\begin{algorithm}[ht]
  \caption{Existence of a connected strongly-proportional allocation for $n$ hungry agents.}
  \label{alg:hungry}
  \begin{algorithmic}[1]
    \For{$t = 1, \ldots, n-1$}
      \State $z \leftarrow \textsc{Mark}_1(0, t/n)$ \algorithmiccomment{agent $1$'s $t/n$-mark}
      \For{$i = 2, \ldots, n$}
        \State \algorithmicif \ $\textsc{Mark}_i(0, t/n) \neq z$ \algorithmicthen \ \Return true \algorithmiccomment{agent $1$'s and $i$'s $t/n$-marks do not coincide}
      \EndFor
    \EndFor
    \State \Return false
  \end{algorithmic}
\end{algorithm}

\begin{algorithm}
[ht]
\caption{Existence of a connected strongly-proportional allocation for $n$ hungry agents.}
\begin{algorithmic}
[1]
\For{$t = 1, \ldots, n-1$}
\State $z \leftarrow \textsc{Mark}_1(0, t/n)$ \algorithmiccomment{agent $1$'s $t/n$-mark}
\For{$i = 2, \ldots, n$}
\State \algorithmicif \ $\textsc{Mark}_i(0, t/n) \neq z$ \algorithmicthen \ \Return true \algorithmiccomment{agent $1$'s and $i$'s $t/n$-marks do not coincide}
\EndFor
\EndFor
\State \Return false
\end{algorithmic}
\end{algorithm}
{
    "title": "on connected strongly-proportional cake-cutting",
    "id": "2312.15326",
    "abstract": "we investigate the problem of fairly dividing a divisible heterogeneous resource, also known as a cake, among a set of agents. we characterize the existence of an allocation in which every agent receives a contiguous piece worth strictly more than their proportional share, also known as a *strongly-proportional allocation*. the characterization is supplemented with an algorithm that determines the existence of a connected strongly-proportional allocation using at most $n \\cdot 2^{n-1}$ queries. we provide a simpler characterization for agents with strictly positive valuations, and show that the number of queries required to determine the existence of a connected strongly-proportional allocation is in $\\theta(n^2)$. our proofs are constructive and yield a connected strongly-proportional allocation, when it exists, using a similar number of queries.",
    "categories": "math.co cs.gt econ.th",
    "doi": "",
    "created": "2023-12-23",
    "updated": "2024-02-13",
    "authors": [
        "zsuzsanna jank\u00f3",
        "attila jo\u00f3",
        "erel segal-halevi",
        "sheung man yuen"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2312.15326"
}
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