Random Walk on Graphs and its Algorithmic Applications. Shengyu Zhang Winter School, [email protected], 2009. Random walk on graphs. On an undirected graph G: Starting from vertex v 0 Repeat for a number of steps: Go to a random neighbor. Simple but powerful. Road map: Random walk. Parameters
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Random Walk on Graphs and its Algorithmic Applications
Shengyu Zhang
Winter School, [email protected], 2009
On an undirected graph G:
Parameters
/Properties
Algorithms
Hitting time
Mixing time
Short intro to math model of quantum mechanics
Types
Algorithms
Element Distinctness
Discrete QW
Formula Evaluation
Continuous QW
PART I. Random Walk
Key parameter 1: Hitting time
j
i
0
1
2
…
n1
i
j
(n/2)line
(n/2)complete graph
Algorithm 1: 2SAT
① http://www.claymath.org/millennium/P_vs_NP/
(x1∨x2)∧(x2∨¬x3) ∧(¬x4∨x3) ∧(x5∨x1)
x1, x2, x3, x4, x5
0, 1, 0, 1, 0
1
Algorithm 2: stconnectivity
Key parameter 2: Mixing time
v
u
w
v1
v2
v3
v4
v5
v1
v2
v3
v4
v5
v6
v
u
w
π’(v) = ∑u: (u,v)∊E p(u)∙[1/d(u)]
= ∑u: (u,v)∊E [d(u)/2m]∙[1/d(u)]
= ∑u: (u,v)∊E 1/2m
= d(v)/2m
= π(v)
So π is the stationary distribution.
Algorithm 3: PageRank
… Tsinghua, Pku, Fudan, IIT(B): 9
R(A) = (1α)/N + α∑B: B→A R(B)/d(B)
is exactly the random walk on the graph, where at each point A, we
= ∑i,jλiξiξiT λjξjξjT = ∑i,jλi λj ξi,ξi ξjξjT
= ∑iλi2ξjξiT
Algorithm 4: Approximately counting
Algorithm 5: Error Reduction with efficient randomness
B
{0,1}m
B
{0,1}m
Nonexpander
{0,1}m: all
random strings
B
Boundary
PART II. Quantum Walk
1
1
1
1
0
0
0
0
Physics
Math

1
α0+β1
(α2+β2=1)
β
Physical System
Unit Vector
α,β: amplitudes

0
α
Evolution
Unitary Matrix
A classical bit
Measurement
Projection
A quantum bit (qubit)
Composition
Tensor Product
Measure by 0and1:
 get 0 w.p. α2; system →0;
 get 1 w.p. β2; system →1.
State space for 2 bits:
all combinations {00, 01, 10, 11}
Classical:
State space for 2 qubits:
the space span{00,01,10,11}
Quantum:
Type 1: Discrete Quantum Walk
µ
¶
1
1
1
p
1
1
2
¡
V
T
V = {S⊆[n]: S = r}; E = {(S,T): S∩T = r2}
Type 2: Continuous Quantum Walk
which has solution ψ(t) = eiHt ψ(0).
∧
¬
∨
∧
∧
∨
∨
General ANDORNOT Formula
(General Game Tree)
(Balanced) ANDOR Tree
max
min
min
Q( f ) ≤n1/2+o(1)
(in both PAC or adversarially generated examples)
~
thr( f ) ≤
deg( f ) ≤
Definition of thr
[BBCMdW, JACM’01]
n1/2+o(1)
x1 x2
x3 x4 x5
:0 :1
:0 :1 :1
∧
∧
∧
∧
∧
∧
0
≠0
1
=0
1
1
0
0
≠0
0
≠0
1
=0
1
=0
If a function evaluates to 0,
then an 0eigenvector
If a function evaluates to 1,
then 0eigenvector has no support on the root.
The quantum walk is here!