Online Algorithms to Minimize Resource Reallocations and Network Communication

In this paper, we consider two new online optimization problems (each with several variants), present similar online algorithms for both, and show that one reduces to the other. Both problems involve a control trying to minimize the number of changes that need to be made in maintaining a state that satisfies each of many users’ requirements. Our algorithms have the property that the control only needs to be informed of a change in a users needs when the current state no longer satisfies the user. This is particularly important when the application is one of trying to minimize communication between the users and the control.

The Resource Allocation Problem (RAP) is an abstraction of scheduling malleable and evolving jobs on multiprocessor machines. A scheduler has a fixed pool of resources of total size

T

. There are

n

users, and each user

j

has a resource requirement for

r

$_{j,{\it t}}$

resources. The scheduler must allocate resources ℓ

$_{j,{\it t}}$

to user

j

at time

t

such that each allocation satisfies the requirement (

r

$_{j,{\it t}}$

≤ℓ

$_{j,{\it t}}$

) and the combined allocations do not exceed

T

(∑

j

ℓ

j

,

t

≤

T

). The objective is to minimize the total number of changes to allocated resources (the number of pairs

j

,

t

where ℓ

$_{j,{\it t}}$

≠ℓ

$_{j, {\it t}+1}$

).

We consider online algorithms for RAP whose resource pool is increased to

sT

and obtain an online algorithm which is

O

(log

s

n

)- competitive. Further we show that the increased resource pool is crucial to the performance of the algorithm by proving that there is no online algorithm using

T

resources which is

f

(

n

)-competitive for any

f

(

n

). Note that our upper bounds all have the property that the algorithms only know the list of users whose requirements are currently unsatisfied and never learn the precise requirements of users. We feel this is important for many applications, since users rarely report underutilized resources as readily as they do unmet requirements. On the other hand, our lower bounds apply to online algorithms that have complete knowledge about past requirements.

The Transmission-Minimizing Approximate Value problem is a generalization of one defined in [1], in which low-power sensors monitor real-time events in a distributed wireless network and report their results to a centralized cache. In order to minimize network traffic, the cache is allowed to maintain approximations to the values at the sensors, in the form of intervals containing the values, and to vary the lengths of intervals for the different sensors so that sensors with fluctuating values are measured less precisely than more stable ones. A constraint for the cache is that the sum of the lengths of the intervals must be within some precision parameter

T

. Similar models are described in [2,3]. We adapt the online randomized algorithm for the RAP problem to solve TMAV problem with similar competitive ratio: an algorithm can maintain

sT

precision and be

O

(log

s

n

)-competitive in transmissions against an adversary maintaining precision

T

.

Further we show that solving TMAV is as hard as solving RAP, by reducing RAP to TMAV. This proves similar lower bounds for TMAV as we had for RAP, when

s

is near 1.