1. Many random walks are faster than one Noga AlonTel Aviv University Chen AvinBen Gurion University Michal KouckyCzech Academy of Sciences Gady KozmaWeizmann Institute Zvi LotkerBen Gurion University Mark R. TuttleIntel

2. Random walks Random step: –Move to an adjacent node chosen at random (and uniformly) Random walk: –Take an infinite sequence of random steps Random walks are cool! Who needs applications?

3. Applications Graph exploration –Randomization avoids need to know topology –Randomization rules when the graph is changing or unknown Tracking –Hunters and prey start on different nodes –Hunters must locate and track prey Communication: devices send messages at random –Exhibits locality, simplicity, low-overhead, robustness –Becoming a popular approach to mobile devices And querying, searching, routing, self-stabilization in wireless ad hoc, peer-to-peer, and distributed systems –Example: find the max node when the edges go up and down –Can’t use depth-first search: can’t backtrack over a missing edge

4. Latency is a problem There are many measures of latency: –Hitting time: Expected time E(H) to visit a given node –Cover time: Expected time E(C) to visit all nodes –Mixing time: Expected time to reach the stationary distribution A walks spends an  v fraction of time at node v on average After the mixing time, a walk is at node v with probability  v

5. Our question Can multiple walks reduce the latency? Choose a node v in a graph. Start k random walks from node v. Can k walks cover the graph k times faster than 1 walk? Our answer: Many times yes, but not always.

6. 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? –Simple formulation of our linear speed-up result –General formulations of our linear speed-up result In terms of the ratio cover-time/hitting-time In terms of mixing time Conclusions and open problems

7. Calculating speed-ups for simple graphs

8. Computer science probability Coin flipping –p is the probability a coin lands heads –1/p is the expected waiting time until a coin lands heads Markov inequality Chernoff inequality –When  = log(n) and  = 4, this probability is very small! –If you expect log(n) samples to be bad, then with high probability fewer than 5log(n) really are bad

9. Calculating speed-ups for simple graphs Let’s calculate E(C 1 ) and E(C k ) 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 A B

12. Clique cover time: n log(n) Random walk visits nodes a 1, a 2, …, a n Assume i visited nodes, n-i unvisited nodes –(n-i)/n is probability the next node chosen is unvisited –n/(n-i) is the expected timed until an unvisted node is chosen Cover time is E i = time to visit i+1 st node after visiting i th node a 1 a 2 … a 3 … … a i … a i+1 … … a n i nodes visited

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:

14. The barbell

15. Barbell cover time: n 2 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 B L instead of O –From B L, 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 B L ORL BLBL BRBR (trust me)

16. Barbell speed-up: 2 k Start k=c log(n) walks on O (but let’s ignore ugly constants) Expect half to move to B L, half to B R : that’s log(n) in each Expect log(n) walks in B L and B R to stay there for n steps –Remember hitting time for the clique is n Expect log(n) walks in B L and B R 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 n 2 –Trust me: Proof must turn each “expect” into “with high probability” –Rejoice with me: That’s a speed up of n=2 log(n) = 2 k O BLBL BRBR RL

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

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

19. Cycle speed-up: log(k) Theorem: If C k  n 2 /s then s  log k Proof: We will show the following: 0 1 n n/2

21. Walk w takes n/2 more steps in one direction than the other –Let S i = +1 or -1 indicate whether w moves left or right at step i –Let D t = S 1 + S 2 +  + S t be the difference in steps left – steps right We can show using Chernoff So

22. These speed-ups are all over the map! (linear, exponential, logarithmic) What is the right answer? When is linear speed-up possible? A simple answer. A general answer.

23. Matthews’ Theorem Theorem: For any graph G C 1  H 1 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 n 2

24. Matthews’ Theorem for k walks Theorem: For any graph G and k  log(n) C k  (e/k) H 1 log (n) + noise Think of a random walk of length eH 1 as a trial –Starting from any node, either the walk hits v or it doesn’t Bound the probability that log(n) trials fail –A walk hits v in H 1 expected time (hitting time definition) –A walk of length eH 1 fails to hit v with probability < 1/e (Markoff) –So log(n) walks of length eH 1 fails with probability < (1/e) log n = 1/n Obtain log(n) trials using k random walks –k walks of length (log n/k) eH 1 amount to log n trials So the k-walk cover time is (e/k) H 1 log (n) + noise

25. Simple speed-up Theorem: When Matthews’ bound is tight, we have linear speed up for k  log(n) Proof: –C 1  H 1 log (n) when Matthew’s bound is tight –C k  (e/k) H 1 log (n) by previous result –C k  (e/k) C 1 Observations: –Matthews is tight for many important graphs: cliques, expanders, torus, hypercubes, d-dimensional grids, d-regular balanced trees, certain random graphs, etc. –We can prove a speed-up even when Matthews is not tight …

26. General speed-up Speed-up in terms of cover-time/hitting-time ratio: –Theorem: If R(n) = E(C 1 )/E(H 1 )   and k  R(n) 1- , then E(C k )  (1/k) E(C 1 ) + noise –When Matthews is tight, R(n) = log(n) Replaces that constant e with 1, but at cost of slightly smaller k Speed-up in terms of mixing time: –Theorem: If G is a d-regular graph with mixing time M, then E(C k )  (M log(n)/k) E(C 1 ) + noise

27. Expanders Expanders are highly-connected, sparse graphs: –Every nodes has degree d –Every set of at least half the nodes has at least  n neighbors Expanders have many applications: –Robust communication networks –Error correcting codes, random number generators, cryptography –Distributed memories, sorting networks, topology, physics… 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!

28. Conclusions Linear speed-ups are possible for many important graphs –Speed-ups are related to the ratio C 1 /H 1 of cover and hitting times –Linear speed-ups occur when this ratio is large –This result is tight… Open problems: –Is the speed-up always at most k? always at least log k? –Is there a property characterizing speed-up better than C 1 /H 1 ? –What if random walks start at different nodes, not the same node? –What is random walks can communicate or leave “breadcrumbs”? –What if the prey can move, not just the hunters? –What if the graph is actually changing dynamically?