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Hop, Jump, and Leap

Hop, Jump, and Leap. Dhruv Matani and Gaurav Menghani. Hop, Jump, and Leap *. Dhruv Matani and Gaurav Menghani. * In O(lg n) with high probability. Hop, Jump, and Leap. Skip Lists are a simple and powerful data-structure.

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Hop, Jump, and Leap

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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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