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Article . 2022
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https://dx.doi.org/10.48550/ar...
Article . 2019
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Article . 2022
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Token Swapping on Trees

Token swapping on trees
Authors: Ahmad Biniaz; Kshitij Jain 0001; Anna Lubiw; Zuzana Masárová; Tillmann Miltzow; Debajyoti Mondal; Anurag Murty Naredla; +2 Authors

Token Swapping on Trees

Abstract

The input to the token swapping problem is a graph with vertices $v_1, v_2, \ldots, v_n$, and $n$ tokens with labels $1, 2, \ldots, n$, one on each vertex. The goal is to get token $i$ to vertex $v_i$ for all $i= 1, \ldots, n$ using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge. We present some results about token swapping on a tree, also known as "sorting with a transposition tree": 1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a "happy leaf"), disproving a conjecture of Vaughan. 2. Any algorithm that fixes happy leaves -- as all known approximation algorithms for the problem do -- has approximation factor at least $4/3$. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2. 3. A generalized problem -- weighted coloured token swapping -- is NP-complete on trees, even when they are restricted to be subdivided stars, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.

Country
Netherlands
Keywords

FOS: Computer and information sciences, 03D15, 05C05, 68R05, Analysis of algorithms and problem complexity, Computational Complexity (cs.CC), Cayley graph, 05C05, Graph algorithms (graph-theoretic aspects), 03d15, 05c05, 68r05, sorting with transposition tree, Computer Science - Data Structures and Algorithms, QA1-939, Data Structures and Algorithms (cs.DS), token swapping, f.2.0, cs.CC, reconfiguration, 68R05, computer science - data structures and algorithms, Computer Science - Computational Complexity, cs.DS, Graph theory (including graph drawing) in computer science, 03D15, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), computer science - computational complexity, F.2.0, Mathematics

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    popularity
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    Top 10%
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
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    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
Top 10%
Average
Top 10%
Green
Published in a Diamond OA journal