Efficient D-2-D via Leader Election: Arbitrary Initial Configuration and No Global Knowledge

TL;DR

The D-2-D problem using n mobile agents is equivalent to finding a maximal independent set of the graph as size of any maximal independent set must be less than $n and the algorithm solves the leader election problem that elects an agent as a leader in rounds with $O(\log n)$ bits of memory at each agent without requiring any prior global knowledge.

Abstract

Distance-2-Dispersion (D-2-D) problem aims to disperse $k$ mobile agents starting from an arbitrary initial configuration on an anonymous port-labeled graph $G$ with $n$ nodes such that no two agents occupy adjacent nodes in the final configuration, though multiple agents may occupy a single node if there is no other empty node whose all adjacent nodes are also empty. In the existing literature, this problem is solved starting from a rooted configuration for $k$ $(\geq 1)$ agents in $O(mΔ)$ synchronous rounds with a total of $O(\log n)$ memory per agent, where $m$ is the number of edges and $Δ$ is the maximum degree of the graph. In this work we study the D-2-D problem using $n$ mobile agents starting from an arbitrary initial configuration. Solving D-2-D with $n$ agents is equivalent to finding a maximal independent set of the graph as size of any maximal independent set must be less than $n$. We solve this problem and terminate in $O(max\{n\log^2 n, m\})$ rounds using $O(\log n)$ memory per agent. The agents do not have any prior knowledge of any graph parameters. During the run of our algorithm, we also solve the leader election problem that elects an agent as a leader in $O(max\{n\log^2 n, m\})$ rounds with $O(\log n)$ bits of memory at each agent without requiring any prior global knowledge.

Figures (2)

• Figure 2: (a) The initial configuration of the agents. Let red('r'), blue('b'), and green('g') dots represent three different groups with $prm\_leader$ values as r, g, and b respectively ($r<b<g$). Let the dot represent the $prm\_leader$ and the outer circle represent its group in the dispersion phase, i.e. phase $1$ (b) The group of agents begin dispersion. The outer circle represents the group it is a part of (c) The $leading$ agents from each group start their respective DFS (d) The 'b' $leading$ agent meets with a settled agent that has $prm\_leader=r$. Thus, its $prm\_leader$ value is changed to 'b' (e) The 'r' $leading$ agent waits at a node as it meets with a settled agent with $prm\_leader=b$(this is represented with the agent waiting outside the node). The 'g' $leading$ agent meets with the settled agents with $prm\_leader=b$. Thus, these settled agents update their $prm\_leader=g$. While the 'b' $leading$ agent waits at the node where it meets with the settled agent having $prm\_leader=g$ (f) Finally all the waiting $leading$ agents are picked by the 'g' $leading$ agent and positioned at the virtually settled nodes. All the settled agents now have $prm\_leader=g$ and there is a single DFS tree(darkened lines represent the tree edge and the dotted lines represent the non-tree edges) in the graph corresponding to the DFS by 'g' $leading$ agent.
• Figure 3: (a) agents are initially positioned arbitrarily in the graph (b) After the completion of phase $1$ and phase $2$ of the algorithm, there exists a single DFS tree with tree edges (shown as darkened lines) and non-tree edges (shown as dotted lines). The DFS tree corresponds to the DFS traversal by the agent $r_{st}$ with the largest value of $prm\_leader$ (denoted with the green colour) (c) $r_{st}$ moves through $crnt\_port$ values of the agents settled at $v_1$ and $v_3$ eventually reaching node $v_5$ which has no child ports. The agent that is settled at $v_5$ sets $decision=1$ and checks if there is any settled agent in its one-hop neighbourhood with $final\_set=1$. Failing to find one, it finally settles at its node by setting $final\_set=1$ which is shown by a blue dot (d) $r_{st}$ backtracks to $v_3$ and further moves through $crnt\_port=2$ to reach $v_4$. Following the decision of the agent settled at $v_4$, $r_{min}$ backtracks to node $v_3$ (where $crnt\_port=\bot$), the decision for this node is now taken. The agent settled at $v_3$ finds an agent at $v_3$ that has $final\_set=1$. Thus, the agent settled at $v_3$ vacates its position and settles at the node $v_5$. (e) $r_{st}$ arrives at $v_1$ which now has $crnt_port=3$ and moves through this port to reach $v_2$. Post the decision for this node, finally $r_{st}$ backtracks to $v_1$ (which now has $crnt\_port=\bot$). The decision for the agent $r_{st}$ is taken. It finds an agent with $final\_set=1$ in its one-hop. Thus, it moves through $final\_port=0$ and sets $r_{st}.final\_set=1$ (f) The D-2-D configuration.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof