Introduction to Quasirandomness
Benjamin Doerr
Max-Planck-Institut für Informatik
Saarbrücken
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, ...
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|.
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.
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?
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?
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)
Benjamin Doerr
Random Walks
 Rule: Move to a neighbor chosen at random
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(n1/2)
moves to each neighbor.
 Quasirandomness: Ensure that neighbors are
served evenly!
Benjamin Doerr
Quasirandom Random Walks
 Rule: Follow the rotor. Rotor updates after each move.
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(n1/2)
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)
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 
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!
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
Benjamin Doerr
Example: Discrepancies on the Line
Random Walk
(Expectation):
2.375
4.75
7.125
9.5
19
9.5
5
8
9
19
9
0.25
0.875
-0.5
-0.5
7.125
9.5
4.75
2.375
Quasirandom:
2
Difference:
-0.375
Benjamin Doerr
10
6
5
3
But: You’ll A
never
discrepancy
have more
> 1!
than
2.29 [Cooper, D, Spencer, Tardos]
-1.125
0.5
0.25
0.625
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
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]
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(n1/6) [Lawler’95]
– Experiment: outradius – inradius ≈ log2(n) [Moore, Machta’00]
 With quasirandom walks:
– Proof: outradius – inradius = O(n1/4+eps) [Levine, Peres]
– Experiment: outradius – inradius < 2
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?
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 !
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”?
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)
4:
Each
informed
node
informs
a random
node
RoundRound
5: Let‘s
remaining
two
get informed...
Round
2:hope
3:
1:the
Starting
Round
node
0: Starting
informs
node
random
is informed
node
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
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(Kn) = O(log(n)) w.h.p.
– Hypercubes: T({0,1}d) = O(log(n)) w.h.p.
– Random graphs: T(Gn,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)]
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]
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...
1
2
3
4
5
6
List: 2 3 4 5 6
34561
45612
56123
61234
12345
 Here: n-1 rounds .
 No hope for quasirandomness here?
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
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):
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 Gn,p, p > (1+eps) log(n)
still hold...
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 Gn,p, p > (1+eps) log(n)
still hold independent from the structure of the lists
[2 good news: (a) results hold, (b) things can be analyzed]
[D, Friedrich, Sauerwald]
Benjamin Doerr
Semi-Deterministic Rumor Spreading
 Results (2):
– Random graphs Gn,p, p = (log(n)+log(log(n)))/n:
 fully randomized: T(Gn,p) = Θ(log(n)2) w.h.p.
 semi-deterministic: T(Gn,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
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)
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!
Benjamin Doerr