Segment Tree Basics

Transcript

1. Segment Tree
   基础篇
   

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2. Range Minimum/Maximum Query
   •需求：在饿了么的⼀个⽤户的历史订单中，找出该⽤户指定⼀段
   时间内单笔订单最贵/最便宜的订单
   One problem
   •O(n), when query K —> ∞ times ?
   

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3. Range Minimum/Maximum Query
   •需求：在饿了么的⼀个⽤户的历史订单中，求出找出该⽤户指定
   ⼀段时间内累积消费总⾦额
   Another problem
   •O(n), when query K —> ∞ times ?
   •prefixSum, time O(n), query O(1)
   

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4. Range Minimum/Maximum Query
   •需求：在 12306 ⽹站上，统计⼀条线路上任意指定两个站点之间
   的可售出的票数总和
   One more thing
   •O(n), when query K —> ∞ times ?
   •prefixSum, time O(n), update O(n), query O(1)
   

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5. How to ?
   Range Minimum/Maximum Query
   5
   O(log n) or O(1) ？
   

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6. Presentation agenda
   6
   What
   How to use Leetcode example
   What is segment tree
   2
   3
   1
   How Example
   

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7. Define
   Segment Tree
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   1. ❌ Complete Binary Tree
   2. ∈ Balanced Binary Tree
   3. As Full binary tree
   

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8. How many node
   Segment Tree
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   0-level ： 1
   1-level ： 2
   2-level ： 4
   3-level ： 8
   4-level ： 16
   (n-1)—level ：2^(n-1)
   n-level：2^n
   Σ 0-(n-1) level：Σ = 2^n - 1 ≈ n-level => 2n
   Mathematical Induction：Σ = 4n (n = 2^k or n = 2^k + 1)
   

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9. How many node
   Segment Tree
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   •When n≥3, [1,n] segment split [1,n] into 2*⌊log (n-1)⌋ sub-
   Interval
   •4n ≥ 2*⌊log (n-1)⌋ <= 2n ≥ log (n-1) <= 4^n ≥ n-1
   

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10. How to create
    Segment Tree
    10
    

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11. How to query
    Segment Tree
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    ❗
    

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12. How to update
    Segment Tree
    12
    ❗
    

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13. How to update
    Segment Tree
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    •Update [2,3] ？
    •❌ O(n)
    

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14. How to update
    Segment Tree
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    •Update [2,3] ？
    •✅O(log n)
    

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15. How to update
    Segment Tree
    15
    

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16. How to query
    Segment Tree
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    •push_down
    

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17. How to update
    Segment Tree
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    •push_down + push_up
    

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18. How to low space
    Segment Tree
    18
    •Leetcode 715
    

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19. Compared
    ST VS Array
    19
    20%
    80%
    60%
    40%
    2585
    785
    Update:
    Array O(n) | Segment Tree O(log n)
    Query:
    Array O(n) | Segment Tree O(log n)
    

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20. Segment Tree
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    Example
    

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21. Segment Tree
    21
    Example
    

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22. Segment Tree
    22
    Example
    

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23. Boyer-Moore Majority Vote Algorithm
    23
    Example
    

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24. Segment Tree
    24
    Example
    

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25. Segment Tree
    25
    Example
    •judge threshold
    •count = map[1:[0 1 4 5] 2:[2 3]]
    •eg: 1 in [0,5] count , twice binary search, find [lowerBound，
    upperBound)
    •(upperBound - lowerBound) ≥ threshold
    

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26. Segment Tree
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    •Leetcode 327. Count of Range Sum
    •Leetcode 715. Range Module
    •Leetcode 699. Falling Squares & 732. My Calendar III
    •Leetcode 850. Rectangle Area II
    •Leetcode 218. The Skyline Problem
    Exercise
    

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27. Advance
    27
    15% 35% 50%
    •One point update (update: min/change, query: sum/min)
    •Interval update (update: min/change, query: sum/hash)
    •Discretization
    •Interval merge
    •Scan line
    •Binary Index Tree
    

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