What is the linear acceleration of many random walks?

1. Many random walks
are faster than one
Noga Alon Tel Aviv University
Chen Avin Ben Gurion University
Michal Koucky Czech Academy of Sciences
Gady Kozma Weizmann Institute
Zvi Lotker Ben Gurion University
Mark R. Tuttle Intel

2. Random walks
• Random step:
– Move to an adjacent node chosen at random (and uniformly)
• Random walk:
– Take an infinite sequence of random steps
Why random
walks are
good

3. Many Applications
• Graph exploration
– Randomization avoids need to know topology
– Randomization rules when the graph is changing or
unknown
• Communication: devices send messages at random
– Exhibits locality, simplicity, low-overhead, robustness
– Becoming a popular approach to mobile devices
What is bad
about
Random
walk?

4. Latency is a problem
• There are many measures of latency:
– Hitting time: Expected time E(Hi,j) to visit a given node
– Cover time: Expected time E(C) to visit all nodes
– Mixing time: Expected time to reach the stationary distribution

5. • Best cases:
– Dense, highly connected graphs such as the complete
graph, expanders, hypercube.
– Optimal cover time: Θ(n∙log n).
• Worst cases
– Connectivity “decreases” and bottlenecks exist.
– Examples: the lollipop graph Θ(n3), the line Θ(n2).
Cover Time - known results

6. Our question
Can multiple walks reduce the latency?
Choose a node v in a graph.
Start k random walks from node v.

7. Speed up
We define the speed-up of k-random walks on a graph G.
Our Questions:
Can The speed be k?
Can k walks cover the graph k time faster than 1
walk?
Our Answer:
Many times yes, but not always.

8. Outline
• First some fun: Calculate speed-ups for simple graphs
– Clique (complete graph): linear speed-up
– Barbell: exponential speed-up
– Cycle: logarithmic speed-up
• Then some answers: When is linear speed-up possible?
– General formulations of our linear speed-up result
• In terms of the ratio cover-time/hitting-time
• Conclusions and open problems

9. Calculating speed-ups for
simple graphs
Let’s calculate E(C1) and E(Ck) for
the clique, the barbell, the cycle.

10. The clique

11. Clique hitting time: n
• A random walk starting at node A
– Chooses a random node each step
– Chooses node B with probability 1/n
– Expected waiting time until choosing B is n
• Hitting time from A to B is n
• Example {1,2,1,3,1,2,5,4}
A
B
What about
cover time?

12. Coupon Collectors problem
• Let n types of coupons and each at
each trile a coupon is picked at uniform
random and independing .
• Find the earliest time at which all n
objects have been picked at least
once.
• We need nlog(n) time.

13. Clique speed-up: k
• A k-walk chooses nodes k times faster
– 1 step of a k-walk chooses k nodes at random
– k steps of a 1-walk chooses k nodes at random
• Calculate expectations, then regroup terms:
• Example {1,2,1,3,1,2,5,4}={1,1,1,5},{2,3,2,4}
)()( 1
ktCPtCP k

)()()()()( 111 k
t
k
tt
CkEtCkPtkCkPtCPCE  

14. The barbell

15. Barbell cover time: n2
• The walk starts at O and moves to L or R: let’s say to L
• The walk must move back to O in order to cover R
• How long do we expect to wait for this L  O transition?
– From L, the walk moves to O with probability 1/(n+1)
• Expect to fail n times and move to BL instead of O
– From BL, the walk takes a long time to return to L
• Remember the hitting time in the clique is n
• Expect n steps to return to L from inside BL
O RL
BL BR
21
))(()()( nnnOLECE 

16. Barbell speed-up: 2k
• Start k=c log(n) walks on O (but let’s ignore ugly constants)
• Expect half to move to BL, half to BR: that’s log(n) in each
• Expect log(n) walks in BL and BR to stay there for n steps
– Remember hitting time for the clique is n
• Expect log(n) walks in BL and BR to cover them in n steps
– Remember k-walk cover time for the clique is n log(n)/k
• So expect log(n) walks to cover barbell in n steps, not n2
– Trust me: Proof must turn each “expect” into “with high probability”
– Rejoice with me: That’s a speed up of n=2log(n) = 2k
O
BL BR
RL

17. The cycle
0
1n
i
i-1i+1

18. Cycle cover time: n2
• Let Ei be expected time to reach 0 from i
– E0 = 0
– Ei = 1 + Ei+1/2 + Ei-1/2
– En = E1
• Solve these recurrence relations
– Show Ei = (i-1)E1 - (i-1)i
• Notice Ei+1 – Ei = Ei – Ei-1 – 2
• Define Di+1 = Ei+1 – Ei and notice Di+1 = Di – 2 = E1 – 2i
– Show E1 ≈ n
• Notice E1 = En = (n-1)E1 - (n-1)n and solve for E1
• So Ei ≈ (i-1)n – (i-1)i = (i-1)(n-i)
– Maximized at i = n/2 and maximum value is n2/4
0
1n
i
i-1i+1

19. Cycle speed-up: log(k)
The probability for a single walk to cover the Cycle in n/2
steps is 2-n/2
.
So, 2Ω(n) walks are needed for a linear speed up.
Generalize for k, the speed up log(k)
0
1n
i
i-1i+1

20. Matthews’ Theorem
• Theorem: For any graph G
C1  H1 log (n)
• This bound may or may not be tight
– On a clique, the cover time is nlog(n) and hitting time is n
– On a line, the cover time and hitting time are both n2

21. When is Matthews’ Theorem
Tight
• Observations: Matthews is tight for many
important graphs:
– Cliques
– expanders,
– torus,
– hypercubes,
– d-dimensional grids,
– d-regular balanced trees,
– random graphs, etc.
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22. linear speed-up
Matthews’ bound [MAT88]: for any graph G
Theorem: For any graph G and k ≤ log n
Corollary: linear speed up when Matthews’ bound is
tight.

23. Proof
(independent trails)
(Markov inequalit)

24. Main Results
Theorem: For a large collection of graphs, as long as k is not
too big there is a speed up of k-o(1).
The collection includes all graphs for which there is a gap
between the cover time C and hmax and k such:
24

25. Proof sketch: 2 speed-up
If there is a gap between the cover time C and hmax then
the cover time is concentrated.
Given: a single random walk of length C+o(C) covers the
graph with high probability.
We Show: 2 random walks of length C/2 + h covers the
graph w.h.p.
When h is an order less than C/2 we have a speed up of
2-o(1).
C/2+o(C)
C/2+o(C)
h
≥C+o(C)
v
u

26. Conclusion
• Observations: Matthews is tight for many
important graphs:
– Cliques
– expanders,
– torus,
– hypercubes,
– d-dimensional grids,
– d-regular balanced trees,
– random graphs, etc.
• We can prove a speed-up even when
Matthews is not tight …
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27. The Result is Tight
For the 2-dimensional grid (torus)
– gives
– gives
On the grid could, ?
On the cycle could, ?
Fore a more restricted families of graphs we can have linear
speed up for a larger k.

28. Expanders
• Expanders yield impressive cover time speed-ups:
– We proved linear speed-up for many graphs for k  log n
– We can prove linear speed-up for expanders for k  n

29. Conclusions
• Linear speed-ups are possible for many important graphs
– Speed-ups are related to the ratio C1/H1 of cover and hitting times
– Linear speed-ups occur when this ratio is large
• Open problems:
– Is the speed-up always at most k? always at least log k?
– What is random walks can communicate or leave “breadcrumbs”?
– What if the graph is actually changing dynamically?

30. End