Hop, Jump, and Leap
48
Hop, Jump, and Leap Dhruv Matani and Gaurav Menghani
description
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. - PowerPoint PPT Presentation
Transcript of 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.
• Augmenting skip-lists to achieve decent running times for most problems.
• By decent, we mean O(lg n) w.h.p
Hop, Jump, and Leap
Order Maintenance
O(1) Amortized
Insert
O(1) Query
Static RMQ O(n) pre-processing
O(1) Query
In class, we saw:
Hop, Jump, and Leap
Order Maintenance
O(lg n) w.h.p O(lg n) w.h.p. Query
Dynamic RMQ O(lg n) w.h.p. Insert
O(lg n) w.h.p. Query
Today, we shall see:
Dynamic Range Minimum Query using
Skip Lists
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Find element at Index 4
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Find the element at index 4
71
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Find the element at index 4
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Find the element at index 4
1
71
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Find the element at index 4
6
71
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301
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71
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Find the element at index 4
1
71
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Find the element at index 4
3
71
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Find the element at index 4
6
71
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Find the element at index 4
3
71
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Find the element at index 4
4
Find minimum in range [3, 7]
71
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Find min [3,7]
71
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71
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Find min [3,7]
[3,7]
71
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Find min [3,7]
[3,7[ = ]3,6[ ,]7,7]
71
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71
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71
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Find min [3,7]
[3,6]
[7,7]
71
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71
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71
85
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71
85
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Find min [3,7]
[3,6]
[7,7]
71
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301
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191
241
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71
262
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71
85
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71
85
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Find min [3,7]
[3,6]
[7,7]
71
81
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301
261
191
241
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71
262
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71
85
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71
85
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Find min [3,7]
[7,7]
[3,6]
71
81
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301
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71
262
83
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71
85
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71
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Find min [3,7]
[7,7]
[3,6[ = ]3,3[ ,]4,6]
71
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71
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71
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Find min [3,7]
[7,7]
[4,6]
71
81
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301
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241
271
71
262
83
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71
85
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71
85
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Find min [3,7]
[7,7]
[4,6]
[3,3]
71
81
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301
261
191
241
271
71
262
83
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71
85
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71
85
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Find min [3,7]
[7,7]
[4,6]
[3,3]
71
81
131
301
261
191
241
271
71
262
83
192
71
85
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71
85
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Find min [3,7]
[7,7]
[4,6]
[3,3]
71
81
131
301
261
191
241
271
71
262
83
192
71
85
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71
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Find min [3,7]
[7,7]
[4,6]
[3,3]
71
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241
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71
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Find min [3,7]
[7,7]
[4,6]
[3,3]
Insert element with Value 5
at Index 5
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Insert 5 at index 5
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Insert 5 at index 5
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Insert 5 at index 5
51
]8,2[]5,2[
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Insert 5 at index 5
51
]8,2[]5,2[52]5,2[
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Insert 5 at index 5
51
]8,2[
52
]5,4[
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Insert 5 at index 5
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]8,2[
52
]5,4[54]5,4[
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Insert 5 at index 5
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]8,2[
52
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]5,4[
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Insert 5 at index 5
51
]8,2[
52
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]5,4[
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Insert 5 at index 5
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Order Maintenance using
Skip Lists
• 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
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Does 13 precede 19?
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Does 13 precede 19? Rank(13) = 1 + 2 + 1 = 4
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Does 13 precede 19? Rank(19) = 1 + 5 + 1 = 7
The End
