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When Local and Non-Local Meet: Quadratic Improvement for Edge Estimation with Independent Set Queries

Tomer Adar, Yahel Hotam, Amit Levi

TL;DR

This work analyzes edge estimation on unknown graphs under a hybrid query model that fuses independent-set queries with local degree and neighbor queries. The authors introduce an advice-based, two-sequence guessing algorithm that partitions edges by vertex degree and estimates each part with IS-assisted sampling, achieving a tight $(1\pm\varepsilon)$-approximation of the edge count $m$ with query complexity $O\left(\min\left(\sqrt{m}, \sqrt{\frac{n}{\sqrt{m}}}\right)\cdot \frac{\log n}{\varepsilon^{5/2}}\right)$. They also prove a nearly matching lower bound via a string-distinction reduction and a canonical graph construction, showing a quadratic improvement in the hybrid model relative to each individual model. The results demonstrate the complementary strengths of IS and local queries and establish near-optimal trade-offs for edge counting in sublinear time, with significant implications for sublinear graph analysis where multiple query primitives are available.

Abstract

We study the problem of estimating the number of edges in an unknown graph. We consider a hybrid model in which an algorithm may issue independent set, degree, and neighbor queries. We show that this model admits strictly more efficient edge estimation than either access type alone. Specifically, we give a randomized algorithm that outputs a $(1\pm\varepsilon)$-approximation of the number of edges using $O\left(\min\left(\sqrt{m}, \sqrt{\frac{n}{\sqrt{m}}}\right)\cdot\frac{\log n}{\varepsilon^{5/2}}\right)$ queries, and prove a nearly matching lower bound. In contrast, prior work shows that in the local query model (Goldreich and Ron, \textit{Random Structures \& Algorithms} 2008) and in the independent set query model (Beame \emph{et al.} ITCS 2018, Chen \emph{et al.} SODA 2020), edge estimation requires $\widetildeΘ(n/\sqrt{m})$ queries in the same parameter regimes. Our results therefore yield a quadratic improvement in the hybrid model, and no asymptotically better improvement is possible.

When Local and Non-Local Meet: Quadratic Improvement for Edge Estimation with Independent Set Queries

TL;DR

This work analyzes edge estimation on unknown graphs under a hybrid query model that fuses independent-set queries with local degree and neighbor queries. The authors introduce an advice-based, two-sequence guessing algorithm that partitions edges by vertex degree and estimates each part with IS-assisted sampling, achieving a tight -approximation of the edge count with query complexity . They also prove a nearly matching lower bound via a string-distinction reduction and a canonical graph construction, showing a quadratic improvement in the hybrid model relative to each individual model. The results demonstrate the complementary strengths of IS and local queries and establish near-optimal trade-offs for edge counting in sublinear time, with significant implications for sublinear graph analysis where multiple query primitives are available.

Abstract

We study the problem of estimating the number of edges in an unknown graph. We consider a hybrid model in which an algorithm may issue independent set, degree, and neighbor queries. We show that this model admits strictly more efficient edge estimation than either access type alone. Specifically, we give a randomized algorithm that outputs a -approximation of the number of edges using queries, and prove a nearly matching lower bound. In contrast, prior work shows that in the local query model (Goldreich and Ron, \textit{Random Structures \& Algorithms} 2008) and in the independent set query model (Beame \emph{et al.} ITCS 2018, Chen \emph{et al.} SODA 2020), edge estimation requires queries in the same parameter regimes. Our results therefore yield a quadratic improvement in the hybrid model, and no asymptotically better improvement is possible.
Paper Structure (33 sections, 46 theorems, 66 equations, 23 figures, 1 table)

This paper contains 33 sections, 46 theorems, 66 equations, 23 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Results
  3. Technical overview
  4. Upper-Bound
  5. Lower bound
  6. Canonical graph construction
  7. Graph distributions
  8. Intuition
  9. Related work
  10. Preliminaries
  11. Classifying vertices and edges
  12. Upper Bound
  13. Estimating $m_{LL}$ (with advice)
  14. Estimating $m_{LH}$ (with advice)
  15. Handling the case where $mbar < m$
  16. ...and 18 more sections

Key Result

Theorem 1

Consider the query model consisting of IS, degree, and neighbor queries to an unknown graph $G=(V,E)$ with $|V|=n$ and $|E|=m$. For every $0 < \varepsilon < 1$, there exists a randomized algorithm that uses queries in expectation and outputs a $(1\pm\varepsilon)$-approximation of $m$ with probability at least $2/3$.

Figures (23)

  • Figure 1: A diagram of the graph structure
  • Figure : A diagram of the graph structure
  • Figure : A diagram of the graph structure
  • Figure : A diagram of the graph structure
  • Figure : A diagram of the graph structure
  • ...and 18 more figures

Theorems & Definitions (55)

  • Theorem 1: Upper Bound
  • Theorem 2: Lower Bound
  • Definition 2.1: Degree oracle
  • Definition 2.2: Random Neighbor oracle
  • Definition 2.3: Independent set oracle
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 3
  • Lemma 3.1: BHRRS18
  • ...and 45 more