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On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

Tsvi Kopelowitz, Ariel Korin, Liam Roditty

TL;DR

The paper investigates when subquadratic-space approximate distance oracles can beat stretch 2, by tying the achievable stretch to the graph’s maximum degree $\\Delta_G$. It introduces a new ADO construction that leverages a degree-sensitive parameter and a refined neighborhood-intersection strategy to obtain a $(2,1-k)$-stretch with subquadratic space when $\\Delta_G \,\le\, O(n^{1/k-epsilon})$. The authors pair this upper bound with tight conditional lower bounds based on a set-intersection hypothesis, showing that for graphs with $\\Delta_G=\Theta(n^{1/k})$, any such ADO requires $ ilde{\Omega}(n^2)$ space, and that improving multiplicative or additive guarantees beyond the stated regime would contradict the hypothesis. They also introduce an edge-splitting technique within a butterfly-like infrastructure to realize the conditional lower bounds and to connect distance-oracle hardness to standard set-intersection problems. The results delineate a clear boundary between subquadratic-space sub-2-stretch ADOs and inherent quadratic-space necessity, offering both constructive methods and hardness implications with practical relevance for sparse graphs and degree-bounded networks.

Abstract

For an undirected unweighted graph G = (V, E) with n vertices and m edges, let d(u, v) denote the distance from u in V to v in V in G. An (alpha, beta)-stretch approximate distance oracle (ADO) for G is a data structure that, given u, v in V, returns in constant time a value d-hat (u, v) such that d(u, v) <= d-hat (u, v) <= alpha * d(u, v) + beta, for some reals alpha > 1, beta. If beta = 0, we say that the ADO has stretch alpha. Thorup and Zwick (2005) showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Patrascu and Roditty (2010) showed that one can obtain stretch 2 using O(m^(1/3)n^(4/3)) space, and so if m is subquadratic in n, then the space usage is also subquadratic. Moreover, Patrascu and Roditty (2010) showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = O-tilde(n), based on the set-intersection hypothesis. In this paper, we investigate the minimum possible stretch achievable by an ADO as a function of the graph's maximum degree, a study motivated by the question of identifying the conditions under which an ADO can be stored with subquadratic space while still ensuring a sub-2 stretch. In particular, we show that if the maximum degree in G is Delta_G <= O(n^(1/k - epsilon)) for some 0 < epsilon <= 1/k, then there exists a (2, 1 - k)-stretch ADO for G that uses O-tilde(n^(2 - (k * epsilon) / 3)) space. For k = 2, this result implies a subquadratic sub-2 stretch ADO for graphs with Delta_G <= O(n^(1/2 - epsilon)). We provide tight lower bounds for the upper bound under the same set intersection hypothesis, showing that if Delta_G = Theta(n^(1/k)), a (2, 1 - k)-stretch ADO requires Omega-tilde(n^2) space. Moreover, we show that for constants epsilon, c > 0, a (2 - epsilon, c)-stretch ADO requires Omega-tilde(n^2) space even for graphs with Delta_G = Theta-tilde(1).

On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

TL;DR

The paper investigates when subquadratic-space approximate distance oracles can beat stretch 2, by tying the achievable stretch to the graph’s maximum degree . It introduces a new ADO construction that leverages a degree-sensitive parameter and a refined neighborhood-intersection strategy to obtain a -stretch with subquadratic space when . The authors pair this upper bound with tight conditional lower bounds based on a set-intersection hypothesis, showing that for graphs with , any such ADO requires space, and that improving multiplicative or additive guarantees beyond the stated regime would contradict the hypothesis. They also introduce an edge-splitting technique within a butterfly-like infrastructure to realize the conditional lower bounds and to connect distance-oracle hardness to standard set-intersection problems. The results delineate a clear boundary between subquadratic-space sub-2-stretch ADOs and inherent quadratic-space necessity, offering both constructive methods and hardness implications with practical relevance for sparse graphs and degree-bounded networks.

Abstract

For an undirected unweighted graph G = (V, E) with n vertices and m edges, let d(u, v) denote the distance from u in V to v in V in G. An (alpha, beta)-stretch approximate distance oracle (ADO) for G is a data structure that, given u, v in V, returns in constant time a value d-hat (u, v) such that d(u, v) <= d-hat (u, v) <= alpha * d(u, v) + beta, for some reals alpha > 1, beta. If beta = 0, we say that the ADO has stretch alpha. Thorup and Zwick (2005) showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Patrascu and Roditty (2010) showed that one can obtain stretch 2 using O(m^(1/3)n^(4/3)) space, and so if m is subquadratic in n, then the space usage is also subquadratic. Moreover, Patrascu and Roditty (2010) showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = O-tilde(n), based on the set-intersection hypothesis. In this paper, we investigate the minimum possible stretch achievable by an ADO as a function of the graph's maximum degree, a study motivated by the question of identifying the conditions under which an ADO can be stored with subquadratic space while still ensuring a sub-2 stretch. In particular, we show that if the maximum degree in G is Delta_G <= O(n^(1/k - epsilon)) for some 0 < epsilon <= 1/k, then there exists a (2, 1 - k)-stretch ADO for G that uses O-tilde(n^(2 - (k * epsilon) / 3)) space. For k = 2, this result implies a subquadratic sub-2 stretch ADO for graphs with Delta_G <= O(n^(1/2 - epsilon)). We provide tight lower bounds for the upper bound under the same set intersection hypothesis, showing that if Delta_G = Theta(n^(1/k)), a (2, 1 - k)-stretch ADO requires Omega-tilde(n^2) space. Moreover, we show that for constants epsilon, c > 0, a (2 - epsilon, c)-stretch ADO requires Omega-tilde(n^2) space even for graphs with Delta_G = Theta-tilde(1).
Paper Structure (30 sections, 9 theorems, 9 equations, 1 figure)

This paper contains 30 sections, 9 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Approximate Distance Oracles.
  3. ADO for general graphs.
  4. ADO for sparse graphs.
  5. Conditional lower bounds and set-intersection.
  6. Main results: When can we beat stretch 2 with subquadratic space?
  7. Organization
  8. Overview of Results and Techniques
  9. Upper Bound: A New ADO
  10. Intuition for the new ADO.
  11. Conditional Lower bound
  12. The challenges.
  13. Combining approaches.
  14. Applying the edge splitting technique.
  15. Additional related work
  16. ...and 15 more sections

Key Result

Theorem 1.4

For any graph $G$, positive real constant $c$, and positive integer $k$ for which $\Delta_G \leq c n^{\frac{1}{k}-\varepsilon}$ for some real $0 < \varepsilon \leq 1/k$, there exists an ADO for $G$ that uses $\tilde{O}(c^k n^{2-\frac{k \varepsilon}{3}})$ space and has a $(2, 1-k)$-stretch.

Figures (1)

  • Figure 1: A query for $u, v \in V$ in the case that $N(u, 16 \hat{c} \, n^{1/3 + 2\alpha}) \cap B(v) = \emptyset$. Since $x \in N(u, 16 \hat{c} \, n^{1/3 + 2\alpha})$, $b \in B(v)$ and $x$ and $b$ are both on a shortest path between $u$ and $v$, it must be that $d(u,x) \leq d(u,b) - 1$.

Theorems & Definitions (22)

  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 1.7
  • Corollary 3.2
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 12 more