randomized computation Sometimes randomness helps in computation. - - PowerPoint PPT Presentation

randomized computation

Sometimes randomness helps in computation.

randomized computation

Augment our usual Turing machines with a read-only tape of random bits. 𝑟0 𝑟1 𝑟start 𝑟accept 𝑟2

𝑏 𝑏 # # 𝑐 𝑏 1 1 1 1

randomized computation

The notion of acceptance changes. One-sided error [two variants]: Two-sided error: 𝑦 ∈ 𝑀 ⇒ Pr 𝑁 accepts 𝑦 = 1 𝑦 ∉ 𝑀 ⇒ Pr 𝑁 accepts 𝑦 <

9 10

What does it mean for a TM 𝑁 with access to random bits to decide a language 𝑀? 𝑦 ∈ 𝑀 ⇒ Pr 𝑁 accepts 𝑦 >

2 3

𝑦 ∉ 𝑀 ⇒ Pr 𝑁 accepts 𝑦 <

1 3

𝑦 ∈ 𝑀 ⇒ Pr 𝑁 accepts 𝑦 >

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𝑦 ∉ 𝑀 ⇒ Pr 𝑁 accepts 𝑦 = 0 In both cases, we can make the error much smaller by repeating a few times.

randomized computation

For instance, the two-sided error variant of 𝐐 is called 𝐂𝐐𝐐. 𝐂𝐐𝐐 stands for “bounded-error probabilistic polynomial time” Derandomization problem: Can every efficient randomized algorithm be replaced by a deterministic one? We don’t know the answer, though it is expected to be “yes.” This is the question of 𝐐 vs. 𝐂𝐐𝐐.

randomized computation

Recall the complexity class 𝐌 = SPACE log 𝑜 The randomized one-sided error variant of 𝐌 is called 𝐒𝐌. 𝑦 ∈ 𝐵 ⇒ Pr 𝑁 accepts 𝑦 ≥

1 2

𝑦 ∉ 𝐵 ⇒ Pr 𝑁 accepts 𝑦 = 0 A language 𝐵 is in 𝐒𝐌 if there is a randomized 𝑃 log 𝑜 space Turing machine such that [one way access to random tape]: The question of whether 𝐌 = 𝐒𝐌 is also open. We know that 𝐒𝐌 ⊆ 𝐎𝐌 ⊆ SPACE log 𝑜 2

Nisan showed how to do this using a pseudo-random number generator for space-bounded machines.

undirected connectivity

Recall the language: DIRPATH = 𝐻, 𝑡, 𝑢 ∶ 𝐻 is a directed graph with a directed 𝑡 → 𝑢 path We saw that DIRPATH is 𝐎𝐌-complete. Thus 𝐌 = 𝐎𝐌 if and only if DIRPATH ∈ 𝑴. What about the language? PATH = 𝐻, 𝑡, 𝑢 ∶ 𝐻 is an 𝐯𝐨𝐞𝐣𝐬𝐟𝐝𝐮𝐟𝐞 graph with an 𝑡 ↔ 𝑢 path

undirected connectivity

What about the language? PATH = 𝐻, 𝑡, 𝑢 ∶ 𝐻 is an 𝐯𝐨𝐞𝐣𝐬𝐟𝐝𝐮𝐟𝐞 graph with an 𝑡 ↔ 𝑢 path Is PATH ∈ 𝐌?

random walks

Let’s show that PATH ∈ 𝐒𝐌. One step of random walk: Move to a uniformly random neighbor.

random walks

What we will show: If 𝐻 = (𝑊, 𝐹) is a graph with 𝑜 vertices and 𝑛 edges, then a random walker started at 𝑡 ∈ 𝑊 who takes 4𝑛𝑜 steps will visit every vertex in the connected component of 𝑡 with probability at least ½. In particular, if there is an 𝑡 ↔ 𝑢 path in 𝐻, she will visit 𝑢 with probability at least ½.

random walks

What we will show: If 𝐻 = (𝑊, 𝐹) is a graph with 𝑜 vertices and 𝑛 edges, then a random walker started at 𝑡 ∈ 𝑊 who takes 4𝑛𝑜 steps will visit every vertex in the connected component of 𝑡 with probability at least ½. In particular, if there is an 𝑡 ↔ 𝑢 path in 𝐻, she will visit 𝑢 with probability at least ½. 𝐒𝐌 Algorithm: Start a random walker at 𝑡 and use the random bits to simulate a random walk for 4𝑛𝑜 steps (can count this high in 𝑃 log 𝑜 space). If we visit 𝑢 at any point, accept. Otherwise, reject. If there is an 𝑡 ↔ 𝑢 path in 𝐻: Pr accept 𝐻, 𝑡, 𝑢 ≥

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If no 𝑡 ↔ 𝑢 path in 𝐻: Pr accept 𝐻, 𝑡, 𝑢 = 0

cover time of a graph

Cover time of a graph = expected # of steps before all vertices are visited.

cover time of a graph

We will show that if 𝐻 has 𝑜 vertices and 𝑛 then the expected time before the random walk covers 𝐻 (from any starting point) is 2𝑛𝑜. Then it must be that with probability at least ½, we cover 𝐻 after 4𝑛𝑜 steps. (This fact is called “Markov’s inequality.”)

heat dispersion on a graph

spectral embedding

𝑤2

hitting times and commute times

For vertices 𝑣, 𝑤 ∈ 𝑊, the hitting time is 𝐼 𝑣, 𝑤 = expected time for random walk to hit 𝑤 starting at 𝑣 𝑣 𝑤 The commute time between 𝑣 and 𝑤 is 𝐷 𝑣, 𝑤 = 𝐼 𝑣, 𝑤 + 𝐼(𝑤, 𝑣)

hitting times and commute times

For vertices 𝑣, 𝑤 ∈ 𝑊, the hitting time is 𝐼 𝑣, 𝑤 = expected time for random walk to hit 𝑤 starting at 𝑣 𝑣 𝑤 The commute time between 𝑣 and 𝑤 is 𝐷 𝑣, 𝑤 = 𝐼 𝑣, 𝑤 + 𝐼(𝑤, 𝑣) Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛.

hitting times and commute times

𝑡 Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛. Let’s use the claim to show that the cover time of 𝐻 is at most 2𝑛(𝑜 − 1).

hitting times and commute times

𝑡 Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛. Let’s use the claim to show that the cover time of 𝐻 is at most 2𝑛(𝑜 − 1). Choose an arbitrary spanning tree of 𝐻.

hitting times and commute times

𝑡 Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛. Let’s use the claim to show that the cover time of 𝐻 is at most 2𝑛(𝑜 − 1). Choose an arbitrary spanning tree of 𝐻. Consider the “tour around the spanning tree”

This is a path 𝑡 = 𝑣0, 𝑣1, 𝑣2, … , 𝑣2𝑜 = 𝑡 Where 𝑣𝑗, 𝑣𝑗+1 is an edge for each 𝑗, and each edge appears exactly twice.

hitting times and commute times

𝑡 Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛. Let’s use the claim to show that the cover time of 𝐻 is at most 2𝑛(𝑜 − 1). Choose an arbitrary spanning tree of 𝐻. Consider the “tour around the spanning tree”

Claim: Cover time of 𝐻 is at most 𝐷 𝑣0, 𝑣1 + 𝐷 𝑣1, 𝑣2 + 𝐷 𝑣2, 𝑣3 + ⋯ + 𝐷 𝑣2𝑜−1, 𝑣2𝑜

hitting times and commute times

𝑡 Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛. Let’s use the claim to show that the cover time of 𝐻 is at most 2𝑛(𝑜 − 1). Choose an arbitrary spanning tree of 𝐻. Consider the “tour around the spanning tree”

Claim: Cover time of 𝐻 is at most

𝑣,𝑤 ∈𝑈

𝐷(𝑣, 𝑤) ≤ 2𝑛(𝑜 − 1)

high school physics

Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛.

high school physics

Claim: If {𝑣, 𝑤} is an edge of 𝐻, then 𝐷 𝑣, 𝑤 ≤ 2𝑛. View the graph 𝐻 as an electrical network with unit resistances on the edges.

When a potential difference is applied between the nodes from an external source, current flows in the network in accordance with Kirchoff’s and Ohm’s laws. K1: The total current flowing into a vertex = total current flowing out K2: The sum of potential differences around any cycle is zero. Ohm: The current flowing along any edge {𝑣, 𝑤} is equal to (potential 𝑣 − potential 𝑤 )/resistance(𝑣, 𝑤) The effective resistance between 𝑣 and 𝑤 is defined as the potential difference required to send one unit of current from 𝑣 to 𝑤.

𝑣 𝑤

presentation agenda

Error-correcting codes: Simple uses of randomness in computation: 1. Intro [Amr] 2. Reed-Solmon codes [Jie] 3. Expander codes [Ali] 1. Checking matrix multiplication [Mahmoud] 2. Karp-Rabin matching algorithm [Stanislav] random pseudo-random