Fenwick Trees Dr. Mattox Beckman University of Illinois at - - PowerPoint PPT Presentation

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Fenwick Trees Dr. Mattox Beckman University of Illinois at - - PowerPoint PPT Presentation

Nov 22, 2022 126 likes •337 views

Introduction Navigation Fenwick Trees Dr. Mattox Beckman University of Illinois at Urbana-Champaign Department of Computer Science Introduction Navigation Objectives Your Objectives: Describe and implement a Fenwick Tree Compare a

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SLIDE 1

Introduction Navigation

Fenwick Trees

  • Dr. Mattox Beckman

University of Illinois at Urbana-Champaign Department of Computer Science

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Introduction Navigation

Objectives

Your Objectives: ◮ Describe and implement a Fenwick Tree ◮ Compare a Fenwick Tree to a Segment Tree

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SLIDE 3

Introduction Navigation

Motivating Example

Exam scores = {2, 4, 5, 5, 6, 6, 6, 7, 7, 8, 9}

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SLIDE 4

Introduction Navigation

Motivating Example

Exam scores = {2, 4, 5, 5, 6, 6, 6, 7, 7, 8, 9} index 1 2 3 4 5 6 7 8 9 10 value 1 1 2 3 2 1 1

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SLIDE 5

Introduction Navigation

Motivating Example

Exam scores = {2, 4, 5, 5, 6, 6, 6, 7, 7, 8, 9} index 1 2 3 4 5 6 7 8 9 10 value 1 1 2 3 2 1 1 cumulative 1 1 2 4 7 9 10 11 11

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SLIDE 6

Introduction Navigation

Motivating Example

Exam scores = {2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 8, 9} index 1 2 3 4 5 6 7 8 9 10 value 1 1 1 2 3 2 1 1 cumulative 1 2 3 5 8 10 11 12 12

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SLIDE 7

Introduction Navigation

Fenwick Tree

value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10

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Introduction Navigation

Fenwick Tree

value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010

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SLIDE 9

Introduction Navigation

Fenwick Tree

ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010

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SLIDE 10

Introduction Navigation

Fenwick Tree

ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010

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SLIDE 11

Introduction Navigation

Fenwick Tree

ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010

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SLIDE 12

Introduction Navigation

Fenwick Tree

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010

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SLIDE 13

Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. E.g.: 5 = 101 4 = 100

ft ft

E.g.: 10 = 1010 8 = 1000

ft ft

Update 5: Visit 5=101 6=110 8=1000 LSOne(n) = n & -n

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Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. ◮ E.g.: 5 = 101 → 4 = 100 → 0

◮ ft(5) + ft(4) = 2 + 2 = 4

E.g.: 10 = 1010 8 = 1000

ft ft

Update 5: Visit 5=101 6=110 8=1000 LSOne(n) = n & -n

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SLIDE 15

Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. ◮ E.g.: 5 = 101 → 4 = 100 → 0

◮ ft(5) + ft(4) = 2 + 2 = 4

E.g.: 10 = 1010 8 = 1000

ft ft

Update 5: Visit 5=101 6=110 8=1000 LSOne(n) = n & -n

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SLIDE 16

Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. ◮ E.g.: 5 = 101 → 4 = 100 → 0

◮ ft(5) + ft(4) = 2 + 2 = 4

◮ E.g.: 10 = 1010 → 8 = 1000 → 0

◮ ft(10) + ft(8) = 1 + 10 = 11

Update 5: Visit 5=101 6=110 8=1000 LSOne(n) = n & -n

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SLIDE 17

Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. ◮ E.g.: 5 = 101 → 4 = 100 → 0

◮ ft(5) + ft(4) = 2 + 2 = 4

◮ E.g.: 10 = 1010 → 8 = 1000 → 0

◮ ft(10) + ft(8) = 1 + 10 = 11

Update 5: Visit 5=101 6=110 8=1000 LSOne(n) = n & -n

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SLIDE 18

Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. ◮ E.g.: 5 = 101 → 4 = 100 → 0

◮ ft(5) + ft(4) = 2 + 2 = 4

◮ E.g.: 10 = 1010 → 8 = 1000 → 0

◮ ft(10) + ft(8) = 1 + 10 = 11

◮ Update 5: Visit 5=101 → 6=110 → 8=1000 LSOne(n) = n & -n

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SLIDE 19

Introduction Navigation

Queries

ft 4 10 ft 3 2 ft 2 1 5 1 ft 1 2 2 1 value 1 1 2 3 2 1 1 index 1 2 3 4 5 6 7 8 9 10 index 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 ◮ To Query sum for position n, fjrst read n. ◮ Then, subtract the lowest order bit, and repeat until n = 0. ◮ E.g.: 5 = 101 → 4 = 100 → 0

◮ ft(5) + ft(4) = 2 + 2 = 4

◮ E.g.: 10 = 1010 → 8 = 1000 → 0

◮ ft(10) + ft(8) = 1 + 10 = 11

◮ Update 5: Visit 5=101 → 6=110 → 8=1000 ◮ LSOne(n) = n & -n