6] 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  23. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [3,6] 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  24. Find min [3,7] [7,7] 7 1 8 5 19 2 [3,6] 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  25. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [3,6] 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  26. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [3,6] = [3,3], [4,6] 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  27. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [4,6] 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  28. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [4,6] 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1 [3,3]

  29. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [4,6] 7 1 26 2 8 3 19 2 [3,3] 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  30. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [4,6] 7 1 26 2 8 3 19 2 [3,3] 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  31. Find min [3,7] [7,7] 7 1 8 5 19 2 7 1 8 5 19 2 [4,6] 7 1 26 2 8 3 19 2 [3,3] 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  32. Find min [3,7] 7 1 8 5 19 2 7 1 8 5 19 2 [4,6] 7 1 26 2 8 3 19 2 [3,3] [7,7] 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  33. Insert element with Value 5 at Index 5

  34. Insert 5 at index 5 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  35. Insert 5 at index 5 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  36. Insert 5 at index 5 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 [5,2] [8,2] 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  37. Insert 5 at index 5 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 5 2 8 2 19 2 [5,2] [5,2] [8,2] 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  38. Insert 5 at index 5 7 1 8 5 19 2 7 1 8 5 19 2 [5,4] [8,2] 7 1 26 2 5 2 8 2 19 2 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  39. Insert 5 at index 5 7 1 8 5 19 2 7 1 5 4 8 2 19 2 [5,4] [5,4] [8,2] 7 1 26 2 5 2 8 2 19 2 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  40. Insert 5 at index 5 7 1 8 5 19 2 [5,4] [8,2] 7 1 5 4 8 2 19 2 7 1 26 2 5 2 8 2 19 2 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  41. Insert 5 at index 5 7 1 5 6 19 2 [5,4] [8,2] 7 1 5 4 8 2 19 2 7 1 26 2 5 2 8 2 19 2 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  42. Insert 5 at index 5 7 1 5 6 19 2 7 1 5 4 8 2 19 2 7 1 26 2 5 2 8 2 19 2 5 1 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  43. Order Maintenance using Skip Lists

  44. To perform Order Maintenance, we insert an element after a given element. • This costs O(lg n) w.h.p • Does X precede Y? • Is the rank(X) < rank(Y)? • This costs O(lg n) w.h.p

  45. Does 13 precede 19? 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  46. Does 13 precede 19? Rank(13) = 1 + 2 + 1 = 4 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  47. Does 13 precede 19? Rank(19) = 1 + 5 + 1 = 7 7 1 8 5 19 2 7 1 8 5 19 2 7 1 26 2 8 3 19 2 7 1 30 1 26 1 13 1 8 1 24 1 19 1 27 1

  48. The End

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