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Balls and Bins Oblivious Routing

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Balls and Bins Oblivious Routing Powered By Docstoc
					EECS 495: Randomized Algorithms                                                        Lecture 7
Using Chernoff



Reading: Text: Chapter 4; Balanced Allo-              Oblivious Routing
cations by Azar, Broder, Karlin, and Upfal
                                                      Given:

                                                        • hypercube network
Balls and Bins
                                                        • permutation destinations
Problem: n balls, place obliviously into n Output:
bins
                                            • set of oblivious routes with min # steps
Algorithm: Random.
Claim: W/prob. (1 − n−c ), fullest bin has Algorithm: (deterministic): Bit fixing (left
            ln n                           to right)
(1 + o(1)) ln ln n balls.
Proof: Let Ejk be event that bin j has more Question: Randomized alg.?
than k balls                                  Recall load-balancing: m jobs, n = m
                                              machines; to distribute load oblivioiusly,
               n            i          n−i
                   n     1         1          we randomly routed jobs to machines.
  Pr[Ejk ] =                   1−
              i=k
                   i     n         n        Idea: Load-balance paths!
                 n                            i
                             ne       i   1
           ≤
               i=k
                              i           n           First try:      random destination,
                n
                             e    i
                                                      bit-fixing
           =
               i=k
                             i
                                 ∞                      • T (el ) = # paths using el
                 e       k        e i
           ≤
                 k                k                     • By symmetry, all T (el ) equal
                             i=0
                 e       k        1                     • Expected path length n/2
           =
                 k             1 − e/k
                     e       k                          • LOE, total expected path length N n/2
           ≤ 2
                     k                                  • N n edges in hypercube
                     1
           ≤ O
                     n2                               So E[T (el )] = 1/2, so delay at most n/2.

for k = O(log n/ log log n). Result follows by Claim: Delay ≤ 6n with high prob.
union bound.                                   Proof: Chernoff: Delay X ≤ l T (el ), so

                                                  1
  • Pr[X > (1 + δ)µ] ≤ exp(−(1 + δ)µ)                 Intuition:
  • Pr[X > 6n] ≤ exp(−6n)
                                                        • At most n/4 bins have 4 balls → prob.
WRONG!                                                    get bin with 5 balls at most (1/4)(1/4) =
                                                          1/16
Proof: Chernoff: Fix packet i, let Hij indi-
cate if routes for i and j share an edge. Inde-         • Expect n/16 bins have 5 balls → prob.
pendent and j Hij ≤ i T (el ).                            get bin with 6 balls at most 1/162
                                                                       k−3
  • delay of fixed packet i ≥ 6n with prob.              • Expect n/22        bins have k balls
    ≤ exp(−6n) ≤ 2−6n
                                                        • Whp, no bin has more than log log n
  • prob. any N = 2n packets gets delay                   balls
    more than 2−6n at most 2−5n (union
    bound)
                                            Problem: Assume system behaves as ex-
  • time to route any packet at most length pects to analyze next layer, must cope with
     plus delay, at most 7n                 conditioning.
                                            Claim: Bound binomial: Pr[B(n, p) >
→ w/prob. ≥ 1 − 2−5n , every packet reaches 2np] < 2−np/3
random dest. in 7n or fewer steps.
                                            Proof: B(n, p) = n Xi where
                                                                i=1
But wanted to reach d(i)!
Idea: Route to random intermediate desti-                     1 : w/prob. p
                                                      Xi =
nation.                                                       0 : otherwise

  • doubles path length                                 • Chernoff:
                                                                                                     µ
  • destroys bad perms.                                                                   eδ
                                                            Pr[X > (1 + δ)µ] <
                                                                                     (1 + δ)(1+δ)
Run backwards, same time bound, so packets
fail to reach final dest. in at most 14n steps
with prob. at most 2−5n+1 ≤ 1/N .                       • µ = np, δ = 1, implies claim

Claim: With prob. ≥ 1 − 1/N , every packet
                                             Note: Let Yi depend on X1 , . . . , Xi−1 . If
reaches dest. in at most 14n steps.
                                             Pr[Yi |X1 , . . . , Xi−1 ] < p for all i, then X sto-
Note: Didn’t allow phase 2 to delay phase 1; castichally dominates Y , so can use above
must have packets wait at intermediate dest. bound.
for 7n steps.
                                             Analysis:

                                                        • vi (t) = # bins of height ≥ i at time t
Power of Two Choices
                                                        • ht = height of t’th ball
Algorithm: For each ball, pick two bins ran-
                                                                                                 i
domly, place in less-loaded bin.                        • bounds β4 = 1/4, βi = 2βi−1 = 2−2
                                                                                  2



                                                  2
Claim: Let Qi be event that vi ≤ βi n. Then                • but then bins no longer grow because
Qi happens whp.
                                                               – prob. ball is tall for any i ≥ i∗ at
Proof: By induction.                                             most ((log n)/n)2
  • Let Yt = 1 if ht ≥ i+1 and vi (t−1) ≤ βi n                 – prob. two particular balls both tall
                                                                 at most ((log n)/n)4 (events nega-
         Yt indicates t’th ball placed in a bad bin
                                                                 tively correlated)
         even though there were enough good bins.
                                                               – union bound, prob. two distinct tall
  • Then Pr[Yi = 1] ≤ βi2                                        balls o(1)
  • Then t Yt stochastically dominated by                    and bins don’t grown beyond i∗ + 1 if
    X with p = βi2                                           only see one tall ball
                                                                        i∗
  • By Chernoff,                                            • βi∗ n = 2−2 ≥ log n → i∗ = log log n −
                                                             log log log n
      Pr[       Yt ≥ 2nβi2 = βi+1 n] < 2−βi+1 n/6
            t                                              • so max load is O(log log n)
                        2
   which is O(1/n ) so long as βi+1 n ≥ 
                                             Relationship to random graphs: bins are
   c log n
                                          nodes, two choices are edges. Bound
                                             comes from low expected degree and
                                                                                  
When Qi holds, then t Yt is # tall balls:
                                             “small” giant component.
 • # tall bins ≤ # tall balls             Note: Extensions:

  • so                                                     • Pick more bins: O(logd log n)
    Pr[¬Qi+1 |Qi ] ≤ Pr[            Yt > βi+1 n|Qi ]     • Pick more bins and consistent tie-
                                t                          breaking: O(log log n/d)
                       ≤ Pr[        Yt > βi+1 n]/ Pr[Qi ]
                                t
                       ≤ 1/(n2 Pr[Qi ])

Deal with conditioning:

Pr[¬Qi+1 ] = Pr[¬Qi+1 |Qi ] Pr[Qi ] + Pr[¬Qi+1 |¬Qi ] Pr[¬Qi ]
              1
           ≤    + Pr[¬Qi ]
             n2
             1
           ≤
             n
by induction.
Deal with large i:

  • ok until that i∗ s.t. bound on # tall bins
    dips below log n

                                                       3

				
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