Randomness in Computing L ECTURE 26 Last time Randomized algorithm - - PowerPoint PPT Presentation

4/23/2020

Randomness in Computing

LECTURE 26

Last time

• Randomized algorithm for 3SAT
• Gambler’s ruin
• Classification of Markov chains

Today

• Stationary distributions
• Random walks on graphs
• Algorithm for 𝑡-𝑢-PATH

Sofya Raskhodnikova;Randomness in Computing; based on slides by Baranasuriya et al.

Classification of Markov chains

• A finite Markov chain is irreducible if its graph representation

consists of one strongly connected component.

• A state 𝑘 is periodic if there exists an integer Δ > 1 such that

Pr 𝑌𝑢+𝑡 = 𝑘 𝑌𝑢 = 𝑘 = 0 unless 𝑡 is divisible by Δ ; otherwise, it is aperiodic.

• A Markov chain is aperiodic if all its states are aperiodic.

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Stationary Distributions

Recall: ҧ 𝑞 𝑢 + 1 = ҧ 𝑞 𝑢 𝑸, where ҧ 𝑞 𝑢 is the distribution of the state

• f the chain at time 𝑢 and 𝑸 is its transition probability matrix.
• A stationary distribution of a Markov chain is a probability

distribution ത 𝜌 such that ത 𝜌 = ത 𝜌𝑸.

(Describes steady state behavior of a Markov chain.) Example: Define Markov chain by the following random walk on the nodes of an 𝑜-cycle. At each step, stay at the same node w.p. ½; go left w.p. ¼ and right w.p. ¼.

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Fundamental theorem

• A stationary distribution of a Markov chain is a probability

distribution ത 𝜌 such that ത 𝜌 = ത 𝜌𝑸.

(Describes steady state behavior of a Markov chain.)

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Fundamental Theorem of Markov Chains (selected items)

Every finite, irreducible and aperiodic Markov chain satisfies the following: 1. There is a unique stationary distribution ത

𝜌 = (𝜌0, 𝜌_1, … , 𝜌𝑜), where

𝜌𝑗 > 0 for all 𝑗 ∈ [𝑜]. 2. For all 𝑗 ∈ [𝑜], the hitting time ℎ𝑗𝑗 = 1/𝜌𝑗.

Random walks on undirected graphs

Given a connected, undirected graph 𝐻 = (𝑊, 𝐹), define the following Markov chain

• states = vertices of the graph
• from each state 𝑤, the chain moves to a uniformly random

neighbor of 𝑤 𝑄

𝑣𝑤 = ቐ

1 𝑒 𝑣 if 𝑣, 𝑤 ∈ 𝐹

• therwise
• Observation: This Markov chain is aperiodic iff G isn’t bipartite.

4/23/2020

Sofya Raskhodnikova; Randomness in Computing

Stationary distribution

• Assume 𝐻 is not bipartite.

4/23/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem

A random walk on 𝐻 has stationary distribution ത 𝜌, where ෍

𝑤∈𝑊

𝜌𝑤 = 𝑒(𝑤) 2|𝐹|

Hitting time, commute time, cover time

• The hitting time from 𝑣 to 𝑤, denoted ℎ𝑣,𝑤, is the expected time to

reach state 𝑤 from state 𝑣.

• The commute time between 𝑣 and 𝑤 is ℎ𝑣,𝑤 + ℎ𝑣,𝑤.
• The cover time of a graph 𝐻 = (𝑊, 𝐹) is the maximum over 𝑤 ∈ 𝑊
• f the expected time for a random walk starting at 𝑤 to visit all

nodes in 𝑊.

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Bound on commute time

Proof: Let 𝐸 = set of 2|𝐹| directed edges 𝒋 → 𝒌 𝑗, 𝑘 ∈ 𝐹}

• Random walk on 𝐻 corresponds to Markov Chain with states 𝐸,

where state at time 𝑢 is the directed edge taken by transition 𝑢.

• This Markov Chain has uniform stationary distribution.

4/23/2020

Sofya Raskhodnikova; Randomness in Computing

Commute Time Lemma

If 𝑣, 𝑤 ∈ 𝐹, the commute time ℎ𝑣,𝑤 + ℎ𝑤,𝑣 is at most 2 𝐹 .

𝒋 𝒌 𝒋 𝒌 𝒋 𝒌 𝒍 𝒋 → 𝒌 𝒌 → 𝒍 𝟐 𝒆 𝒌

Bound on commute time

Proof: This Markov Chain has uniform stationary distribution.

• By Fundamental Thm of Markov Chains, ℎ𝑣→𝑤,𝑣→𝑤 =

1 2|𝐹|

= expected time to traverse 𝑣 → 𝑤 starting at 𝑣 → 𝑤 = expected time to go from 𝑤 to 𝑣 and then traverse 𝑣, 𝑤

• But this is only one way to go from 𝑤 to 𝑣 to 𝑤:

ℎ𝑤,𝑣 + ℎ𝑣,𝑤 ≤ 1 2|𝐹|

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Commute Time Lemma

If 𝑣, 𝑤 ∈ 𝐹, the commute time ℎ𝑣,𝑤 + ℎ𝑤,𝑣 is at most 2 𝐹 .

𝒗 𝒘 𝒗 → 𝒘 𝒘 → 𝒙

Bound on cover time

Proof: Choose a spanning tree 𝑈 of 𝐻.

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Cover Time Lemma

The cover time of 𝐻 with 𝑜 nodes and 𝑛 edges is at most 2𝑛(𝑜 − 1).

Application: 𝒕-𝒖-PATH

Problem: Given an undirected graph 𝐻 with 𝑜 nodes and 𝑛 edges and two nodes, 𝑡 and 𝑢, determine if 𝐻 contains a path from 𝑡 to 𝑢.

• Can be solved by BFS in 𝑃(𝑛 + 𝑜) time
• This approach requires Ω(𝑜) space.
• Today: a randomized algorithm that uses 𝑃(log 𝑜) space.

Less space than it takes to store a path!

4/26/2020

Sofya Raskhodnikova; Randomness in Computing

• 1. Start a random walk from 𝑡.
• 2. If the walk reaches 𝑢 in 2𝑜3 steps, accept; otherwise, reject.

Algorithm for 𝑡-𝑢-PATH

Correctness of 𝒕-𝒖-PATH algorithm

Proof: If there is no path, the algorithm correctly rejects.

• Suppose there is an 𝑡-𝑢 path.
• The expected time to reach 𝑢 from 𝑡 is at most the expected cover

time of the connected component, which is, by Cover Lemma is ≤ 2𝑛𝑜 ≤ 𝑜3.

• By Markov’s inequality, the probability that the walk takes more

than 2𝑜3 steps to reach 𝑢 is at most 1/2. Space analysis: Need to keep

• current position: 𝑃(log 𝑜) bits
• counter for the number of steps: 𝑃(log 𝑜) bits

4/28/2020

Sofya Raskhodnikova; Randomness in Computing

Theorem

The algorithm uses 𝑃(log 𝑜) bits and has error probability ≤ 1/2.