Dynamic%20connectivity%20algorithms%20for%20Feynman%20diagrams

1
Dynamic connectivity algorithms for Feynman
diagrams
Rubao Ji
Advisor Dr. Robert W.
Robinson Committee Dr. E.
Rodney Canfield
Dr. Daniel M. Everett
? 2002.11 ?
2
Outline of Thesis

• Introduction
• Algorithm design and time complexity analysis
• Data structure and implementation
• Experimental results and discussions

3
Feynman Diagram
Introduction
5
A Feynman diagram with order of n can be viewed
as a matching on 2n nodes (edges in the matching
are undirected and called V-lines) along with a
permutation of the 2n nodes (represented by
directed edges called G-lines)
3
2
4
0
1
G line
V line
4
Feynman Diagram
Introduction
2
A Feynman diagram with order of n can be viewed
as a matching on 2n nodes (edges in the matching
are undirected and called V-lines) along with a
permutation of the 2n nodes (represented by
directed edges called G-lines)
3
5
4
1
0
G line
V line
5
Feynman Diagram
Introduction
2
A Feynman diagram with order of n can be viewed
as a matching on 2n nodes (edges in the matching
are undirected and called V-lines) along with a
permutation of the 2n nodes (represented by
directed edges called G-lines)
5
3
4
1
0
G line
V line
6
Switching and Connectivity
Introduction
a
a
Connected
b
b
Switch(a,b)
Not Connected
a
a
b
b
7
Switching and Connectivity
Introduction
a
a
5
b
b
1
2
Switch(2,4)
Switch(a,b)
0
3
4
a
a
b
b
8
Dynamic vs. Static
Introduction

• Dynamic graph problem
• Answer queries on graphs that are
  undergoing a sequence of updates (e.g. insertions
  and deletions of edges and vertices)

• Fully-dynamic
• Semi-dynamic (incremental, decremental)

• Dynamic algorithm
• Update the solution of a problem more
  efficiently after dynamic changes rather than
  having to re-compute it from scratch each time
  (static).

9
Amortized Expected
Introduction

• Running time of one single operation averaged
  over a
• sufficiently long sequence of operations
• Amortized Time
• any sequence long enough
• worst case bond

• Expected Time
• sequence has some distribution or is
  random
• probabilistic worst case

10
A Brief History
Introduction
Deterministic algorithms Frederickson, 1985.
O(m1/2)
O(1) Epstein et al., 1992. O(n1/2)
O(1) HenzingerKing,
1997. O(n1/3 log n)
O(log n/loglog n) Holm et al., 1998.
O(log2 n)
O(log n/loglog n)
Update
Query
Randomized algorithms HenzingerKing, 1995.
O(log3 n)
O(log n/loglog n) HenzingerThorup, 1997.
O(log2 n) O(log
n/loglog n) Thorup, 2000.
O(log n(loglog n)3) O(log
n/logloglog n)
Update
Query
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Features that matter
Algorithm design and time complexity analysis
O (log n) O (log n) O (log? n)

1. Number of G-cycles
2. Update time
3. Query time

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Lemma 2.1
Algorithm design and time complexity analysis
For a randomly generated Feynman diagram of order
n, the expected number of G-cycles is O(log n).
Cl G-cycle containing vertex v has length l
v
Prove Pr Cl 1/2n
Pr C1 1/2n Pr C2(2n-1)/2n 1/(2n-1)
1/2n Pr C3(2n-1)/2n (2n-2)/(2n-1)1/(2n-2)
1/2n
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Lemma 2.1 (contd)
Algorithm design and time complexity analysis
Xi the size of the G-cycle containing node i S
total number of G-cycles
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Update
Algorithm design and time complexity analysis

• Splay Tree (Self-Adjusting BST)
• Treap (BST Heap)

G-cycle In-order traversal V-line Implicit
numbering
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Update
Algorithm design and time complexity analysis

• Splay Tree (Self-Adjusting BST)
• Treap (BST Heap)

G-cycle In-order traversal V-line Implicit
numbering
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Update
Algorithm design and time complexity analysis
Lemma 2.2 Each splay operation has an amortized
time of O(log n) over ?(n) operations such that
the tree size is always O(n). (From
SleatorTarjan,1985)
Lemma 2.3 Each treap operation has an expected
time of O(log n) averaged over all permutations
of the nodes for the priority attribute. (From
Seidel Aragon, 1996)
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Update Splay Tree
Algorithm design and time complexity analysis
x
Splay(x)
x
x
t1
t2
t1
t2
Join(t1,t2)
x
x
t1
t2
Split(x,t)
x
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Update Treap
Algorithm design and time complexity analysis
delete(x)
x
Insert(x)
x
x
t1
t2
t1
t2
delete(x)
Join(t1,t2)
b
a
Rotate left
b
a
y
t1
t2
Split(x,t)
x
Rotate right
x
x
y
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Query
Algorithm design and time complexity analysis

• SCQ (Simplified Connectivity Query)
• ACQ (Advanced Connectivity Query)
• IDQ (Integrated DFS Connectivity Query)

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Query SCQ
Algorithm design and time complexity analysis

• Straightforward approach
• Simple DFS or pre-order traversal
• Linear

21
Query ACQ
Algorithm design and time complexity analysis

• More complicated than SCQ
• Start from smallest component
• O(log2 n)

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Lemma 2.4
Algorithm design and time complexity analysis
ACQ algorithm has an expected cost of O(log2 n)
time to query the connectivity of Feynman diagram.

• Di The probability that, in examing vertices in
  the smallest component, a V -edge joining it to
  another component will be found in ? i attempts.

D1 gt 1/2, D2 gt 1/2 1/4 Dn gt 1/2 1/4
1/2(n-1)

• Si The probability that at least i attempts will
  be needed to find another component.

Si gt 1 - Di
ExSteps lt 1 1/2 1/4 2

• O(log n) expected component, O(log n) expected
  joins, after each join, take O(log n) expected
  time to find the smallest component.

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Query IDQ
Algorithm design and time complexity analysis

• Integrated with update process
• Maintain a G-cycle graph
• O(log n)

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IDQ (contd)
Algorithm design and time complexity analysis
Switch(a,b)
a
a
b
b
Case 2
Case 1
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IDQ (contd)
Algorithm design and time complexity analysis
Switch(a,b)
a
a
b
b
Case 2
Case 1
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IDQ (contd)
Algorithm design and time complexity analysis
Switch(a,b)
a
a
b
b
Case 2
Case 1
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IDQ (contd)
Algorithm design and time complexity analysis
IDQ algorithm has an expected cost of O(log n)
time to query the connectivity of Feynman diagram.

• DFS (Linear in number of G-cycles)
• Chance of disconnected O(1/n)
• If connected, O(1) to update the edge

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Lemma 2.5
Algorithm design and time complexity analysis
Using a splay forest data structure for Feynman
diagram update combined with ACQ for connectivity
queries gives an amortized expected time per
operation of O(log2 n). Using a treap forest
data structure for Feynman diagram update
combined with ACQ for connectivity queries gives
an expected time per operation of O(log2 n).
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Lemma 2.6
Algorithm design and time complexity analysis
Using a splay forest data structure for Feynman
diagram update combined with IDQ for connectivity
queries gives an amortized expected time per
operation of O(log n). Using treap a forest
data structure for Feynman diagram update
combined with IDQ for connectivity queries gives
an expected time per operation of O(log n).
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Update -ASF
Data structures and implementations
ASF (Array-based Splay Forest)
i
lci
rci
pi
si
0 1 2 3 4 5
0 2
5 1 3
2 4 4 5 4 2
1 1 4 1 6 2
4
2
1
0
5
3
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ASF Switch(a,b) a ? T1, b ? T2
Data structures and implementations
a
b
T1
T2
Splay(a)
Splay(b)
a
b
a
b
a
b
a
Switch
b
a
a
b
w
b
Splay(w)
b
a
a
w
b
Switch (a,b)
b
Join
a
w
b
a
a
b
a
b
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ASF Switch(a,b) a ,b ? T
Data structures and implementations
T
T
a
a
a
a
b
b
a
b
a
b
Splay(a)
Splay(a)
b
b
a
a
Switch (a,b)
b
b
a
b
b
b
a
b
a
Splay(b)
Splay(b)
a
b
b
Reattach
Reattach
b
a
b
a
a
b
b
b
b
a
a
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Update -ATF
Data structures and implementations
ATF (Array-based Treap Forest)
i
lci
rci
pi
si
prioi
0 1 2 3 4 5
0 2
5 1 3
2 4 4 5 4 2
1 1 4 1 6 2
1 2 4 0 5 3
4
2
1
0
5
3
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ATF Switch(a,b) a ? T1, b ? T2
Data structures and implementations
Split(a)
w
w
Split(b)
T1
T2
a
a
b
b
a
b
a
b
Switch
w
w
Delete(w)
Delete(w)
b
a
a
b
a
b
b
a
Join
x
Delete (x)
a
b
b
a
a
b
b
a
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ATF Switch(a,b) a ,b ? T
Data structures and implementations
T
Split(a)
T
Split(a)
a
a
b
b
w
a
a
b
b
w
b
b
a
a
b
b
a
a
Split(b)
Split(b)
w
w
w
w
Reattach
Reattach
a
b
b
b
a
b
b
a
a
b
b
b
a
a
Delete(w)
Delete(w)
b
a
a
b
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IDQ Adjacency list
Data structures and implementations
0
1
2
3
4
2
1
2
0
0
4
0
3
2
1
node
down
next
Type A node
visited
0
3
4
node
p1
link
Type B node
p2
adj_next
visited
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IDQ Switch(a,b) a ? T1, b ? T2
Data structures and implementations
2
1
5
0
3
0
1
2
3
4
4
2
1
2
0
0
5
5
4
0
1
5
3
5
0
4
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IDQ Switch(a,b) a ,b ? T
Data structures and implementations
0
1
5
3
4
6
0
1
2
3
4
2
1
2
0
0
5
6
1
2
1
2
0
0
4
0
4
0
3
3
6
2
1
1
5
0
3
0
3
4
4
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Exp 1 ASF vs. ATF
Experimental results and discussion

• n 10-100,000
• 5000 switches
• ACQ
• random

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Exp 2 Update vs. Query
Experimental results and discussion
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Exp 3 Frequency of Dis-connection
Experimental results and discussion
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Exp 4 Time per Switching
Experimental results and discussion
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Exp 5 IDQ/ACQ/SCQ
Experimental results and discussion
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Discussion
Experimental results and discussion

• Splay tree vs. Treap
• IDQ vs. ACQ vs. SCQ
• Further study

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Thanks !
Animation downloaded from http//www.egglescliffe
.org.uk/physics/particles/parts/parts1.html
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ATF Insert
Data structures and implementations
0
Insert(2)
2
Rotate(2)
2
Insert(5)
0
2
0
0
5
i
Wi
Pi
Insert(3)
0 1 2 3 4 5
2 0 5 4 1 3
3 2 4 0 5 1
2
Insert(4)
2
Rotate(4)
2
0
5
0
5
0
5
3
3
4
4
3
Rotate(4)
4
4
2
2
2
Rotate(4)
Insert(1)
1
0
4
0
0
5
5
5
3
3
3