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Benjamin Doerr Max-Planck-Institut für Informatik Saarbrücken Introduction to Quasirandomness

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Benjamin Doerr Randomness in Computer Science Revolutionary Discovery (~1970): –Algorithms making random decisions (“randomized algorithms”) are extremely powerful. –Often, making all decisions at random, suffices. –Such algorithms are easy to find easy to implement ( “algorithm engineering”) often easy to analyse. –Numerous applications: Quicksort primality testing randomized rounding,...

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Benjamin Doerr Randomness in Computer Science Example: Exploring unknown environment. –Environment: Graph G = (V,E) –Task: Explore! E.g.: “Is there a way from A to B?” –Simple randomized solution: Do a random walk: Choose the next vertex to go to uniformly at random from your neighbors. –Analysis: For all graphs, the expected time to visit all vertices (“cover time”) is at most 2 |V| |E|.

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Benjamin Doerr Beyond Randomness... Pseudorandomness (not the topic of this talk): –Aim: Have something look completely random Cope with the fact that true randomness is difficult to get. –Example: Pseudorandom numbers shall look random independent of the particular application. Quasirandomness (topic of this talk): –Aim: Imitate a particular property of a random object. Extract the good featrure from the random object. –Example: Quasi-Monte-Carlo Methods (numerics) Monte-Carlo Integration: Use random sample points. QMC Integration: Use evenly distributed sample points.

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Benjamin Doerr Outline of the Talk Part 0: From Randomness to Quasirandomness [done] –Quasirandom: Imitate a particular aspect of randomness Part 1: Quasirandom random walks –also called “Propp machine” or “rotor router model” Part 2: Quasirandom rumor spreading –the right dose of randomness?

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Benjamin Doerr Reminder Quasirandomness Imitate a particular property of a random object! Why study this? –Understanding randomness (basic research): Is it truely the randomness that makes a random object useful, or just a particular property it usually has? –Learn from the random object and make it better (custom it to your application) without randomness! –Combine random and quasirandom elements: What is the right dose of randomness?

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Benjamin Doerr Part 1: Quasirandom Random Walks Random Walk How to make it quasirandom? Three results: –Cover times –Discrepancies: Massive parallel walks –Internal diffusion limited aggregation (physics)

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Benjamin Doerr Random Walks Rule: Move to a neighbor chosen at random

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Benjamin Doerr Quasirandom Random Walks Simple observation: If the random walk visits a vertex many times, then it leaves it to each neighbor approximately equally often –n visits to a vertex of constant degree d: n/d + O(n 1/2 ) moves to each neighbor. Quasirandomness: Ensure that neighbors are served evenly!

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Benjamin Doerr Quasirandom Random Walks Rule: Follow the rotor. Rotor updates after each move.

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Benjamin Doerr Quasirandom Random Walks Simple observation: If the random walk visits a vertex many times, then it leaves it to each neighbor approx. equally often –n visits to a vertex of constant degree d: n/d + O(n 1/2 ) moves to each neighbor. Quasirandomness: Ensure that neighbors are served evenly! –Put a rotor on each vertex, pointing to its neighbors –The quasirandom walk moves in the rotor direction –After each step, the rotor turns and points to the next neighbor (following a given permutation of the neighbors)

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Benjamin Doerr Quasirandom Random Walks Other names –Propp machine (after Jim Propp) –Rotor router model –Deterministic random walk Some freedom in the design –Initial rotor directions –Order, in which the rotors serve the neighbors –Also: Alternative ways to ensure fairness in serving the neighbors Fortunately: No real difference

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Benjamin Doerr Result 1: Cover Times Cover time: How many steps does the (quasi)random walk need to visit all vertices? Classical result [AKLLR’79]: For all graphs G=(V,E) and all vertices v, the expected time a random walk started in v needs to visit all vertices, is at most 2 |E|(|V|-1). Quasirandom: Same bound of 2|E|(|V|-1), independent of the particular set-up of the Propp machine. Note: Same bound, but ‘sure’ instead of ‘expected’. “Lesson to learn”: Not randomness yields the good cover time, but the fact that each vertex serves its neighbors evenly!

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Benjamin Doerr Result 2: Discrepancies Model: Many chips do a synchronized (quasi)random walk Discrepancy: Difference between the number of chips on a vertex at a time in the quasirandom model and the expected number in the random model

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Benjamin Doerr Example: Discrepancies on the Line 19 1095528639 9.5 4.75 9.52.375 7.125 0.5-0.5 0.25 -0.3750.875-1.1250.625 Random Walk (Expectation): Quasirandom: Difference: A discrepancy > 1!But: You’ll never have more than 2.29 [Cooper, D, Spencer, Tardos]

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Benjamin Doerr Result 2: Discrepancies Model: Many chips do a synchronized (quasi)random walk Discrepancy: Difference between the number of chips on a vertex at a time in the quasirandom model and the expected number in the random model Cooper, Spencer [CPC’06]: If the graph is an infinite d-dim. grid, then (under some mild conditions) the discrepancies on all vertices at all times can be bounded by a constant (independent of everything!) –Note: Again, we compare ‘sure’ with expected

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Benjamin Doerr Result 3: Internal Diffusion Limited Aggregation Model: –Start with an empty 2D grid. –Each round, insert a particle at ‘the center’ and let it do a (quasi)random walk until it finds an empty grid cell, which it then occupies. –What is the shape of the occupied cells after n rounds? 100, 1600 and 25600 particles in the random walk model: [Moore, Machta]

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Benjamin Doerr Result 3: Internal Diffusion Limited Aggregation Model: –Start with an empty 2D grid. –Each round, insert a particle at ‘the center’ and let it do a (quasi)random walk until it finds an empty grid cell, which it then occupies. –What is the shape of the occupied cells after n rounds? With random walks: –Proof: outradius – inradius = O(n 1/6 ) [Lawler’95] –Experiment: outradius – inradius ≈ log 2 (n) [Moore, Machta’00] With quasirandom walks: –Proof: outradius – inradius = O(n 1/4+eps ) [Levine, Peres] –Experiment: outradius – inradius < 2

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Benjamin Doerr Result 3: Internal Diffusion Limited Aggregation Model: –Start with an empty 2D grid. –Each round, insert a particle at ‘the center’ and let it do a (quasi)random walk until it finds an empty grid cell, which it then occupies. –What is the shape of the occupied cells after n rounds?

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Benjamin Doerr Summary Quasirandom Walks Model: –Follow the rotor and rotate it –Simulates: Balanced serving of neighbors 3 particular results: Surprising good simulation (“or better”) of many random walk aspects. –Graph exploration in 2|E|(|V|-1) steps –Many chips: For grids, we have the same number of chips on each vertex at all times as expected in the random walk (apart from constant discrepancy) –IDLA: Almost perfect circle. Try a quasirandom walk instead of a random one !

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Benjamin Doerr Part 2: Quasirandom Rumor Spreading Classical: “Randomized Rumor Spreading” –protocol to distribute information in a network How to make this quasirandom? “The right dose of randomness”?

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Benjamin Doerr Randomized Rumor Spreading Model (on a graph G): –Start: One node is informed –Each round, each informed node informs a neighbor chosen uniformly at random –Broadcast time T(G): Number of rounds necessary to inform all nodes (maximum taken over all starting nodes) Round 0: Starting node is informedRound 1: Starting node informs random nodeRound 2: Each informed node informs a random nodeRound 3: Each informed node informs a random node Round 4: Each informed node informs a random node Round 5: Let‘s hope the remaining two get informed...

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Benjamin Doerr Randomized Rumor Spreading Model (on a graph G): –Start: One node is informed –Each round, each informed node informs a neighbor chosen uniformly at random –Broadcast time T(G): Number of rounds necessary to inform all nodes (maximum taken over all starting nodes) Application: –Broadcasting updates in distributed replicated databases Properties: –simple –robust –self-organized

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Benjamin Doerr Randomized Rumor Spreading Model (on a graph G): –Start: One node is informed –Each round, each informed node informs a neighbor chosen uniformly at random –Broadcast time T(G): Number of rounds necessary to inform all nodes (maximum taken over all starting nodes) Results [n: Number of nodes] : –Easy: For all graphs G, T(G) ≥ log(n) –Complete graphs: T(K n ) = O(log(n)) w.h.p. –Hypercubes: T({0,1} d ) = O(log(n)) w.h.p. –Random graphs: T(G n,p ) = O(log(n)) w.h.p., p > (1+eps)log(n)/n [Frieze&Grimmet (1985), Feige, Peleg, Raghavan, Upfal (1990), Karp, Schindelhauer, Shenker, Vöcking (2000)]

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Benjamin Doerr Deterministic Rumor Spreading? As above except: –Each node has a list of its neighbors. –Informed nodes inform their neighbors in the order of this list (starting at the top if done). Simulates 2 “neighbor fairness” properties: –Within few rounds, a vertex informs distinct neighbors only. –Within many rounds, each neighbor is informed equally often [clearly the less useful property ]

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Benjamin Doerr Deterministic Rumor Spreading? As above except: –Each node has a list of its neighbors. –Informed nodes inform their neighbors in the order of this list (starting at the top if done). Problem: Might take long... Here: n-1 rounds . No hope for quasirandomness here? 134562 List: 2 3 4 5 63 4 5 6 14 5 6 1 25 6 1 2 36 1 2 3 41 2 3 4 5

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Benjamin Doerr Semi-Deterministic Rumor Spreading As above except: –Each node has a list of its neighbors. –Informed nodes inform their neighbors in the order of this list, but start at a random position in the list

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Benjamin Doerr Semi-Deterministic Rumor Spreading As above except: –Each node has a list of its neighbors. –Informed nodes inform their neighbors in the order of this list, but start at a random position in the list Results (1):

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Benjamin Doerr Semi-Deterministic Rumor Spreading As above except: –Each node has a list of its neighbors. –Informed nodes inform their neighbors in the order of this list, but start at a random position in the list Results (1): The log(n) bounds for –complete graphs, –hypercubes, –random graphs G n,p, p > (1+eps) log(n) still hold...

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Benjamin Doerr Semi-Deterministic Rumor Spreading As above except: –Each node has a list of its neighbors. –Informed nodes inform their neighbors in the order of this list, but start at a random position in the list Results (1): The log(n) bounds for –complete graphs, –hypercubes, –random graphs G n,p, p > (1+eps) log(n) still hold independent from the structure of the lists [D, Friedrich, Sauerwald] [2 good news: (a) results hold, (b) things can be analyzed]

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Benjamin Doerr Semi-Deterministic Rumor Spreading Results (2): –Random graphs G n,p, p = (log(n)+log(log(n)))/n: fully randomized: T(G n,p ) = Θ(log(n) 2 ) w.h.p. semi-deterministic: T(G n,p ) = Θ(log(n)) w.h.p. –Complete k-regular trees: fully randomized: T(G) = Θ(k log(n)) w.h.p. semi-deterministic: T(G) = Θ(k log(n)/log(k)) w.p.1 Algorithm Engineering Perspective: –need fewer random bits –easy to implement: Any implicitly existing permutation of the neighbors can be used for the lists

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Benjamin Doerr Outlook: The Right Dose of Randomness? Broadcasting results also indicate that the right dose of randomness can be important! –Alternative to the classical “everything independent at random” –May-be an interesting direction for future research? Related: –Dependent randomization, e.g., dependent randomized rounding: Gandhi, Khuller, Partharasathy, Srinivasan (FOCS’01+02) D. (STACS’06+07) –Randomized search heuristics: Combine random and other techniques, e.g., greedy Example: Evolutionary algorithms –Mutation: Randomized –Selection: Greedy (survival of the fittest)

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Benjamin Doerr Summary Quasirandomness: –Simulate a particular aspect of a random object Surprising results: –Quasirandom walks –Quasirandom rumor spreading For future research: –Good news: Quasirandomness can be analyzed (in spite of ‘nasty’ dependencies) –Many open problems –“What is the right dose of randomness?” Grazie!

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