Randomized Algorithms Markov Chain X 0 , X 1 , X 2 , . . . 1 1 - - PowerPoint PPT Presentation

Randomized Algorithms

南京大学 尹一通

1/3 1/3 1/3 1/3 2/3 1

1 2 3

Markov Chain

M = (Ω, P)

π(t) = π(0)P t

X0, X1, X2, . . .

1/3 1/3 1/3 1/3 2/3 1

1 2 3

1 2 1 2 1 3 2 3 3 4 1 4 3 4 1 4

reducible

1 1

periodic

If a finite Markov chain is irreducible and aperiodic, then ∀ initial distribution M = (Ω, P)

π(0) lim

t→∞ π(0)P t = π

where is a unique stationary distribution satisfying

π πP = π

Fundamental Theorem of Markov Chain:

If a finite Markov chain is irreducible and aperiodic, then ∀ initial distribution M = (Ω, P)

π(0) lim

t→∞ π(0)P t = π

where is a unique stationary distribution satisfying

π πP = π

Fundamental Theorem of Markov Chain:

finiteness existence irreducibility uniqueness ergodicity convergence

(Perron-Frobenius)

) )

(coupling)

If a Markov chain is irreducible and ergodic, then ∀ initial distribution M = (Ω, P)

π(0) lim

t→∞ π(0)P t = π

where is a unique stationary distribution satisfying

π πP = π

Fundamental Theorem of Markov Chain:

ergodic : aperiodic + non-null persistent

PageRank

• A page has higher rank

if pointed by more high- rank pages.

• High-rank pages have

greater influence.

• Pages pointing to few
• thers have greater

influence.

Rank: importance of a page

PageRank (simplified)

r(v) =

• u:(u,v)∈E

r(u) d+(u) d+(u) : out-degree of u G(V, E) the web graph rank of a page: r(v) P(u, v) =

• 1

d+(u)

if (u, v) ∈ E,

• therwise.

random walk: rP = r stationary distribution: a tireless random surfer

Random Walk on Graph

• undirected graph G(V, E)
• walk:
• random walk:

v1, v2, . . . ∈ V that vi+1 ∼ vi

P = D−1A adjacency matrix A

D(u, v) =

• d(u)

u = v u = v

P(u, v) =

• 1

d(u)

u v u ⇥ v vi+1 is uniformly chosen from N(vi)

Random Walk on Graph

• stationary:
• convergence;
• stationary distribution;
• hitting time: time to reach a vertex;
• cover time: time to reach all vertices;
• mixing time: time to converge.

random walk on G(V,E)

Random Walk on Graph

• for finite chain:

irreducible and aperiodic ⇒ converge

• irreducible ⇔ G is connected
• aperiodic ⇔ G is non-bipartite

P(u, v) =

• 1

d(u)

u v u ⇥ v

G(V,E)

At(u, v) > 0 P t(u, v) > 0 bipartite ⇒ no odd cycle ⇒ period =2 non-bipartite ⇒ ∃(2k+1)-cycle undirected ⇒ ∃ 2-cycle

• ⇒ aperiodic

Random Walk on Graph

P(u, v) =

• 1

d(u)

u v u ⇥ v

G(V,E)

Stationary distribution π: ∀v ∈ V, πv = d(v) 2m

• v∈V

πv =

• v∈V

d(v) 2m = 1 2m

• v∈V

d(v) = 1 =

• u∈V

πuP(u, v) =

• u∈N(v)

d(u) 2m 1 d(u) = d(v) 2m

= πv (πP)v

regular graph uniform distribution

Lazy Random Walk

• undirected graph G(V, E)
• lazy random walk: flip a coin to decide

whether to stay adjacency matrix A

D(u, v) =

• d(u)

u = v u = v

P(u, v) =     

1 2

u = v

1 2d(u)

u ∼ v

• therwise

P = 1 2(I + D−1A)

always aperiodic!

Lazy Random Walk

G(V,E)

Stationary distribution π: ∀v ∈ V, πv = d(v) 2m

=

• u∈V

πuP(u, v)

= πv (πP)v

P(u, v) =     

1 2

u = v

1 2d(u)

u ∼ v

• therwise

= 1 2 d(v) 2m + 1 2 X

u∈N(v)

d(u) 2m 1 d(u) = d(v) 2m

Hitting and Covering

• hitting time: expected time to reach v from u
• cover time: expected time to visit all vertices

τu,v = E ⌅ min

• n > 0

⇤ ⇤ Xn = v ⇥ ⇤ ⇤ ⇤ X0 = u ⇧ Cu = E ⌅ min

• n

⇤ ⇤ {X0, . . . , Xn} = V ⇥ ⇤ ⇤ ⇤ X0 = u ⇧ C(G) = max

u∈V Cu

consider a random walk on G(V,E)

C(G) = Θ(n log n) G = Kn C(G) = Θ(n2) C(G) = Θ(n3) G : path or cycle G : “lollipop” K n

2

n 2

Hitting Time

τu,v = E ⌅ min

• n > 0

⇤ ⇤ Xn = v ⇥ ⇤ ⇤ ⇤ X0 = u ⇧

stationary distribution π

πv = d(v) 2m τv,v = 1 πv = 2m d(v)

Renewal Theorem:

τv,v = 1 πv = 2m d(v)

random walk on G(V,E)

uv ∈ E τu,v < 2m

Lemma: u v

τv,v = X

wv∈E

1 d(v)(1 + τw,v) 2m = X

wv∈E

(1 + τw,v)

τu,v < 2m

Cover Time

C(G) = max

u∈V Cu

Theorem: C(G) ≤ 4nm

Cu = E h min

• n
• {X0, . . . , Xn} = V
• X0 = u

i

pick a spanning tree T of G

Cover Time

C(G) = max

u∈V Cu

Theorem: C(G) ≤ 4nm

Cu = E h min

• n
• {X0, . . . , Xn} = V
• X0 = u

i

pick a spanning tree T of G Eulerian tour:

v1 → v2 → · · · → v2(n−1) → v2n−1 = v1

C(G) ≤

2(n−1)

• i=1

τvi,vi+1 < 4nm

USTCON

(undirected s-t connectivity)

• Instance:
• undirected G(V, E);
• vertices: s, t
• s-t connected ?
• deterministic:
• traverse: linear space
• log-space ?

G s t

undirected

USTCON can be solved by a poly-time Monte Carlo randomized algorithm with bounded one-sided error, which uses O(log n) extra space. Theorem (Aleliunas-Karp-Lipton-Lovász-Rackoff 1979)

• start a random walk at s;
• if reach t in steps

return “yes”

• else return “no”

4n3 C(G) ≤ 4nm < 2n3

Cover time: Markov’s inequality ⇒ Pr[“no”] < 1/2 unconnected ⇒ “no” space: O(log n) connected ⇒

Electric Network

edge uv : resistance Ruv vertex v : potential

φv

edge orientation :

u → v Cu→v

current flow Kirchhoff’s Law: ∀ vertex v, flow-in = flow-out Ohm’s Law:

∀ edge uv, Cu→v = φu,v

Ruv

potential difference

φu,v = φu − φv

Effective Resistance

electrical network: u effective resistance R(u,v):

1 1

potential difference between u and v required to send 1 unit of flow current from u to v

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

u

each edge e resistance

Re = 1

for every edge e : each vertex u :

Re = 1

inject d(u) units current flow into u remove all 2m units current flow from v

∀u ∈ V, φu,v = τu,v

Lemma: graph G(V,E) construct an electrical network: a special vertex v :

∀u ∈ V, φu,v = τu,v

Lemma:

d(u) = X

uw∈E

Cu→w (Kirchhoff)

(Ohm)

= X

uw∈E

(φu,v − φw,v) = X

uw∈E

φu,w = d(u)φu,v − X

uw∈E

φw,v

φu,v = 1 + 1 d(u) X

uw∈E

φw,v

∀u ∈ V, φu,v = τu,v

Lemma:

τu,v = E ⌅ min

• n > 0

⇤ ⇤ Xn = v ⇥ ⇤ ⇤ ⇤ X0 = u ⇧

τu,v = X

wu∈E

1 d(u)(1 + τw,v) = 1 + 1 d(u) X

wu∈E

τw,v φu,v = 1 + 1 d(u) X

uw∈E

φw,v

∀u ∈ V, φu,v = τu,v

Lemma:

= 1 + 1 d(u) X

wu∈E

τw,v φu,v = 1 + 1 d(u) X

uw∈E

φw,v τu,v

)

has the same unique solution

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

u

each edge e resistance

Re = 1

Scenario B

A: B:

∀u ∈ V, φu,v = τu,v

Lemma:

= 1 + 1 d(u) X

wu∈E

τw,v φu,v = 1 + 1 d(u) X

uw∈E

φw,v τu,v

)

has the same unique solution

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

Scenario B

A: B:

φA

u,v = τu,v

φB

v,u = τv,u

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

Scenario B

Scenario C

A: B: C:

φA

u,v = τu,v

φB

v,u = τv,u

φC

u,v = φB v,u = τv,u

Scenario D

D: D = A + C

φD

u,v = φA u,v + φC u,v = τu,v + τv,u

φD

u,v : potential difference between u and v

to send 2m units current flow from u to v

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

Scenario B

Scenario C

A: B: C:

φA

u,v = τu,v

φB

v,u = τv,u

φC

u,v = φB v,u = τv,u

Scenario D

D: D = A + C

φD

u,v = φA u,v + φC u,v = τu,v + τv,u

R(u, v) = φD

u,v

2m

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

pick a spanning tree T of G Eulerian tour:

v1 → v2 → · · · → v2(n−1) → v2n−1 = v1

C(G) ≤

2(n−1)

• i=1

τvi,vi+1 = X

uv∈T

(τu,v + τv,u)

= 2m X

uv∈T

(Ru,v + Rv,u) = 2mn

Theorem: C(G) ≤ 2nm

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v)

G: path

1 n u v

C(G) ≤ 2nm = 2n2 = τv,u τu,v + τv,u = 2mR(u, v) = 2n(n − 1) = Ω(n2) = O(n2) C(G) ≥ τu,v C(G) = Θ(n2)

Theorem (Chandra-Raghavan-Ruzzo-Smolensky-Tiwari 1989)

∀u, v, ∈ V, τu,v + τv,u = 2mR(u, v) K n

2

n 2

C(G) ≥ τu,v

G: lollipop

C(G) ≤ 2nm = O(n3)

u v

τu,v + τv,u = 2mRu,v = Ω(n3) τv,u = O(n2) = Ω(n3) C(G) = Θ(n3)