Randomness in Computing L ECTURE 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 4/23/2020 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 𝑌 𝑢+𝑡 = 𝑘 𝑌 𝑢 = 𝑘 = 0 unless 𝑡 is divisible by Δ ; otherwise, it is aperiodic. • A Markov chain is aperiodic if all its states are aperiodic. Sofya Raskhodnikova; Randomness in Computing 4/28/2020
ҧ Stationary Distributions 𝑞 𝑢 + 1 = 𝑞 𝑢 𝑸 , where ҧ 𝑞 𝑢 is the distribution of the state Recall: ҧ of 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. Sofya Raskhodnikova; Randomness in Computing 4/28/2020
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.) 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 = (𝜌 0 , 𝜌_1, … , 𝜌 𝑜 ) , where 𝜌 𝑗 > 0 for all 𝑗 ∈ [𝑜] . For all 𝑗 ∈ [𝑜] , the hitting time ℎ 𝑗𝑗 = 1/𝜌 𝑗 . 2. Sofya Raskhodnikova; Randomness in Computing 4/28/2020
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 1 if 𝑣, 𝑤 ∈ 𝐹 𝑄 𝑣𝑤 = ቐ 𝑒 𝑣 0 otherwise • Observation: This Markov chain is aperiodic iff G isn’t bipartite. Sofya Raskhodnikova; Randomness in Computing 4/23/2020
Stationary distribution • Assume 𝐻 is not bipartite. Theorem A random walk on 𝐻 has stationary distribution ത 𝜌 , where 𝜌 𝑤 = 𝑒(𝑤) 2|𝐹| 𝑤∈𝑊 Sofya Raskhodnikova; Randomness in Computing 4/23/2020
Hitting time, commute time, cover time • The hitting time from 𝑣 to 𝑤 , denoted ℎ 𝑣,𝑤 , is the expected time to reach state 𝑤 from state 𝑣. • The commute time between 𝑣 and 𝑤 is ℎ 𝑣,𝑤 + ℎ 𝑣,𝑤 . • The cover time of a graph 𝐻 = (𝑊, 𝐹) is the maximum over 𝑤 ∈ 𝑊 of the expected time for a random walk starting at 𝑤 to visit all nodes in 𝑊 . Sofya Raskhodnikova; Randomness in Computing 4/28/2020
Bound on commute time Commute Time Lemma If 𝑣, 𝑤 ∈ 𝐹 , the commute time ℎ 𝑣,𝑤 + ℎ 𝑤,𝑣 is at most 2 𝐹 . Proof: Let 𝐸 = set of 2 |𝐹| directed edges 𝒋 → 𝒌 𝑗, 𝑘 ∈ 𝐹} 𝒋 𝒌 𝒋 𝒌 • Random walk on 𝐻 corresponds to Markov Chain with states 𝐸, where state at time 𝑢 is the directed edge taken by transition 𝑢. 𝟐 𝒋 𝒌 𝒆 𝒌 𝒍 𝒌 → 𝒍 𝒋 → 𝒌 • This Markov Chain has uniform stationary distribution. Sofya Raskhodnikova; Randomness in Computing 4/23/2020
Bound on commute time Commute Time Lemma If 𝑣, 𝑤 ∈ 𝐹 , the commute time ℎ 𝑣,𝑤 + ℎ 𝑤,𝑣 is at most 2 𝐹 . Proof: This Markov Chain has uniform stationary distribution. 𝒘 → 𝒙 𝒗 𝒘 𝒗 → 𝒘 1 • By Fundamental Thm of Markov Chains, ℎ 𝑣→𝑤,𝑣→𝑤 = 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 : 1 ℎ 𝑤,𝑣 + ℎ 𝑣,𝑤 ≤ 2|𝐹| Sofya Raskhodnikova; Randomness in Computing 4/28/2020
Bound on cover time Cover Time Lemma The cover time of 𝐻 with 𝑜 nodes and 𝑛 edges is at most 2𝑛(𝑜 − 1). Proof: Choose a spanning tree 𝑈 of 𝐻 . Sofya Raskhodnikova; Randomness in Computing 4/28/2020
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! Algorithm for 𝑡 - 𝑢 -PATH 1. Start a random walk from 𝑡 . 2. If the walk reaches 𝑢 in 2𝑜 3 steps, accept ; otherwise, reject . Sofya Raskhodnikova; Randomness in Computing 4/26/2020
Correctness of 𝒕 - 𝒖 -PATH algorithm Theorem The algorithm uses 𝑃(log 𝑜) bits and has error probability ≤ 1/2. 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 ≤ 𝑜 3 . • By Markov’s inequality, the probability that the walk takes more than 2𝑜 3 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 Sofya Raskhodnikova; Randomness in Computing 4/28/2020
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