Parallel Algorithms for Trees

Ahmad KhayyatDepartment of Electrical & Computer Engineering

ahmad.khayyat@ece.queensu.ca

Parallel Computation: Models and MethodsSelim G. Akl Chapter 6

CISC 879 — Algorithms and ApplicationsQueen’s University

October 29, 2008

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 1

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Outline

1 Introduction

2 Building Euler Tours

3 Applications of Euler Tours

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Graphs

Outline

1 IntroductionGraphsEuler Tours

2 Building Euler Tours

3 Applications of Euler Tours

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Graphs

Graphs

A Graph G = (V,E) is a set of vertices V connected by a setof edges E.

If the edges have orientation, the graph is directed.

A path is an ordered list of edges in the form(vi, vj), (vj , vk), (vk, vl).

A cycle is a path that begins and ends at the same vertex.

The degree of a vertex in an undirected graph is the numberof edges adjacent to the vertex.

The in-degree of a vertex v in a directed graph is the numberof edges entering v,The out-degree of a vertex v in a directed graph is thenumber of edges leaving v.

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Outline

1 IntroductionGraphsEuler Tours

2 Building Euler Tours

3 Applications of Euler Tours

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

An undirected graph has an Euler tour if each vertex has an evendegree.

(v1, v7), (v7, v2), (v2, v3), (v3, v6)

(v6, v7), (v7, v5), (v5, v6)

(v6, v4), (v4, v5), (v5, v1)

v1

v5

v7

v4 v6 v3

v2

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

An undirected graph has an Euler tour if each vertex has an evendegree.

(v1, v7), (v7, v2), (v2, v3), (v3, v6)

(v6, v7), (v7, v5), (v5, v6)

(v6, v4), (v4, v5), (v5, v1)

v1

v5

v7

v4 v6 v3

v2

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

An undirected graph has an Euler tour if each vertex has an evendegree.

(v1, v7), (v7, v2), (v2, v3), (v3, v6)

(v6, v7), (v7, v5), (v5, v6)

(v6, v4), (v4, v5), (v5, v1)

v1

v5

v7

v4 v6 v3

v2

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

An undirected graph has an Euler tour if each vertex has an evendegree.

(v1, v7), (v7, v2), (v2, v3), (v3, v6)

(v6, v7), (v7, v5), (v5, v6)

(v6, v4), (v4, v5), (v5, v1) v1

v5

v7

v4 v6 v3

v2

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

A directed graph has an Euler tour if the in-degree of each vertexis equal to the out-degree of the vertex.

(v1, v2), (v2, v3), (v3, v4)

(v4, v2), (v2, v4)

(v4, v5), (v5, v6), (v6, v4), (v4, v6)

(v6, v5), (v5, v1)

v1 v2

v3 v4

v5 v6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 6

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

A directed graph has an Euler tour if the in-degree of each vertexis equal to the out-degree of the vertex.

(v1, v2), (v2, v3), (v3, v4)

(v4, v2), (v2, v4)

(v4, v5), (v5, v6), (v6, v4), (v4, v6)

(v6, v5), (v5, v1)

v1 v2

v3 v4

v5 v6

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

A directed graph has an Euler tour if the in-degree of each vertexis equal to the out-degree of the vertex.

(v1, v2), (v2, v3), (v3, v4)

(v4, v2), (v2, v4)

(v4, v5), (v5, v6), (v6, v4), (v4, v6)

(v6, v5), (v5, v1)

v1 v2

v3 v4

v5 v6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 6

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

A directed graph has an Euler tour if the in-degree of each vertexis equal to the out-degree of the vertex.

(v1, v2), (v2, v3), (v3, v4)

(v4, v2), (v2, v4)

(v4, v5), (v5, v6), (v6, v4), (v4, v6)

(v6, v5), (v5, v1)

v1 v2

v3 v4

v5 v6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 6

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Euler Tours

Euler Tour (ET)

A cycle where every edge of the graph appears in the cycle exactlyonce.

A directed graph has an Euler tour if the in-degree of each vertexis equal to the out-degree of the vertex.

(v1, v2), (v2, v3), (v3, v4)

(v4, v2), (v2, v4)

(v4, v5), (v5, v6), (v6, v4), (v4, v6)

(v6, v5), (v5, v1)

v1 v2

v3 v4

v5 v6

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Euler Tours

Trees

Tree

An undirected graph that is connected and contains no cycles.

A tree with n vertices has exactly n− 1 edges.

A directed tree (DT) with n vertices has 2n− 2 edges.

ë Its ET is a sequence of 2n− 2 edges.

v1v3 v4

v5

v6v7

v2

v1v3 v4

v5

v6v7

v2

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Data Structure

Outline

1 Introduction

2 Building Euler ToursData StructureThe AlgorithmAnalysis

3 Applications of Euler Tours

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Data Structure

Directed Trees (DT) as Linked Lists

n linked lists, one for each vertex.

The nodes of a list are edges leaving that vertex.

(v1, v3)head(v1) (v1, v4) (v1, v5)

(v2, v7)head(v2)

(v3, v1)head(v3)

(v4, v1)head(v4)

(v5, v1)head(v5) (v5, v6) (v5, v7)

(v6, v5)head(v6)

(v7, v2)head(v7) (v7, v5)

v1v3 v4

v5

v6v7

v2

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Outline

1 Introduction

2 Building Euler ToursData StructureThe AlgorithmAnalysis

3 Applications of Euler Tours

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Algorithm Input and Output

Input:n linked lists representing a directed tree (DT).

Output:Compute the Euler tour (ET) for the given DT:

ë Arrange all the edges of the DT in a single linked list.ë Each edge (vi, vj) is followed by an edge (vj , vk).ë The first edge leaves some vertex vl, the last edge enters vl.

(v1, v3) (v3, v1) (v1, v4) (v4, v1)

(v1, v5) (v5, v6) (v6, v5) (v5, v7)

(v7, v2) (v2, v7) (v7, v5) (v5, v1)

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Building the Euler Tour in Parallel

Assuming the shared-memory model, with n− 1 processors.

Each processor Pij , i < j, is in charge of two edges: (vi, vj)and (vj , vi).

Pij determines the position (actually, the successor) of thetwo nodes holding (vi, vj) and (vj , vi) (or ij and ji) — in ET.

Successor of (vi, vj)

if next(ji) = jk thensucc(ij) ← jk

elsesucc(ij) ← head(vj)

end if

Successor of (vj , vi)

if next(ij) = im thensucc(ji) ← im

elsesucc(ji) ← head(vi)

end if

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Example: Pij = P13Successor of (v1, v3)

Successor of (vi, vj)

if next(ji) = jk thensucc(ij) ← jk

elsesucc(ij) ← head(vj)

end if

(v1, v3) (v3, v1)

v1v3 v4

v5

v6v7

v2

(v1, v3)head(v1) (v1, v4) (v1, v5)...

(v3, v1)head(v3)

31

13

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Example: Pij = P13Successor of (v1, v3)

Successor of (vi, vj)

if next(ji) = jk thensucc(ij) ← jk

elsesucc(ij) ← head(vj)

end if

next(31) = null

(v1, v3) (v3, v1)

v1v3 v4

v5

v6v7

v2

(v1, v3)head(v1) (v1, v4) (v1, v5)...

(v3, v1)head(v3)

31

13

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Example: Pij = P13Successor of (v1, v3)

Successor of (vi, vj)

if next(ji) = jk thensucc(ij) ← jk

elsesucc(ij) ← head(vj)

end if

succ(13) ← head(v3)

(v1, v3) (v3, v1)

v1v3 v4

v5

v6v7

v2

(v1, v3)head(v1) (v1, v4) (v1, v5)...

(v3, v1)head(v3)

31

13

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Example: Pij = P13Successor of (v3, v1)

Successor of (vj , vi)

if next(ij) = im thensucc(ji) ← im

elsesucc(ji) ← head(vi)

end if

(v1, v3) (v3, v1) (v1, v4)

v1v3 v4

v5

v6v7

v2

(v1, v3)head(v1) (v1, v4) (v1, v5)...

(v3, v1)head(v3)

13

31

14

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Example: Pij = P13Successor of (v3, v1)

Successor of (vj , vi)

if next(ij) = im thensucc(ji) ← im

elsesucc(ji) ← head(vi)

end if

next(13) = 14

(v1, v3) (v3, v1) (v1, v4)

v1v3 v4

v5

v6v7

v2

(v1, v3)head(v1) (v1, v4) (v1, v5)...

(v3, v1)head(v3)

13

31

14

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 14

Introduction Building Euler Tours Applications of Euler Tours Conclusion

The Algorithm

Example: Pij = P13Successor of (v3, v1)

Successor of (vj , vi)

if next(ij) = im thensucc(ji) ← im

elsesucc(ji) ← head(vi)

end if

succ(31) ← 14

(v1, v3) (v3, v1) (v1, v4)

v1v3 v4

v5

v6v7

v2

(v1, v3)head(v1) (v1, v4) (v1, v5)...

(v3, v1)head(v3)

13

31

14

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 14

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Analysis

Outline

1 Introduction

2 Building Euler ToursData StructureThe AlgorithmAnalysis

3 Applications of Euler Tours

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Analysis

Correctness

Visits all the edges:

ë Every pair of edges is assigned to a processor.

Produces a single cycle, not several small cycles:

(vi, vj) (vj , vk)

(vj , vi) (vk, vj)

(vj , vi)head(vj) (vj , vk)

next(ji) = jk

next(jk) = ji, or head(vj) = ji

vi vj vk

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Analysis

Correctness

Visits all the edges:

ë Every pair of edges is assigned to a processor.

Produces a single cycle, not several small cycles:

(vi, vj) (vj , vk)

(vj , vi) (vk, vj)

(vj , vi)head(vj) (vj , vk)

next(ji) = jk

next(jk) = ji, or head(vj) = ji

vi vj vk

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Analysis

Complexity

t(n) = O(1)p(n) = n− 1 = O(n)

⇒ c(n) = O(n)

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Outline

1 Introduction

2 Building Euler Tours

3 Applications of Euler ToursUtility ComputationsSimple Applications of Euler ToursComputing Minima

4 Conclusion

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Pointer Jumping parent(vi) ← parent(parent(vi))

Find the root of v8.

Time Complexity

Sequential: O(n)

Parallel: O(log n)

v0

v1

v2

v3

v4

v5

v6

v7

v8

1

1

1

1

1

1

1

1

2

2

1

1

1

1

2

2

3

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Pointer Jumping parent(vi) ← parent(parent(vi))

Find the root of v8.

Time Complexity

Sequential: O(n)

Parallel: O(log n)

v0

v1

v2

v3

v4

v5

v6

v7

v8

1

1

1

1

1

1

1

1

2

2

1

1

1

1

2

2

3

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Pointer Jumping parent(vi) ← parent(parent(vi))

Find the root of v8.

Time Complexity

Sequential: O(n)

Parallel: O(log n)

v0

v1

v2

v3

v4

v5

v6

v7

v8

1

1

1

1

1

1

1

1

2

2

1

1

1

1

2

2

3

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 19

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Pointer Jumping parent(vi) ← parent(parent(vi))

Find the root of v8.

Time Complexity

Sequential: O(n)

Parallel: O(log n)

v0

v1

v2

v3

v4

v5

v6

v7

v8

1

1

1

1

1

1

1

1

2

2

1

1

1

1

2

2

3

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Parallel Prefix Computation for Linked Lists

Step 1: forall i do in parallelnext(i) ← succ(i)

Step 2: finished ← falseStep 3: while not finished do

(3.1) finished ← true(3.2) forall i do in parallel

(i) if next(i) 6= nil then(a) val(next(i)) ← val(i) ◦ val(next(i))(b) next(i) ← next(next(i))

(ii) if next(i) 6= nil then

finished common←−−−−− false

x0 x1

x0-1

x2

x1-2

x3

x2-3

x4

x3-4

x5

x4-5

x6

x5-6

x7

x6-7x0 x0-1 x0-2 x0-3 x1-4 x2-5 x3-6 x4-7x0-4 x0-5 x0-6 x0-7

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Input & Output

Input: A linked listOutput: Prefix computation

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Parallel Prefix Computation for Linked Lists

Step 1: forall i do in parallelnext(i) ← succ(i)

Step 2: finished ← falseStep 3: while not finished do

(3.1) finished ← true(3.2) forall i do in parallel

(i) if next(i) 6= nil then(a) val(next(i)) ← val(i) ◦ val(next(i))(b) next(i) ← next(next(i))

(ii) if next(i) 6= nil then

finished common←−−−−− false

x0 x1

x0-1

x2

x1-2

x3

x2-3

x4

x3-4

x5

x4-5

x6

x5-6

x7

x6-7x0

x0-1 x0-2 x0-3 x1-4 x2-5 x3-6 x4-7x0-4 x0-5 x0-6 x0-7

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 20

Notation

xa-b = xa ◦ xa+1 ◦ · · · ◦ xb

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Parallel Prefix Computation for Linked Lists

Step 1: forall i do in parallelnext(i) ← succ(i)

Step 2: finished ← falseStep 3: while not finished do

(3.1) finished ← true(3.2) forall i do in parallel

(i) if next(i) 6= nil then(a) val(next(i)) ← val(i) ◦ val(next(i))(b) next(i) ← next(next(i))

(ii) if next(i) 6= nil then

finished common←−−−−− false

x0 x1

x0-1

x2

x1-2

x3

x2-3

x4

x3-4

x5

x4-5

x6

x5-6

x7

x6-7

x0 x0-1 x0-2 x0-3 x1-4 x2-5 x3-6 x4-7

x0-4 x0-5 x0-6 x0-7

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 20

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Parallel Prefix Computation for Linked Lists

Step 1: forall i do in parallelnext(i) ← succ(i)

Step 2: finished ← falseStep 3: while not finished do

(3.1) finished ← true(3.2) forall i do in parallel

(i) if next(i) 6= nil then(a) val(next(i)) ← val(i) ◦ val(next(i))(b) next(i) ← next(next(i))

(ii) if next(i) 6= nil then

finished common←−−−−− false

x0 x1

x0-1

x2

x1-2

x3

x2-3

x4

x3-4

x5

x4-5

x6

x5-6

x7

x6-7

x0 x0-1 x0-2 x0-3

x1-4 x2-5 x3-6 x4-7

x0-4 x0-5 x0-6 x0-7

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 20

Complexity

t(n) = O(log n)p(n) = n

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

Parallel Suffix Computation for Linked Lists

Step 1: forall i do in parallelnext(i) ← succ(i)

Step 2: finished ← falseStep 3: while not finished do

(3.1) finished ← true(3.2) forall i do in parallel

(i) if next(i) 6= nil then(a) val(i) ← val(i) ◦ val(next(i))(b) next(i) ← next(next(i))

(ii) if next(i) 6= nil then

finished common←−−−−− false

x0 x1 x2 x3 x4 x5 x6 x7

x0-7 x1-7 x2-7 x3-7 x4-7 x5-7 x6-7 x7

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 21

Complexity

t(n) = O(log n)p(n) = n

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

List Sequencing

List Sequencing

Computing the distance of each node from the beginning of thelist (node position).

Input: Linked list L Output: Sequence numbers

forall i do in parallel

val(i) ← 1

Parallel-Prefix(L, +)

1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 22

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

List Sequencing

List Sequencing

Computing the distance of each node from the beginning of thelist (node position).

Input: Linked list L Output: Sequence numbers

forall i do in parallel

val(i) ← 1

Parallel-Prefix(L, +)

1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

List Ranking

List Ranking

Computing the distance of each node from the end of the list.

Input: Linked list L Output: Node ranks

forall i do in parallel

(3.1) val(i) ← 1

(3.2) if succ(i) 6= nil then

succ(succ(i)) ← i

Parallel-Prefix(L, +)

1 1 1 1 1 1 1 1

8 7 6 5 4 3 2 1

Complexityt(n) = O(log n), p(n) = O(n)

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Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

List Ranking

List Ranking

Computing the distance of each node from the end of the list.

Input: Linked list L Output: Node ranks

forall i do in parallel

(3.1) val(i) ← 1

(3.2) if succ(i) 6= nil then

succ(succ(i)) ← i

Parallel-Prefix(L, +)

1 1 1 1 1 1 1 1

8 7 6 5 4 3 2 1

Complexityt(n) = O(log n), p(n) = O(n)

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 23

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Utility Computations

List Ranking

List Ranking

Computing the distance of each node from the end of the list.

Input: Linked list L Output: Node ranks

forall i do in parallel

(3.1) val(i) ← 1

(3.2) if succ(i) 6= nil then

succ(succ(i)) ← i

Parallel-Prefix(L, +)

1 1 1 1 1 1 1 1

8 7 6 5 4 3 2 1

Complexityt(n) = O(log n), p(n) = O(n)

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 23

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Outline

1 Introduction

2 Building Euler Tours

3 Applications of Euler ToursUtility ComputationsSimple Applications of Euler ToursComputing Minima

4 Conclusion

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 24

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

An Euler Tour is a . . .

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

An Euler Tour returns to the root of a subtree only after allvertices in the subtree have been visited.

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 25

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Depth-First Traversal

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

An Euler Tour returns to the root of a subtree only after allvertices in the subtree have been visited.

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 25

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Finding Parents

Recall:

ë pos(i) is node i’s sequence number.ë In an Euler Tour, a node looks like (vi, vj).ë If (vi, vj) is a node, then (vj , vi) is also a node.

pos(vi, vj) < pos(vj , vi) ⇒ (vi, vj) is an advance edgeotherwise (vi, vj) is a retreat edge.

parent(root) ← nilforall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) thenparent(vj) ← vi

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 26

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Counting Descendants des(vi)

The subtree of vi edges appear between the advance edge(parent(vi), vi) and the retreat edge (vi, parent(vi)) in ET.

Half of these edges are advance edges and half are retreat.

des(vi) is the number of advance edges in the subtree + 1.

des(vi) =pos(vi, parent(vi))− 1− pos(parent(vi), vi)

2+ 1

(v1, v3) (v3, v1) (v1, v4) (v4, v1)

(v1, v5) (v5, v6) (v6, v5) (v5, v7)

(v7, v2) (v2, v7) (v7, v5) (v5, v1)

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 27

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Preorder (top-down)

The preorder number of a vertex vj is the number of advanceedges in ET traversed before reaching vj for the first time + 1.

preorder(root) ← 1forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 1

else

val(ij) ← 0

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

preorder(vj) ← val(ij) + 1

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 28

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Preorder (top-down)

The preorder number of a vertex vj is the number of advanceedges in ET traversed before reaching vj for the first time + 1.

preorder(root) ← 1forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 1

else

val(ij) ← 0

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

preorder(vj) ← val(ij) + 1

1, 3

1

1

3, 1

0

1

1, 4

1

1

4, 1

0

2

1, 5

1

1

5, 6

1

1

6, 5

0

4

5, 7

1

1

7, 2

1

1

2, 7

0

6

7, 5

0

6

5, 1

0

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 28

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Preorder (top-down)

The preorder number of a vertex vj is the number of advanceedges in ET traversed before reaching vj for the first time + 1.

preorder(root) ← 1forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 1

else

val(ij) ← 0

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

preorder(vj) ← val(ij) + 1

1, 3

11

3, 1

01

1, 4

12

4, 1

02

1, 5

13

5, 6

14

6, 5

04

5, 7

15

7, 2

16

2, 7

06

7, 5

06

5, 1

06

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 28

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Preorder (top-down)

The preorder number of a vertex vj is the number of advanceedges in ET traversed before reaching vj for the first time + 1.

preorder(root) ← 1forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 1

else

val(ij) ← 0

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

preorder(vj) ← val(ij) + 1

1, 3

12

3, 1

0

1

1, 4

13

4, 1

0

2

1, 5

14

5, 6

15

6, 5

0

4

5, 7

16

7, 2

17

2, 7

0

6

7, 5

0

6

5, 1

0

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 28

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Postorder (bottom-up)

The postorder number of a vertex vi is the number of retreat edgesin ET traversed before reaching vi for the last time + 1.

postorder(root) ← nforall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 0

else

val(ij) ← 1

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) > pos(vj , vi) then

postorder(vi) ← val(ij)

v1v3 v4

v5

v6v7

v2

1

2 3

4

5 12

6

78

11

9

10

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 29

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Postorder (bottom-up)

The postorder number of a vertex vi is the number of retreat edgesin ET traversed before reaching vi for the last time + 1.

postorder(root) ← nforall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 0

else

val(ij) ← 1

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) > pos(vj , vi) then

postorder(vi) ← val(ij)

1, 3

0

0

3, 1

1

1

1, 4

0

1

4, 1

1

2

1, 5

0

2

5, 6

0

2

6, 5

1

3

5, 7

0

3

7, 2

0

3

2, 7

1

4

7, 5

1

5

5, 1

1

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 29

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Postorder (bottom-up)

The postorder number of a vertex vi is the number of retreat edgesin ET traversed before reaching vi for the last time + 1.

postorder(root) ← nforall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 0

else

val(ij) ← 1

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) > pos(vj , vi) then

postorder(vi) ← val(ij)

1, 3

00

3, 1

11

1, 4

01

4, 1

12

1, 5

02

5, 6

02

6, 5

13

5, 7

03

7, 2

03

2, 7

14

7, 5

15

5, 1

16

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 29

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Numbering Vertices Postorder (bottom-up)

The postorder number of a vertex vi is the number of retreat edgesin ET traversed before reaching vi for the last time + 1.

postorder(root) ← nforall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← 0

else

val(ij) ← 1

Parallel-Prefix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) > pos(vj , vi) then

postorder(vi) ← val(ij)

1, 3

0

0

3, 1

11

1, 4

0

1

4, 1

12

1, 5

0

2

5, 6

0

2

6, 5

13

5, 7

0

3

7, 2

0

3

2, 7

14

7, 5

15

5, 1

16

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 29

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Tree Binary Relations

“An Euler Tour returns to the root of a subtree only after allvertices in the subtree have been visited.”

vi is an ancestor of vj (vj is a descendant of vi) iff:

preorder(vi) ≤ preorder(vj) < preorder(vi) + des(vi)

Examples:

v7 is an ancestor of v2:

6 ≤ 7 < 6 + 2 = 8

v6 is not an ancestor of v2:

5 ≤ 7 ≮ 5 + 1 = 6

v1

1

v3

2

v4

3

v5

v6

5

v7

6

v2

74

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 30

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Levels of Vertices

level(vj) = retreat edges − advance edges, following first vj in ET

= advance edges − retreat edges, up to first vj in ET

level(root) ← 0forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← −1

else

val(ij) ← 1

Parallel-Suffix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

level(vj) ← val(ij) + 1

1, 3

−1

1

3, 1

1

1

1, 4

−1

1

4, 1

1

1

1, 5

−1

1

5, 6

−1

1

6, 5

1

2

5, 7

−1

1

7, 2

−1

1

2, 7

1

3

7, 5

1

2

5, 1

1

1

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 31

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Levels of Vertices

level(vj) = retreat edges − advance edges, following first vj in ET

= advance edges − retreat edges, up to first vj in ET

level(root) ← 0forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← −1

else

val(ij) ← 1

Parallel-Suffix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

level(vj) ← val(ij) + 1

1, 3

−10

3, 1

11

1, 4

−10

4, 1

11

1, 5

−10

5, 6

−11

6, 5

12

5, 7

−11

7, 2

−12

2, 7

13

7, 5

12

5, 1

11

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 31

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Levels of Vertices

level(vj) = retreat edges − advance edges, following first vj in ET

= advance edges − retreat edges, up to first vj in ET

level(root) ← 0forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← −1

else

val(ij) ← 1

Parallel-Suffix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

level(vj) ← val(ij) + 1

1, 3

−11

3, 1

1

1

1, 4

−11

4, 1

1

1

1, 5

−11

5, 6

−12

6, 5

1

2

5, 7

−12

7, 2

−13

2, 7

1

3

7, 5

1

2

5, 1

1

1

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 31

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Simple Applications of Euler Tours

Levels of Vertices

level(vj) = retreat edges − advance edges, following first vj in ET= advance edges − retreat edges, up to first vj in ET

level(root) ← 0forall (vi, vj) ∈ ET do in parallel

if pos(vi, vj) < pos(vj , vi) then

val(ij) ← −1

else

val(ij) ← 1

Parallel-Suffix(ET, +)

forall (vi, vj) ∈ ET do in parallelif pos(vi, vj) < pos(vj , vi) then

level(vj) ← val(ij) + 1

1, 3

−11

3, 1

1

1

1, 4

−11

4, 1

1

1

1, 5

−11

5, 6

−12

6, 5

1

2

5, 7

−12

7, 2

−13

2, 7

1

3

7, 5

1

2

5, 1

1

1

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 31

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Outline

1 Introduction

2 Building Euler Tours

3 Applications of Euler ToursUtility ComputationsSimple Applications of Euler ToursComputing Minima

4 Conclusion

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 32

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

The Interval Minima Problem

S = {sk : k = preorder(vi)⇒ sk = val(vi)}

Svi = {sk, sk+1, . . . , s`}, ` = k + des(vi)− 1

find min(Svi) ∀ vi

Example:

3 = preorder(v5), s3 = 6S = {10, 5, 6, 7, 3, 4, 2, 1}

Sv2 : k = 2, ` = 2 + 5− 1 = 6Sv2 = {5, 6, 7, 3, 4}

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 33

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

The Interval Minima Problem

S = {sk : k = preorder(vi)⇒ sk = val(vi)}

Svi = {sk, sk+1, . . . , s`}, ` = k + des(vi)− 1

find min(Svi) ∀ vi

Example:

3 = preorder(v5), s3 = 6S = {10, 5, 6, 7, 3, 4, 2, 1}

Sv2 : k = 2, ` = 2 + 5− 1 = 6Sv2 = {5, 6, 7, 3, 4}

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 33

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

The Interval Minima Problem

S = {sk : k = preorder(vi)⇒ sk = val(vi)}

Svi = {sk, sk+1, . . . , s`}, ` = k + des(vi)− 1

find min(Svi) ∀ vi

Example:

3 = preorder(v5), s3 = 6S = {10, 5, 6, 7, 3, 4, 2, 1}

Sv2 : k = 2, ` = 2 + 5− 1 = 6Sv2 = {5, 6, 7, 3, 4}

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 33

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

The Interval Minima Tree

Prefix Minima (PM)

The prefix minima of a sequence {a1, a2, . . . , an} are given by asequence {b1, b2, . . . , bn} where bi = min(a1, a2, . . . , ai).

Suffix Minima (SM)

The suffix minima of a sequence {a1, a2, . . . , an} are given by asequence {d1, d2, . . . , dn} where di = min(ai, ai+1, . . . , an}.

PM 2 1 1 1 1 1

Seq. 2 1 3 4 5 6

SM 1 1 3 4 5 6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 34

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

The Interval Minima Tree

Prefix Minima (PM)

The prefix minima of a sequence {a1, a2, . . . , an} are given by asequence {b1, b2, . . . , bn} where bi = min(a1, a2, . . . , ai).

Suffix Minima (SM)

The suffix minima of a sequence {a1, a2, . . . , an} are given by asequence {d1, d2, . . . , dn} where di = min(ai, ai+1, . . . , an}.

A complete binary tree such that:

Leaves are s1, s2, . . . , sn.

Each internal node x contains two sequences:the prefix minima (PM) and suffix minima (SM) of the valuesheld by leaves in the subtree rooted at x.

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 34

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM =

10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM =

10 ,5 ,5 ,5

SM =

5 ,5 ,6 ,7

PM =

3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1

PM =

10 ,5

SM =

5 ,5

v110

1 v25

2

PM =

6 ,6

SM =

6 ,7

v56

3 v77

4

PM =

3 ,3

SM =

3 ,4

v63

5 v84

6

PM =

2 ,1

SM =

1 ,1

v32

7 v41

8

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM =

10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM =

10 ,5 ,5 ,5

SM =

5 ,5 ,6 ,7

PM =

3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1

PM = 10 ,

5

SM =

5

,5

v110

1 v25

2

PM = 6 ,

6

SM =

6

,7

v56

3 v77

4

PM = 3 ,

3

SM =

3

,4

v63

5 v84

6

PM = 2 ,

1

SM =

1

,1

v32

7 v41

8

l1, l2, . . . , lp

Lr1, r2, . . . , rp

R

{min(lp, ri)}LPM:

{min(r1, li)} RSM:

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM =

10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM =

10 ,5 ,5 ,5

SM =

5 ,5 ,6 ,7

PM =

3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

l1, l2, . . . , lp

Lr1, r2, . . . , rp

R

{min(lp, ri)}LPM:

{min(r1, li)} RSM:

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM =

10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM = 10 ,5 ,

5 ,5

SM =

5 ,5

,6 ,7

PM = 3 ,3 ,

2 ,1

SM =

1 ,1

,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

l1, l2, . . . , lp

Lr1, r2, . . . , rp

R

{min(lp, ri)}LPM:

{min(r1, li)} RSM:

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM =

10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM = 10 ,5 ,5 ,5

SM = 5 ,5 ,6 ,7

PM = 3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

l1, l2, . . . , lp

Lr1, r2, . . . , rp

R

{min(lp, ri)}LPM:

{min(r1, li)} RSM:

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM = 10 ,5 ,5 ,5 ,

3 ,3 ,2 ,1

SM =

1 ,1 ,1 ,1

,1 ,1 ,1 ,1

PM = 10 ,5 ,5 ,5

SM = 5 ,5 ,6 ,7

PM = 3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

l1, l2, . . . , lp

Lr1, r2, . . . , rp

R

{min(lp, ri)}LPM:

{min(r1, li)} RSM:

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Constructing the Tree

PM = 10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM = 10 ,5 ,5 ,5

SM = 5 ,5 ,6 ,7

PM = 3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

l1, l2, . . . , lp

Lr1, r2, . . . , rp

R

{min(lp, ri)}LPM:

{min(r1, li)} RSM:

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 35

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Finding the Minima Using the TreeAn Example: Sv2

k = 2, ` = k + 5− 1 = 6 ⇒ Sv2 = {s2, . . . , s6}

PM = 10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1 ,1 ,1 ,1 ,1

PM = 10 ,5 ,5 ,5

SM = 5 ,5 ,6 ,7

PM = 3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 36

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Finding the Minima Using the TreeAn Example: Sv2

k = 2, ` = k + 5− 1 = 6 ⇒ Sv2 = {s2, . . . , s6}

PM = 10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1 ,1 ,1 ,1 ,1w

PM = 10 ,5 ,5 ,5

SM = 5 ,5 ,6 ,7

PM = 3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 36

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

Finding the Minima Using the TreeAn Example: Sv2

k = 2, ` = k + 5− 1 = 6 ⇒ Sv2 = {s2, . . . , s6}

PM = 10 ,5 ,5 ,5 ,3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1 ,1 ,1 ,1 ,1w

PM = 10 ,5 ,5 ,5

SM = 5 ,5 ,6 ,7x

PM = 3 ,3 ,2 ,1

SM = 1 ,1 ,1 ,1y

PM = 10 ,5

SM = 5 ,5

v110

1 v25

2

PM = 6 ,6

SM = 6 ,7

v56

3 v77

4

PM = 3 ,3

SM = 3 ,4

v63

5 v84

6

PM = 2 ,1

SM = 1 ,1

v32

7 v41

8

v110

1

v32

7v25

2

v56

3

v41

8

v63

5

v77

4v84

6

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 36

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Computing Minima

The Interval Minima Algorithm

Input: A tree T Output: min(Svi) for all vi ∈ T

O(1) ET ← Parallel-Euler-Tour(T )O(log n) Parallel-Preorder(ET)O(log n) Parallel-Descendants(ET)O(log n) T ′ ← Interval-Minima-Tree(T )

forall vi ∈ T do in parallelO(1) k ← preorder(vi), `← k + des(vi)− 1

O(log n) w ← Lowest-Common-Ancestor(sk, s`) in T ′

O(1) x ← Left-Child(w), y ← Right-Child(w);

O(1) min(Svi)← min(SM(x)k , PM

(y)` )

Complexity:

t(n) = O(log n) p(n) = O(n) ⇒ c(n) = n log(n)

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 37

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Summary

Outline

1 Introduction

2 Building Euler Tours

3 Applications of Euler Tours

4 ConclusionSummaryQuestions

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 38

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Summary

Summary

ET

O(1)

DFT

O(1)

Prefix

O(log n)

Pos

O(log n)

Rank

O(log n)

Preorder

O(log n)

Parent

O(1)

Des

O(1)

Levels

O(log n)

Bin

O(1)

IntMin

O(log n)

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 39

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Summary

Thank You

Thank You!

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 40

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Questions

Outline

1 Introduction

2 Building Euler Tours

3 Applications of Euler Tours

4 ConclusionSummaryQuestions

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 41

Introduction Building Euler Tours Applications of Euler Tours Conclusion

Questions

Questions

???

ECE @ Queen’s • Fall 2008 Ahmad Khayyat Parallel Algorithms for Trees • 42