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Benjamin Doerr Max-Planck-Institut für Informatik Saarbrücken Quasirandomness

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Benjamin Doerr Introduction to Quasirandomness Pseudorandomness: –Aim: Have something look completely random –Example: Pseudorandom numbers shall look random independent of the particular application. Quasirandomness: –Aim: Imitate a particular property of a random object. –Example: Low-discrepancy point sets A random point set P is evenly distributed in [0,1] 2 : For all axis-parallel rectangles R, Quasirandom analogue: There are point sets P such that holds for all R Leads to (Quasi)-Monte-Carlo methods ¯ ¯ j P \ R j ¡ vo l ( R ) j P j ¯ ¯ = O ( j P j 1 = 2 + " ) ¯ ¯ j P \ R j ¡ vo l ( R ) j P j ¯ ¯ = O ( l og ( j P j ))

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Benjamin Doerr Why Study Quasirandomness? Basic research: Is it truely the randomness that makes a random object useful, or something else? –Example numerical integration: Random sample points are good because of their sublinear discrepancy (and ‘not’ because they are random) 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 Outline of the Talk Part 0: Introduction 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 Part 1: Quasirandom Random Walks Reminder: 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 Reminder: 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 step 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 [D]: For all graphs, starting vertices, initial rotor directions and rotor permutations, the quasirandom walk needs at most 2|E|(|V|-1) steps to visit all vertices. Note: Same bound, but ‘sure’ instead of ‘expected’.

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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 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 –Cooper, D, Friedrich, Spencer [SODA’08]: Fails for infinite trees –Open problem: Results for any other graphs

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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 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 Summary Quasirandom Walks Model: –Follow the rotor and rotate it –Simulates: Random walks leave each vertex in each direction roughly equally often Results: Surprising good simulation (or “better”) –Graph exploration in 2|E|(|V|-1) steps –Many chips: For grids (but not trees), 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.

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Benjamin Doerr Part 2: Quasirandom Rumor Spreading Reminder: Randomized Rumor Spreading –same as in the talk by Thomas Sauerwald Quasirandom Rumor Spreading –Purely deterministic: Poor –With a little bit of randomness: Great! [simpler and better than randomized] “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 databases 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) –Frieze, Grimmet: T(K n ) = O(log(n)) w.h.p. –Feige, Peleg, Raghavan, Upfal: T({0,1} d ) = O(log(n)) w.h.p. –Feige et al.: T(G n,p ) = O(log(n)) w.h.p., p > (1+eps)log(n)/n

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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. Problem: Might take long... Here: n-1 rounds 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

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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, –random graphs G n,p, p > (1+eps) log(n), –hypercubes 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, –random graphs G n,p, p > (1+eps) log(n), –hypercubes still hold independent from the structure of the lists [D, Friedrich, Sauerwald (SODA ’ 08)]

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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 The Right Dose of Randomness? Results 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?” Thanks!

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