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Online List Labeling with Near-Logarithmic Writes

Martin P. Seybold

TL;DR

This work addresses Online List Labeling in the linear-space regime, where $n\le N$ keys must fit into $m = N + \lceil \varepsilon N \rceil$ slots under adversarial updates. It introduces a skip-list induced interval-tree with a proactive, budget-based smooth allocation that maps keys to array positions and triggers subtree reallocations to maintain order with minimal writes. The main contributions are an algorithm achieving expected amortized writes of $O(\varepsilon^{-1} \log n \log\log n)$ per update against an oblivious adversary, improving upon prior bounds in the linear-space setting, and a dual strategy combining smooth (gamma-based) and non-smooth adaptive splits to control drift. Open questions include extending to non-constant $\varepsilon$ and pursuing tighter bounds such as $\widetilde{O}(\varepsilon^{-1} + \log n)$.

Abstract

In the Online List Labeling problem, a set of $n \leq N$ elements from a totally ordered universe must be stored in sorted order in an array with $m=N+\lceil\varepsilon N \rceil$ slots, where $\varepsilon \in (0,1]$ is constant, while an adversary chooses elements that must be inserted and deleted from the set. We devise a skip-list based algorithm for maintaining order against an oblivious adversary and show that the expected amortized number of writes is $O(\varepsilon^{-1}\log (n) \operatorname{poly}(\log \log n))$ per update.

Online List Labeling with Near-Logarithmic Writes

TL;DR

This work addresses Online List Labeling in the linear-space regime, where keys must fit into slots under adversarial updates. It introduces a skip-list induced interval-tree with a proactive, budget-based smooth allocation that maps keys to array positions and triggers subtree reallocations to maintain order with minimal writes. The main contributions are an algorithm achieving expected amortized writes of per update against an oblivious adversary, improving upon prior bounds in the linear-space setting, and a dual strategy combining smooth (gamma-based) and non-smooth adaptive splits to control drift. Open questions include extending to non-constant and pursuing tighter bounds such as .

Abstract

In the Online List Labeling problem, a set of elements from a totally ordered universe must be stored in sorted order in an array with slots, where is constant, while an adversary chooses elements that must be inserted and deleted from the set. We devise a skip-list based algorithm for maintaining order against an oblivious adversary and show that the expected amortized number of writes is per update.
Paper Structure (7 sections, 6 theorems, 23 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution, Paper Outline, and Open Problems
  3. Using the Skip-List for Maintaining an Allocation
  4. Maintaining Allocations by Reallocating Certain Subtrees
  5. Setting $\gamma=\Theta(1/\log n)$ Prevents Smooth-Allocations from 'Drifting'
  6. Non-Smooth: Countering the Drift with Repeated, Adaptive Splits
  7. Analysis of the Run-Length and the Weight-Balance

Key Result

Lemma 2.1

Let $\gamma \in (0,\tfrac{1}{2}]$. After every update operation, every allocated interval $U$ satisfies the pre-condition $\Delta(U) \geq 1$ of having at least one free slot. In particular, reallocation from any allocated interval remains possible. Thus, the allocation maintained by the interval tre

Figures (1)

  • Figure 1: Interval tree of a skip-list (top) and allocation of keys (bottom). Allocated intervals and their respective index-ranges are shown as solid boxes, non-allocated intervals in gray. The keys on level $\ell$ are $X_\ell \setminus X_{\ell+1}$ and the (open) intervals of level $\ell$ are $\mathbb{R} \setminus X_\ell$. The example shows an initial allocation with five runs.

Theorems & Definitions (12)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2: First&Last are Drift Safe
  • proof
  • Lemma 2.3: Rounds are Drift Safe
  • proof
  • Theorem 3.1
  • Lemma 3.2: Subtree Weight
  • proof
  • ...and 2 more