2,296 research outputs found
On the restricted Hanoi Graphs
Consider the restricted Hanoi graphs which correspond to the variants of the
famous Tower of Hanoi problem with multiple pegs where moves of the discs are
restricted throughout the arcs of a movement digraph whose vertices represent
the pegs of the puzzle and an arc from vertex to vertex exists if and
only if moves from peg to peg are allowed. In this paper, we gave some
notes on how to construct the restricted Hanoi graphs as well as some
combinatorial results on the number of arcs in these graphs.Comment: 8 pages, 2 figure
Fatal Attractors in Parity Games: Building Blocks for Partial Solvers
Attractors in parity games are a technical device for solving "alternating"
reachability of given node sets. A well known solver of parity games -
Zielonka's algorithm - uses such attractor computations recursively. We here
propose new forms of attractors that are monotone in that they are aware of
specific static patterns of colors encountered in reaching a given node set in
alternating fashion. Then we demonstrate how these new forms of attractors can
be embedded within greatest fixed-point computations to design solvers of
parity games that run in polynomial time but are partial in that they may not
decide the winning status of all nodes in the input game.
Experimental results show that our partial solvers completely solve
benchmarks that were constructed to challenge existing full solvers. Our
partial solvers also have encouraging run times in practice. For one partial
solver we prove that its run-time is at most cubic in the number of nodes in
the parity game, that its output game is independent of the order in which
monotone attractors are computed, and that it solves all Buechi games and weak
games.
We then define and study a transformation that converts partial solvers into
more precise partial solvers, and we prove that this transformation is sound
under very reasonable conditions on the input partial solvers. Noting that one
of our partial solvers meets these conditions, we apply its transformation on
1.6 million randomly generated games and so experimentally validate that the
transformation can be very effective in increasing the precision of partial
solvers
Distance-uniform graphs with large diameter
An ϵ-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ϵ-fraction of the remaining vertices are at distance exactly d. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the follow- ing conjecture of independent interest: that every ϵ-distance-uniform graph (and, in fact, a broader class of ϵ-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices n. We disprove this conjecture and characterize the asymptotics of this extremal prob- lem. Speci-cally, for 1/n ≤ ϵ ≤ 1 /log n , we construct ϵ-distance-uniform graphs with critical distance 2ω(log n/log ϵ-1). We also prove an upper bound on the critical distance of the form 2O(log n/log ϵ-1) for all ϵ and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest.Peer ReviewedPostprint (author's final draft
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