Randomized Leaf-Token Eviction (RLT)

• The paper introduces RLT as a distributed election algorithm that iteratively eliminates leaf nodes using lifetime distributions based on local degree, weight, and received neighbor information.
• It details two key parameterizations—the (max,+) algebra family and the 1/2-stable family—that yield closed-form expressions for node election probabilities and expected completion times.
• Notable instances include uniform and weighted-proportional elections, where local parameters are mapped to explicit eviction timings, providing both theoretical insights and practical applications.

Randomized Leaf-Token Eviction (RLT) is a class of distributed randomized election algorithms defined on a tree $T = (V, E)$, in which nodes are iteratively eliminated—specifically, leaves are removed one at a time—until a sole surviving node is selected as leader. Each elimination is determined by a leaf’s random lifetime, sampled according to a probability law that may incorporate both local parameters and information transmitted from previously eliminated neighbors. The RLT paradigm generalizes numerous known election strategies and admits exact analysis for a range of probability law parametrizations, most notably through families derived from the $(\max,+)$ algebra with exponential random variables and from Lévy 1/2-stable laws. Precise closed-form expressions characterize both the election probability for each node and the total expected time to completion in these special cases (Marckert et al., 2015).

1. General RLT Framework

At the initiation of the process ($t=0$), each node $u \in V$ possesses only its degree $\deg(u)$, a prescribed weight $w_u$ (an arbitrary real or integer parameter), and an independent continuous uniform random generator $U_u$. The algorithm proceeds, at each step, as follows:

• A leaf node $u$ (i.e., with $\deg(u) = 1$ in the current subgraph) that is being eliminated at time $t$ has received, from each neighbor that was previously eliminated, a packet of information $(\max,+)$0.
• Based on the collection of information packets $(\max,+)$1, its own $(\max,+)$2, and a "computed value" $(\max,+)$3, the node $(\max,+)$4 formulates its remaining lifetime distribution. This is selected thorough the mapping $(\max,+)$5 on $(\max,+)$6.
• Using $(\max,+)$7, $(\max,+)$8 samples $(\max,+)$9. The node is scheduled for elimination at time $t=0$0.
• Upon elimination, $t=0$1 transmits all collected information (including its own tuple $t=0$2) to its remaining neighbor and disappears.
• The next elimination is always the pending leaf with minimal scheduled elimination time ($t=0$3). All distributions $t=0$4 are atomless, ensuring tie-free evolution.

The procedure continues until only one node $t=0$5 remains. The election probability for $t=0$6, denoted $t=0$7, represents the probability that $t=0$8 is the survivor (Marckert et al., 2015).

2. Probability Law Specification for Leaf Lifetimes

The eviction lifetime distribution $t=0$9 can depend in fully general ways on the history of the eviction process (encoded in the forest of arrival packets $u \in V$0) and on local parameters $u \in V$1. However, two notable families allow exact closed-form results:

(A) $u \in V$2-algebra Family

Each leaf $u \in V$3 is assigned nonnegative integers $u \in V$4, often set by recursion over subtrees. The distribution

$u \in V$5

is used, where $u \in V$6 denotes an exponential random variable with rate $u \in V$7. The maximum of $u \in V$8 independent unit-rate exponentials $u \in V$9 satisfies $\deg(u)$0. For any rooted subtree $\deg(u)$1 the eviction time in the "directed-elimination" version is distributed as $\deg(u)$2 with $\deg(u)$3 (an integer label defined by structural recursion).

In the undirected (original) tree $\deg(u)$4, the election probability for $\deg(u)$5 is

$\deg(u)$6

where $\deg(u)$7 denotes the component containing $\deg(u)$8 in $\deg(u)$9 with edge $w_u$0 removed, and the sum is over all neighbors $w_u$1 of $w_u$2 (Marckert et al., 2015).

(B) 1/2-Stable Family

Here, each leaf samples from i.i.d. Lévy $w_u$3-stable random variables $w_u$4 on $w_u$5, with density

$w_u$6

Sums $w_u$7 satisfy $w_u$8. For elimination processes, the total time for a component of size $w_u$9 is distributed as $U_u$0.

The probability that a sum over $U_u$1 leaves is less than an independent sum over $U_u$2 leaves is

$U_u$3

Thus, in the tree, the election probability becomes

$U_u$4

3. Formal Expressions and Algorithmic Workflow

Upon becoming a leaf, node $U_u$5 computes the cumulative distribution function $U_u$6 via its deterministic map $U_u$7. Actual scheduling is achieved by drawing $U_u$8, based on local uniform randomness.

• $U_u$9 Family:

$u$0

$u$1

$u$2

• 1/2-Stable Family:

$u$3

No finite mean exists for $u$4 due to the heavy tail of the Lévy $u$5-stable law.

4. Closed-Form Results for Election Probabilities and Completion Time

Regardless of the specific lifetime distribution, the election probability is always expressible by the pairwise comparison formula:

$u$6

Plugging in the law for $u$7, for the two notable families:

• $u$8 algebra family:

$u$9

$\deg(u) = 1$0

The completion time $\deg(u) = 1$1 has cumulative distribution function $\deg(u) = 1$2 and expected value $\deg(u) = 1$3.

• 1/2-stable family:

$\deg(u) = 1$4

$\deg(u) = 1$5

For this family, $\deg(u) = 1$6.

5. Principal Algorithmic Instances

Two particularly significant instantiations are:

Instance	Parameter Setting	Election Probability $\deg(u) = 1$7	Expected Completion Time
Uniform election	$\deg(u) = 1$8 for all $\deg(u) = 1$9	$t$0	$t$1
Weighted-proportional	$t$2 with integer $t$3	$t$4	$t$5

For uniform election, all nodes are selected equiprobably. Weighted-proportional election selects each node with probability proportional to a prescribed positive weight $t$6.

6. Structural Interpretation and Pairwise Comparisons

The core analytical approach for RLT algorithms is the reduction of node election probabilities to the probability that a rooted subtree "outlasts" the complementary component in the tree when the edge between them is removed. This reduction is formalized via the pairwise survival-order probability framework, which is enabled by coupling with the "directed-elimination" version of the process.

The $t$7 and 1/2-stable families yield, respectively, rational functions in the $t$8-labels and simple arctangent formulas for $t$9. Arctangent summation identities for special tree topologies emerge from these expressions, such as $(\max,+)$00 and $(\max,+)$01-term analogues in trees constructed from "skeleton+leaves" structures (Marckert et al., 2015).