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Power laws and power-of-two-choices

Amanda Redlich

Abstract

This paper analyzes a variation on the well-known "power of two choices" allocation algorithms. Classically, the smallest of $d$ randomly-chosen options is selected. We investigate what happens when the largest of $d$ randomly-chosen options is selected. This process generates a power-law-like distribution: the $i^{th}$-smallest value scales with $i^{d-1}$, where $d$ is the number of randomly-chosen options, with high probability. We give a formula for the expectation and show the distribution is concentrated around the expectation

Power laws and power-of-two-choices

Abstract

This paper analyzes a variation on the well-known "power of two choices" allocation algorithms. Classically, the smallest of randomly-chosen options is selected. We investigate what happens when the largest of randomly-chosen options is selected. This process generates a power-law-like distribution: the -smallest value scales with , where is the number of randomly-chosen options, with high probability. We give a formula for the expectation and show the distribution is concentrated around the expectation
Paper Structure (7 sections, 5 theorems, 19 equations, 3 figures)

This paper contains 7 sections, 5 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Results
  4. Proof sketch
  5. Initialization Lemmas
  6. Proof of Theorem \ref{['big']}
  7. Conclusions and Future Work

Key Result

Theorem 1

After $m>n^{4d+13}$ balls have been placed under $\mathrm{UNFAIR}(m,n,d)$, the expected load $b_{i(m)}(m)$ in the $i^{th}$ smallest bin is Furthermore the probability that all$n$ bins are within $O(m/n^{d+1})$ of their expectation is $1-O(n^{-d})$.

Figures (3)

  • Figure 1: Experimental result for $n=100$, $m=10^6$, $d=2$
  • Figure 2: Experimental result for $n=100$, $m=10^6$, $d=3$
  • Figure 3: Experimental result for $n=100$, $m=10^6$, $d=4$

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 2: Theorem 4.3 from me
  • proof
  • proof