---
title: Randomized Leaf-Token Eviction (RLT)
url: https://www.emergentmind.com/topics/randomized-leaf-token-eviction-rlt
type: topic
---

# Randomized Leaf-Token Eviction (RLT)

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 [1507.04909].

## 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 $I_i$.
- Based on the collection of information packets $\{I_i\}$, its own $\deg(u),w_u$, and a "computed value" $\Gamma_u$, the node $u$ formulates its remaining lifetime distribution. This is selected thorough the mapping $\Delta: (\text{Received-forest }f, \text{Local-info }l_u) \to \mathcal{D}_u$ on $[0,\infty)$.
- Using $U_u$, $u$ samples $D_u \sim \mathcal{D}_u$. The node is scheduled for elimination at time $t + D_u$.
- Upon elimination, $u$ transmits all collected information (including its own tuple $L_u$) to its remaining neighbor and disappears.
- The next elimination is always the pending leaf with minimal scheduled elimination time ($\tau = t+D_u$). All distributions $\mathcal{D}_u$ are atomless, ensuring tie-free evolution.

The procedure continues until only one node $\,\hat{u}\,$ remains. The election probability for $u$, denoted $q_u = \Pr(\hat{u} = u)$, represents the probability that $u$ is the survivor [1507.04909].

## 2. Probability Law Specification for Leaf Lifetimes

The eviction lifetime distribution $\mathcal{D}_u = \Delta(f, l_u)$ can depend in fully general ways on the history of the eviction process (encoded in the forest of arrival packets $f$) and on local parameters $l_u = (\deg(u), w_u, \Gamma_u)$. However, two notable families allow exact closed-form results:

### (A) $(\max,+)$-algebra Family

Each leaf $u$ is assigned nonnegative integers $C_u, g_u$, often set by recursion over subtrees. The distribution 
$$
Y_{n,k} := \sum_{i=1}^k \mathcal{E}^{[n+i]}
$$
is used, where $\mathcal{E}^{[a]}$ denotes an exponential random variable with rate $a$. The maximum of $m$ independent unit-rate exponentials $M_m$ satisfies $M_{n+k} \overset{d}{=} M_n + Y_{n,k}$. For any rooted subtree $\tau$ the eviction time in the "directed-elimination" version is distributed as $M_{\Theta(\tau)}$ with $\Theta(\tau) = C(\tau) + g(\tau)$ (an integer label defined by structural recursion).

In the undirected (original) tree $T$, the election probability for $u$ is
$$
q_u = 1 - \sum_{v} \frac{\Theta(T[v\llbracket u])}{\Theta(T[u\llbracket v])+\Theta(T[v\llbracket u])}
$$
where $T[u\llbracket v]$ denotes the component containing $u$ in $T$ with edge $(u,v)$ removed, and the sum is over all neighbors $v$ of $u$ [1507.04909].

### (B) 1/2-Stable Family

Here, each leaf samples from i.i.d. Lévy $1/2$-stable random variables $X^v$ on $(0, \infty)$, with density
$$
f_X(t) = \frac{1}{\sqrt{2\pi}} t^{-3/2} \exp\left[-\frac{1}{2t}\right].
$$
Sums $S_m = \sum_{i=1}^m X_i$ satisfy $S_m \overset{d}{=} m^2 X$. For elimination processes, the total time for a component of size $m$ is distributed as $m^2 X$.

The probability that a sum over $m$ leaves is less than an independent sum over $n$ leaves is
$$
\Pr[S_m < S'_n] = \frac{2}{\pi} \arctan\left(\frac{n}{m}\right).
$$
Thus, in the tree, the election probability becomes
$$
q_u = 1 - \sum_v \frac{2}{\pi} \arctan \left( \frac{|T[v\llbracket u]|}{|T[u\llbracket v]|} \right ).
$$

## 3. Formal Expressions and Algorithmic Workflow

Upon becoming a leaf, node $u$ computes the cumulative distribution function $F_u(x) = \Pr[D_u \le x]$ via its deterministic map $\Delta(f, l_u)$. Actual scheduling is achieved by drawing $D_u = F_u^{-1}(U_u)$, based on local uniform randomness.

- **$(\max,+)$ Family**:
  $$
  \mathcal{D}_u = \mathcal{Y}[C_u, g_u] \quad \text{with} \quad Y_{C_u,g_u} = \sum_{i=1}^{g_u} \mathcal{E}^{[C_u + i]}
  $$
  $$
  \Pr[Y_{n,k} \le x] = \sum_{j=0}^{k-1} {k \choose j} (1-e^{-x})^j e^{-(n+1)x}
  $$
  $$
  \Pr[M_n < M_m] = \frac{n}{n+m}
  $$
- **1/2-Stable Family**:
  $$
  X^u \sim \text{Levy}_{1/2}, \quad S_n = \sum X^u \sim \text{Stable}_{1/2} \text{ (scale } n^2 \text{)}
  $$
  No finite mean exists for $S_n$ due to the heavy tail of the Lévy $1/2$-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:
$$
q_u = 1 - \sum_{v \sim u} \Pr\left[ D^\star(T[u\llbracket v]) < D^\star(T[v\llbracket u]) \right]
$$
Plugging in the law for $D^\star$, for the two notable families:

- **$(\max,+)$ algebra family**:
  $$
  D^\star(\tau) \sim M_{\Theta(\tau)} \implies \Pr[M_a < M_b] = \frac{a}{a+b}
  $$
  $$
  q_u = 1 - \sum_{v} \frac{\Theta(T[v\llbracket u])}{\Theta(T[u\llbracket v])+\Theta(T[v\llbracket u])}
  $$
  The completion time $D^\star(T)$ has cumulative distribution function $(1 - e^{-t})^{\Theta(T)}$ and expected value $H_{\Theta(T)} = \sum_{i=1}^{\Theta(T)} \frac{1}{i}$.
- **1/2-stable family**:
  $$
  D^\star(\tau) = \sum_{v \in \tau} X^v, \ \ \Pr[S_m < S'_n] = \frac{2}{\pi} \arctan(n/m)
  $$
  $$
  q_u = 1 - \sum_v \frac{2}{\pi} \arctan \left( \frac{|T[v\llbracket u]|}{|T[u\llbracket v]|} \right)
  $$
  For this family, $E[D^\star(T)]=\infty$.

## 5. Principal Algorithmic Instances

Two particularly significant instantiations are:

| Instance                 | Parameter Setting                                 | Election Probability $q_u$     | Expected Completion Time        |
|--------------------------|---------------------------------------------------|--------------------------------|-------------------------------|
| Uniform election         | $g_u = 1$ for all $u$                             | $q_u = 1/|T|$                  | $H_{|T|}$                     |
| Weighted-proportional    | $g_u = w_u$ with integer $w_u > 0$                | $q_u = w_u/\sum_v w_v$         | $H_{\sum_v w_v}$              |

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

## 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 $(\max,+)$ and 1/2-stable families yield, respectively, rational functions in the $\Theta$-labels and simple arctangent formulas for $q_u$. Arctangent summation identities for special tree topologies emerge from these expressions, such as $\arctan(x) + \arctan(1/x) = \pi/2$ and $k$-term analogues in trees constructed from "skeleton+leaves" structures [1507.04909].

Source: https://www.emergentmind.com/topics/randomized-leaf-token-eviction-rlt