APPROX and RANDOM 2006 Online Algorithms to Minimize Resource Reallocation and Network Communication Sashka Davis, UCSD Jeff Edmonds, York University, Canada Russell Impagliazzo, UCSD

APPROX and RANDOM 2006 Resource Allocation Problems [KKD02, PL95, IRSD99, Edm00] Given: Multi-processor machine with T identical processors. Problem: assign processors to parallel jobs whose requirements are evolving and malleable. Goal: schedule jobs, satisfy processor requirements of each job, minimize preemption.

APPROX and RANDOM 2006 The Weak Department Chair Problem I want 12!

APPROX and RANDOM 2006 RAP: Resource Allocation Problem RAP Instance T identical processors. n users. Input: (i,r t,i ) - at time t user i requests r i,t processors. Output: (l t,i ) - the algorithm must allocate l t,i processors to i, l t,i ≥ r t,i. Constraints: ∑ r t,i ≤ T and ∑ l t,i ≤ T, for all t. Objective: Minimize changes to the global state. Cost = |{(l t,i,l t+1,i ), where l t,i ≠ l t+1,i }|. The algorithm is not notified when users current demands fall bellow their current allocations.

APPROX and RANDOM 2006 The Strong Department Chair Problem I want 30, If not – penalty! You can’t have 30! I take the penalty!

APPROX and RANDOM 2006 RAPP: Resource Allocation Problem with Penalties RAPP Instance T identical processors. n users. Input: (i,r t,i, p t,i ) - at time t user i requests r t,i processors and penalty p t,i. Output: (l t,i ) - allocation of l t,i, processors to i s.t., l t,i ≥ r t,i or do nothing. Constraints: ∑ r t,i ≤ T and ∑ l t,i ≤ T, for all t. Objective: Minimize changes to the global state, i.e., reallocations. Cost: |{(l t,i,l t+1,i ), where l t,i ≠ l t+1,i }| + ∑ p t,i, when the scheduler fails to satisfy the t’th request. The algorithm is not notified when its current demand falls bellow its current allocation.

APPROX and RANDOM 2006 The Humble Chair Problem I want MORE ! ?

APPROX and RANDOM 2006 RRAP: Restricted Resource Allocation Problem RRAP Instance T identical processors n users Input: (i) - at time t user i complains. Output: (l t,i ), such that l t,i ≥ l t-1,i. Constraints: ∑ l j,t ≤ T, for all t. Objective: Minimize changes to the global state, i.e., reallocations. Cost: |{(l t,i,l t+1,i )}|, such that l t,i ≠ l t+1,i. The algorithm never learns the precise demands exactly, only an upper bound for each. ?

APPROX and RANDOM 2006 Network Communication Problem [OLW01, CKA02, CYV06 ] Central cache and a network of low-power sensors. Sensors read values. 1.Cache must know the values read exactly – #sensor reads = #network transmissions. 2.Sensors are low-power devices and we want to minimize network communication. –Solution: Settle for approximation.

APPROX and RANDOM 2006 TMAV: Transmission Minimizing Approximate Value Problem n sensors reading values Sensor 1 [L 1,,H 1 ] Sensor 1 [L 1,,H 1 ] Sensor n [L n,H n ] Sensor n [L n,H n ] v n  [L n,H n ] v1v1 v 1  [L′ 1, H′ 1, ] Precision T ≥ ∑(H i -L i ) Constraints: T ≥ ∑(H i -L i ); v i  [L i,H i ], for all t, i Objective: Minimize network communication. Cost: The number of transmissions between sensors and cache. Central Cache

APPROX and RANDOM 2006 Two Online Problems Minimize Resource Reallocation Minimize Network Communication Central Control Maintains State. Must satisfy the demands of many users. Objective: Minimize changes to the state. A property: online algorithms do NOT know the precise requirements of users. TMAV ? RRAP RAPPRAP

APPROX and RANDOM 2006 Bi-criteria Online Algorithms Adversary uses T resources/precision. Algorithm: –use sT resources/precision. –the precise requirements of users are unknown to the algorithm. Goal: Find randomized, competitive online algorithms for RAP, RRAP, RAPP, and TMAV problems using the smallest possible s. When s=1 then the competitive ratio is infinity.

APPROX and RANDOM 2006 Results: Upper Bounds 1.O(log s n)-competitive algorithm for RRAP, where s is a constant, s≥3. 2.Modified the solution for RRAP and obtained algorithms with similar competitive ratios O(log s n) for RAP, RAPP, and TMAV. ?

APPROX and RANDOM 2006 Results: Lower Bounds 1.For s = 1 no competitive algorithm for RAP and TMAV exists. 2.Defined the notion of competitive ratio preserving online reduction with respect to adaptive online adversary “≤ AD_ON ’’. 1.RAP ≤ AD_ON TMAV 1.RAP ≤ AD_ON RAPP

APPROX and RANDOM 2006 Results: Lower Bounds Using Reductions (h,k)-paging ≤ AD_ON RAP 1.No online algorithm, using (1+ε) resources can achieve competitive ratio better than Ω(1/ ε) against an adaptive online adversary, using resource of size 1. 2.No online algorithm using (1+ ε) resources can achieve competitive ratio better than Ω(log(1/ ε)) against an oblivious adversary using resource of size 1.

APPROX and RANDOM 2006 The Remainder of the Talk 1.Steal From the Rich – a randomized O(log s n)-competitive algorithm for RRAP. 2.For s=1 no competitive algorithm for RAP and TMAV exists.

APPROX and RANDOM 2006 RRAP: Restricted Resource Allocation Problem RRAP Instance : T identical processors, n users. Input: (i) - at time t user i complains. Output: (l i,t ), such that l t,i ≥ l t-1,i. Constraints: ∑ l t,i ≤ T, for all t. Cost: Number of pairs (l t,i,l t+1,i ), such that l t,i ≠ l t+1,i. The algorithm never learns the precise demands exactly, only an upper bound for each. ?

APPROX and RANDOM 2006 Steal From the Rich Algorithm sT/n user 1 sT/n user 2 sT/n user n Initially partition sT resources evenly among the n users. Let s be a constant, and r=Θ(√s), μ be a constants, which depend on s, but not the instance.

APPROX and RANDOM 2006 Steal From the Rich Algorithm l t,1 user 1 l t,2 user 2 l t,k user k At time t+1 user j complains. l t,j user j SFR picks a user k from [n]-{j} with probability l t,k /(sT-l t,j ). l t,n user n δ l t+1,k ← l t,k -δ; l t+1,,j +1←l t,j +δ; δ l t,j user j user k μT/n SFROPT

APPROX and RANDOM 2006 How Much to Steal from the Rich? SFR maintains the following invariants: 1.All users have at least μT/n l t+1,k ≥ μT/n, hence δ ≤ l t,k - μT/n; 2.l t+1,k does not shrink by a factor more than 1/r l t+1,k ≥ l k,t /r, hence δ ≤ l k,t (r-1)/r; 3.l t+1,j does not grow by a factor more than r l t+1,j ≤ rl t,j,, hence δ ≤ l j,t (r-1); δ = min {l t,k -μT/n; l t,k (r-1)/r; l t,j (r-1)}.

APPROX and RANDOM 2006 SFR Analysis Want to show that for any req. sequence σ E(SFRs(σ)) ≤ O(log s n)OPT(σ)+d. Φ: R n  R n → R + ; a t =SFR t +(Φ t -Φ t-1 ) E(SFRs(σ)) = E(∑SFR t )=E(∑a t )-Φ end +Φ 0 Want to prove that for all t: Φ t ≤ O(n log s n), for all t, E(a t ) ≤ O(log s n)OPT t. Then Φ 0 ≤ O(n log s n), and we use d = O(n log s n).

APPROX and RANDOM 2006 SFR Potential Function ΔΦ is small when SFR and OPT have proportional allocations. When SFR has cost and OPT does not, then ΔΦ is negative and compensates for the actual cost of SFR.

APPROX and RANDOM 2006 Amortized Update Cost E(a t ) = E(SFR t + ΔΦ t ) ≤ O(log s n)OPT t Case 1: OPT t ≠ 0, SFR = 0. E(a t ) = E(0 + #changed intervals  O(log s n)) ≤ O(log s n)OPT t Case 2: OPT t = 0, SFR = 2. E(a t ) = E(2+ΔΦ t ) E(ΔΦ t ) ≤ -2. In Case 2, SFR does: –l t,j grows by a factor of r then ΔΦ t )≤-14; –l t,k shrinks by a factor of 1/r then ΔΦ t ≤-14; –Neither: (δ = l t,k -μT/n) then ΔΦ t ≥ 0 (unfortunate but rare event). Concluding: E(SFRs(σ)) ≤ O(log s n)OPT(σ)+d.

APPROX and RANDOM 2006 The Additional Resource is Vital Theorem: There is no online algorithm using T resources that is f(n) competitive against and adversary using T resources, for any function f. Consider RAP with 2 users and T=1.

APPROX and RANDOM 2006 If s=1 then competitive ratio is ∞ 0 1 user1user2 1.Adversary cost is 2. 2.Probability of incurring cost during t’th request is 1/8t. 3.The expected cost of the algorithm diverges as t goes to infinity. S 4,1 <rS 2,1 <r S 1,1 < r r  [0,1] S 4,1 <rS≥r S 3,2 = 1-S

APPROX and RANDOM 2006 Relating the Hardness of the Problems TMAV RAP RAPP SFR RRAP ≤ AD_ON ? SFR

APPROX and RANDOM 2006 Conclusions 1.We obtained O(log s n)-competitive algorithms for four different problems. 2.Justified the need for sT resource. Defined a notion of online reduction with respect to adaptive online adversary. Related the hardness of the problems using online reductions. Reduced (h-k)-Paging to RAP and transferred the standard paging lower bounds to the four problems.

APPROX and RANDOM 2006 New Issues We studied memoryless online algorithms that do not know the current demands exactly. Online reductions to leverage existing lower bounds and relate hardness of online problems.

APPROX and RANDOM 2006 Open problems Close the gap between the upper and lower bounds. Can competitive ratio preserving reductions with respect to adaptive online adversary deliver other lower bounds for other problems? Do other problems have similar memoryless online solutions, where the algorithm does not know the demands exactly, but only an upper bound approximation of it.