Source-linked AI summary
Online Treasure Hunt in Vertex-Permuted Dynamic Rings
Kamran Ayoubi, Bernard Mans, Lata Narayanan
TL;DR
The paper asks when agents can find a treasure in dynamic rings whose vertices are permuted over time. It proves impossibility without dynamicity restrictions and gives bounds and algorithms for restricted VP(δ) settings, including multi-agent search.
Problem
The paper studies online treasure hunt by agents in dynamic rings, where changing vertex positions make search depend on the sequence of snapshots.
Source: §1 · p. 1Method
The paper analyzes unrestricted VP, bounded-dynamicity VP(δ), and random VP, establishing impossibility results and designing algorithms for restricted settings.
Source: §1 · p. 2, §1.1 · p. 2Results
For unrestricted VP, some inputs defeat every online algorithm with k ≤ n−3 agents; in VP(δ), flagged deterministic single-agent bounds are Θ(δn) for δ ≥ 2n, and a k-agent algorithm asymptotically matches its lower bound.
Source: §1.1 · p. 2Takeaways & Limitations
The results give a restricted-dynamicity setting where flags support deterministic single-agent search guarantees and multiple agents achieve an asymptotically matched search-time bound.
Source: §1.1 · p. 2The single-agent lower bound is proved for δ ≥ 2n, leaving its validity across the full feasible range δ ≥ ⌈(n−1)/2⌉ open.
Source: §7 · p. 19
Abstract
from arXiv · showhide
We study the problem of treasure hunt by a group of $k \geq 1$ agents in vertex-permuted dynamic rings (VP). In this model, the $n$ vertices remain on a ring but are permuted at each time step. We first show that treasure hunt is impossible for any $k \leq n-3$ agents, if there are no restrictions on the sequence of permutations used in the dynamic ring. We then study the $VP(\delta)$ setting, in which for every pair $i, j$ of vertices, the edge $(i, j)$ is guaranteed to appear within $\delta$ steps. We show that the class $VP(\delta)$ is feasible only for $\delta \geq \left\lceil \frac{n-1}{2}\right\rceil$. For the one-agent case, we show a tight bound of $\Theta(\delta n)$ on the worst-case search time as well as competitive ratio of any online algorithm for treasure hunt, provided $\delta \geq 2n$. We then give an optimal algorithm for $k$ agents, thereby showing that $k$ agents can obtain a speedup of $k$ on the worst-case search time. Finally, in the R-VP setting, in which in every step, the vertices are arranged as a ring according to a random permutation, we show that treasure hunt takes expected $\Theta(n)$ steps against an oblivious adversary and $\Theta(n \log n)$ steps against an adaptive adversary.
1 Introduction
The paper studies online treasure hunt by agents in dynamic rings whose vertex positions may change arbitrarily. It establishes impossibility without restrictions on the dynamics, then summarizes results for restricted and random models.
1.1 Overview of results
- For unrestricted VP rings, no deterministic or randomized online algorithm can find the treasure with k ≤ n−3 agents on some inputs.The treasure can remain unreachable by every agent.Source: §1.1 · p. 2
- In VP(δ), feasibility begins at δ ≥ ⌈(n−1)/2⌉; for δ ≥ 2n, a flagged deterministic agent has tight Θ(δn) search-time and competitive-ratio bounds.Without flags, no deterministic single-agent algorithm can solve the problem; a randomized algorithm finds the treasure in expected O(δ^2) time.Source: §1.1 · p. 2
- For k agents, a lower bound of (⌈(n−2)/k⌉−2)δ holds when δ ≥ (k+1)(n−k−2), and an algorithm asymptotically matches it.The lower bound also holds when agents have memory, chirality, full node-label knowledge, and whole-graph visibility.Source: §1.1 · pp. 2–3
- Flags provide one-bit persistent memory visible to agents at neighboring nodes, avoiding the need to remember node identities or communicate by messages.The authors identify this flag mechanism as a novel idea to their knowledge.Source: §1.1 · p. 3
- In R-VP, expected search time is Θ(n) against an oblivious adversary and Θ(n log n) against an adaptive adversary.The adaptive adversary knows the input sequence of random graphs.Source: §1.1 · p. 3
1.2 Organization
- The introduction previews the paper’s progression from unrestricted impossibility to single-agent and then multi-agent results.Section 2 covers related work, Section 3 the model, Section 4 unrestricted impossibility, Section 5 single-agent VP(δ) and R-VP, and Section 6 multi-agent VP(δ).Source: §1.2 · p. 3
2 Related work
The related work spans temporal-graph models, dynamic-ring exploration, and search, while distinguishing this paper’s vertex-permuted setting and flag mechanism from neighboring approaches.
2 Related work
- Vertex-permuted dynamism keeps each snapshot isomorphic to a fixed base graph while permuting vertex identities; prior work studied dispersion and restless exploration in this setting.Restless agents must move to a neighboring node at every step.Source: §2 · p. 3
- Temporal-graph exploration seeks walks visiting all vertices, whereas treasure hunt seeks a specified node; most temporal exploration work is offline, and prior online 1-interval ring exploration uses a different dynamicity model.The passages also cite temporal exploration as introduced by Michail and Spirakis.Source: §2 · pp. 3–4
- Flags resemble pebbles but remain visible to agents at neighboring nodes, unlike pebbles visible only at the marked node; they also differ from mobile-agent lights.Lights are carried by robots, while flags are left at nodes.Source: §2 · p. 4
- For random rather than adversarial dynamics, prior analyses often use random-walk tools such as hitting and cover times on temporal graphs.Source: §2 · p. 4
3 Model
The model is a temporal graph whose snapshots are rings formed by arbitrary vertex permutations. Agents search online with local observations, and the paper defines both bounded-dynamicity and flag-visibility variants.
3 Model
- A temporal graph is a sequence of undirected snapshots on a shared vertex set, with possibly changing edge sets and a finite lifetime.Source: §3 · p. 4
- In VP, each snapshot is a cycle formed by an arbitrary permutation of the n nodes, whose identities are unknown to agents and whose port labels need not give a global orientation.Source: §3 · p. 4
- VP(δ) guarantees every pair of vertices is adjacent at least once in each fixed window of δ steps, but not necessarily in every sliding δ-step window.Adjacencies with all other vertices are guaranteed within sliding windows of length 2δ.Source: §3 · pp. 4–5
- In the flag model, visiting agents can mark nodes, and neighboring agents can see those flags; without flags, agents cannot distinguish previously visited nodes from unvisited ones.Agents do not send messages; flags serve as indirect communication.Source: §3 · p. 5
- Agents are anonymous, identical, memoryless, and online: each acts from its current neighborhood without knowing future graphs; success is the first time any agent reaches the fixed, otherwise unseen treasure.Agents work synchronously, and the stated objective is to minimize the first arrival time.Source: §3 · p. 5
- Competitive ratio compares an online algorithm’s cost with that of an optimal offline algorithm that knows the entire input sequence.The lower-bound framework also applies against oblivious adversaries for deterministic algorithms and adaptive offline adversaries for randomized algorithms [8].Source: §3 · pp. 5–6
4 Treasure hunt in unrestricted VP
In unrestricted VP, an adversary can prevent treasure discovery for k agents by confining them to a small block of vertices. The confinement strategy works even against agents with substantial capabilities.
4 Treasure hunt in unrestricted VP
- Treasure hunt is impossible for any randomized or deterministic algorithm with k agents starting together when 1 ≤ k ≤ n − 3.The impossibility holds even with memory, chirality, full graph visibility, and knowledge of node labels.Source: §4 · p. 6
- TrapIn-(k + 2) confines agents to a block by swapping an occupied boundary vertex with a free interior vertex whenever an agent reaches the boundary.The illustrated procedure keeps all agents inside a contiguous block while the treasure lies outside it.Source: §4 · p. 7, Figure 2 · p. 7
- A similar trapping strategy can confine agents who start at different nodes when n is sufficiently large compared with k.The proof sketch traps subsets of agents in separate node sets; details are omitted.Source: §4 · p. 7
5 Treasure hunt by a single agent in restricted models of VP
After unrestricted VP impossibility, the section examines VP(δ) and random R-VP. It establishes a feasibility threshold, one-agent guarantees, and adversary-dependent search times in the random model.
5 Treasure hunt by a single agent in restricted models of VP
- The section turns from unrestricted VP impossibility to restricted dynamic models, specifically VP(δ) and R-VP.The restriction is introduced as a response to the impossibility result.Source: §5 · p. 7
5.1 The VP(δ) model
- The class VP(δ) is non-empty exactly when δ ≥ ⌈(n−1)/2⌉.The lower bound follows from the number of neighbor slots per vertex; Walecki-style Hamiltonian-cycle decompositions attain it.Source: §5.1 · p. 8
- δ(n−1) steps suffice with flags, while deterministic agents require at least δ(n−3) steps for δ ≥ 2n, making Move-to-Unflagged asymptotically optimal.The lower bound holds even with chirality, memory, full visibility, and node labels.Source: Upper bound for one agent · p. 9, Lower bound for one agent · p. 11
- Any deterministic online agent with chirality, memory, and full visibility but no node labels has competitive ratio Ω(δn).The lower-bound construction allows an offline optimum to find the treasure in one step.Source: Lower bound for one agent · p. 11
- Without flags, no deterministic single-agent algorithm solves treasure hunt for δ ≥ n; randomization with positive-probability waiting yields expected O(δ²) search.The randomized strategy relies on a positive probability of waiting at each step.Source: Agents that cannot flag visited vertices · pp. 11–13
5.2 Treasure hunt in R-VP
- In R-VP, clockwise-neighbor search is stochastically equivalent to a random walk on the complete graph, without requiring flags.The model chooses a uniformly random vertex permutation at each step.Source: §5.2 · pp. 14–15
- Expected search time in R-VP is Θ(n) against an oblivious adversary and Θ(n log n) against an adaptive adversary.The adaptive adversary can place the treasure at the last vertex visited, making the search time the clique cover time.Source: §5.2 · p. 15
6 Search by k ≥2 agents
For k ≥ 2 agents in VP(δ), Generalized-Move-to-Unflagged achieves O(nδ/k) search time, and a matching lower bound makes it asymptotically optimal in the stated δ range. The lower-bound adversary realizes every pairwise adjacency while allowing at most k new vertices to be flagged per period.
6.1 Upper bound for k agents in V P (δ)
- Generalized-Move-to-Unflagged moves agents only to unflagged neighbors; mutual-exclusion port access limits duplicate moves, yielding at least min{n_i,⌈k/2⌉} newly flagged nodes in each δ-window.The algorithm orders available ports by each agent’s private local orientation and requests them in sequence.Source: §6.1 · pp. 15–16
6.2 Lower bound for k agents in V P (δ)
- Despite realizing every pairwise adjacency, the adversary limits the agents to at most k newly flagged vertices per period, even with memory, chirality, full visibility, and node labels.The construction traps agents within a block of k + 2 interior nodes, with the treasure placed among exterior nodes.Source: §6.2 · pp. 16, 18
- The lower-bound adversary uses three phases, including Walecki Hamiltonian decompositions, to realize all vertex-pair adjacencies within a period of (k + 1)(n −k −2) steps.The construction covers E-E and I-I pairs in its first phase, then completes the remaining adjacencies.Source: §6.2 · pp. 16–18
- For δ ≥ (k + 1)(n −k −2), Generalized-Move-to-Unflagged is asymptotically optimal: its search time is δ⌈2n/k⌉, against a lower bound of (⌈(n −2)/k⌉−2)δ.The lower bound applies to every online algorithm.Source: §6.1 · p. 16, §6.2 · pp. 18–19
7 Conclusion and open problems
The paper concludes with results on unrestricted and restricted vertex-permuted rings, including a linear speedup from multiple agents, and also studies R-VP. It leaves open the full feasible δ range for the one-agent lower bound, adaptive-adversary randomized competitive ratios, and extensions beyond rings.
7 Conclusion and open problems
- The conclusion covers unrestricted VP impossibility, flagged one-agent bounds in VP(δ), and a linear speedup for k agents.It also identifies R-VP as a studied model.Source: §7 · p. 19
- Whether the one-agent lower bound extends from δ ≥ 2n to the full feasible range δ ≥ ⌈(n −1)/2⌉ remains open.Source: §7 · p. 19
- The competitive ratio of randomized algorithms against an adaptive online adversary remains unstudied.Source: §7 · p. 19
- Whether the results extend beyond rings to vertex-permuted graphs with other topologies remains open.Source: §7 · p. 19