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Almost Linear 3-Spanners of Temporal Cliques
Julia Baligacs, Davide Bilò, Václav Blažej, Maël Dumas, Anna Zych-Pawlewicz
TL;DR
The paper addresses how to sparsify temporal cliques while preserving minimum-hop temporal distances with a constant stretch. It introduces a recursive edge-selection approach and obtains almost-linear-size temporal 3-spanners, including an O(nL) bound for bounded lifetime.
Problem
The paper studies whether temporal cliques can preserve minimum-hop temporal distances with sparse temporal α-spanners, motivated by the communication and failure costs of additional relays.
Source: §1 · p. 2Method
The approach recursively uses Zorro to select carefully structured edges covering large collections of source-target pairs with temporal paths of length at most 3, then processes remaining pairs.
Source: §3 · p. 5Results
Every temporal clique admits an almost-linear-size temporal 3-spanner; with lifetime L, it admits one of size O(nL).
Source: §1 · p. 3Takeaways & Limitations
The bounded-lifetime bound removes the logarithmic factor and replaces exponential lifetime dependence in the previous bound with linear dependence.
Source: §1 · p. 3The analysis reduces temporal cliques to instances in which every edge carries exactly one time label.
Source: §2 · p. 3
Abstract
from arXiv · showhide
Temporal graphs model dynamic networks by assigning positive integer time labels to the edges, while information propagates along temporal paths, whose edge labels are traversed in nondecreasing order. A temporal $α$-spanner of a temporal graph with $n$ vertices is a temporal subgraph that approximates the minimum-hop temporal distance between every pair of vertices within a factor of $α$. While general temporal graphs may not admit sparse temporal $α$-spanners for any value of $α$, temporal cliques are known to admit temporal $(2k-1)$-spanners of size $\widetilde{\mathcal{O}}(kn^{1+1/k})$ for every positive integer $k$. We present a simple recursive algorithm that computes, for every temporal clique on $n$ vertices, a temporal $3$-spanner of size $n^{1+2/\sqrt{\ln n}}=n^{1+o(1)}$, thereby improving the previous best upper bound of $\widetilde{\mathcal{O}}(n^{3/2})$. We also show that a modified version of our algorithm computes temporal $3$-spanners of size $\mathcal{O}(nL)$ when the lifetime is bounded by $L$, i.e., all time labels are in $\{1,\ldots,L\}$, thus improving the previous bound of $\mathcal{O}(2^Ln\log n)$. Both results are particularly striking in light of the known lower bound of $Ω(n^2)$ on the size of temporal $2$-spanners, which already holds for temporal cliques of lifetime $L\geq 3$. Both algorithms rely on a new simple recursive decomposition that certifies temporal connectivity for a large collection of source-target pairs using only $\mathcal{O}(n)$ carefully selected edges and recursively processes only the remaining pairs. Besides yielding substantially improved upper bounds, this approach is significantly simpler than previous constructions.
1 Introduction
Temporal α-spanners seek sparse subgraphs that preserve minimum-hop temporal reachability, with temporal cliques offering a tractable setting but a major sparsity–hop tradeoff remaining open. This work introduces a recursive construction yielding almost-linear temporal 3-spanners, including improved bounds for bounded lifetimes.
Problem setting
- Temporal paths traverse edges in nondecreasing time order, and temporal spanners preserve temporal connectivity using a subgraph.Temporal graph size counts edge labels, while temporal spanners may be dense even when the underlying graph is highly connected.Source: §1 · p. 2
- Temporal α-spanners approximate every pair’s minimum-hop temporal distance within a factor of α, making relay count an explicit sparsification constraint.For temporal cliques, this guarantees paths of at most α edges between every vertex pair.Source: §1 · p. 2
Prior gap
- Temporal 2-spanners can require Ω(n^2) edges for lifetime L ≥ 3, making temporal 3-spanners the first nontrivial sparse constant-hop regime there.The cited prior results also establish O(n log n) temporal 2-spanners when L = 2.Source: §1 · p. 3
Main results
- For lifetime L, Theorem 2 gives temporal 3-spanners of size O(nL), replacing the previous exponential lifetime dependence and logarithmic factor.The new bounded-lifetime spanner can be computed in time O(n^3L), and the results extend to temporal bicliques.Source: §1 · p. 3
Method
- The recursive Zorro procedure selects O(n) edges that certify 3-hop temporal connectivity for many source–target pairs, then recurses on uncovered pairs.This yields a simpler recursive decomposition than previous constructions.Source: §1 · p. 3
Implication
- Allowing one additional hop changes the known regime from potentially quadratic temporal 2-spanners to near-linear temporal 3-spanners.Whether every temporal clique has a linear-size temporal 3-spanner remains open.Source: §1 · p. 3
2 Preliminaries
The preliminaries reduce temporal-clique spanners to temporal biclique bispanners and establish conventions for temporal labels, paths, and neighbor orderings.
Temporal-graph conventions
- The paper may assume every temporal-clique edge has exactly one time label, so a spanner is simply a set of edges.Keeping an arbitrary single label from each multi-labeled edge preserves any temporal α-spanner guarantee.Source: §2 · p. 3
- A temporal graph supports paths whose selected edge labels are traversed in nondecreasing order.This temporal ordering governs reachability and spanner guarantees.Source: §2 · p. 3
- Each vertex’s neighbors are ordered by nondecreasing incident-edge labels, defining its i-th smallest neighbor.Ties may be broken arbitrarily.Source: §2 · p. 3
Temporal bicliques
- A temporal biclique separates vertices into sources and targets and includes every source-target edge with its time labeling.A balanced biclique of size n has n sources and n targets.Source: §2 · p. 4
- A temporal α-bispanner requires at most α-hop temporal paths from every source to every target, but not from targets to sources.This directed source-target requirement is the biclique analogue used in the reduction.Source: §2 · p. 4
Reduction between models
- Sparse temporal spanners for cliques can be obtained from equally sparse balanced biclique bispanners.Observation 3 transfers a size bound f(n,L) and stretch α from bicliques to cliques; the construction duplicates each clique vertex into source and target copies, as illustrated by Figure 1.Source: §2 · p. 4, Figure 1 · p. 4
3 Almost linear 3-bispanners of temporal bicliques
The section introduces Zorro, a simple recursive procedure that builds temporal 3-bispanners for balanced temporal bicliques, yielding almost-linear size and near-quadratic computation time.
Zorro algorithm
- Zorro selects two centers and at most n edges forming a diameter-3 tree that covers k(n−k) source–target pairs.It chooses t∗ among frequently appearing small neighbors, then s∗ and the largest neighbors of s∗.Source: §3 · p. 5
- For every covered pair, the path (s, t∗, s∗, t) is temporal and has at most three hops.The nondecreasing labels follow from the choices of s∗, t∗, S1, and T2.Source: §3 · pp. 5–6, Figure 2 · p. 6
Recursive construction
- RecSpan recursively applies Zorro to uncovered sub-bicliques while preserving the 3-bispanner guarantee.Its worst-case size satisfies K(n) ≤ 2n + 2K(f(n)) + K(n−f(n)).Source: §3 · pp. 6–7
Size bound
- Choosing f appropriately gives balanced temporal bicliques a 3-bispanner of size n^(1+2/√ln n) = n^(1+o(1)).The resulting construction also applies to temporal cliques through the clique–biclique correspondence.Source: §3 · pp. 5, 7, 8, 10
Running time
- The resulting 3-bispanner can be computed in time O(n^(2+o(1))).The running-time analysis combines the cost of Zorro with the recursive decomposition.Source: §3 · pp. 5, 9, 10
4 Temporal cliques of bounded lifetime
For bounded-lifetime temporal bicliques, the construction exploits time labels to cover larger portions at each recursive step, obtaining sparse temporal 3-bispanners with linear dependence on lifetime.
Initial coverage
- The bounded-lifetime assumption allows covered subset sizes exceeding n, unlike the basic Zorro guarantee whose two subset sizes sum to n.This larger coverage drives faster shrinkage in the recursion.Source: §4 · p. 10
- For lifetime L, a balanced biclique contains subsets S′ and T′ with |S′| + |T′| ≥ n + n/L that can be covered by O(n) edges.The covering edges are built from a selected pair and temporal paths through it.Source: §4 · pp. 10–11
Balancing recursion
- When the bipartition is unbalanced, Lemma 10 removes r vertices from the larger side using at most 2n+r edges, provided r < n/(L−1).The resulting edge set covers all pairs between the smaller side and the removed vertices.Source: §4 · p. 11
Final bound
- Every balanced temporal biclique of size n and lifetime at most L admits a temporal 3-bispanner of size at most 12Ln, computable in O(Ln^3) time.The recursion shrinks the instance to size at most (1−1/L)n at each step.Source: §4 · pp. 12–13