Cost-Oblivious Reallocation for Scheduling and Planning

Abstract

In a reallocating-scheduler problem, jobs may be inserted and deleted from the system over time. Unlike in traditional online scheduling problems, where a job's placement is immutable, in reallocation problems the schedule may be adjusted, but at some cost. The goal is to maintain an approximately optimal schedule while also minimizing the reallocation cost for changing the schedule.This paper gives a reallocating scheduler for the problem of assigning jobs to p (identical) servers so as to minimize the sum of completion times to within a constant factor of optimal, with an amortized reallocation cost for a lengthw job of O(f (w) · log 3 log ∆), where ∆ is the length of the longest job and f () is the reallocation-cost function. Our algorithm is cost oblivious, meaning that the algorithm is not parameterized by f (), yet it achieves this bound for any subadditive f (). Whenever f () is strongly subadditive, the reallocation cost becomes O(f (w)).To realize a reallocating scheduler with low reallocation cost, we design a k-cursor sparse table. This data structure stores a dynamic set of elements in an array, with insertions and deletions restricted to k "cursors" in the structure. The data structure achieves an amortized cost of O(log 3 k) for insertions and deletions, while also guaranteeing that any prefix of the array has constant density. Observe that this bound does not depend on n, the number of elements, and hence this data structure, restricted to k n cursors, beats the lower bound of Ω(log 2 n) for general sparse tables.

A Nearly Quadratic Improvement for Memory Reallocation

“…The Memory Reallocation Problem, and its variations, have been studied in a variety of different settings, ranging from history independent data structures [5,9], to storage allocation in databases [4], to allocating time intervals to a dynamically changing set of parallel jobs [2,3,6]. The version considered here [3,5,9] is notable for its choice of cost function: if we model the time needed to allocation/deallocate/move an object of size 𝑠 as 𝑂 (𝑠), then an overhead of 𝑂 (𝑐) implies that the total time spent moving objects around is at most an 𝑂 (𝑐)-factor larger than the time spent simply allocating/deallocating objects. The problem of minimizing movement overhead is especially important in systems with many parallel readers, since objects may need to be locked while they are being moved.…”

“…Suppose an item 𝐼 is deleted. RSUM forms a set 𝑌 containing 𝐼 and roughly 𝑚/2 − 1 other nearby items, with total size 𝑦 ∈ 3 4 𝑚𝛿 + [−𝛿, 𝛿]. In particular, if 𝐼 is in the main-body RSUM arbitrarily adds items contiguous with 𝐼 from the same block to 𝑌 until the total size of 𝑌 lies in 3 4 𝑚𝛿 + [−𝛿, 𝛿].…”

“…RSUM forms a set 𝑌 containing 𝐼 and roughly 𝑚/2 − 1 other nearby items, with total size 𝑦 ∈ 3 4 𝑚𝛿 + [−𝛿, 𝛿]. In particular, if 𝐼 is in the main-body RSUM arbitrarily adds items contiguous with 𝐼 from the same block to 𝑌 until the total size of 𝑌 lies in 3 4 𝑚𝛿 + [−𝛿, 𝛿]. If 𝐼 is in the trash can RSUM simply adds arbitrary trash can items contiguous with 𝐼 to 𝑌 until the total size is appropriate.…”

“…Fix some delete. Let 𝑦 ∈ 3 4 𝑚𝛿 + [−𝛿, 𝛿] be the size of the set of items 𝑌 contiguous with the deleted item which we aim to swap. Suppose we are given a set 𝑋 of 𝑚 items with sizes chosen uniformly randomly and independently from [𝛿, 2𝛿].…”

Cost-oblivious storage reallocation

“…There is a variety of possible extensions to this concept. One such direction is to consider the sum of allocation costs; we address this in a related followup paper [Bender et al 2015].…”