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The Strahler number of a Parity Game K. S. Thejaswini University of Warwick September 16th, 2020 Joint work with: Laure Daviaud City, University of London, Marcin Jurdzinski University of Warwick Highlights - 2020 K. S. Thejaswini

SLIDE 1

The Strahler number of a Parity Game

K. S. Thejaswini

University of Warwick

September 16th, 2020 Joint work with: Laure Daviaud City, University of London, Marcin Jurdzinski University of Warwick Highlights - 2020

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SLIDE 2

Parity Games

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SLIDE 3

Parity Games

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SLIDE 4

Parity Games

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SLIDE 5

Parity Games

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SLIDE 6

Quasipolynomial Algorithms for Parity Games

Parity Game Universal Tree Algorithm Solution

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SLIDE 7

Quasipolynomial Algorithms for Parity Games

An ordered tree is (n, h)-Universal if any ordered tree embeds into it as long as it has height at most h at most n leaves. Example of a (4, 2)-Universal tree

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SLIDE 8

Quasipolynomial Algorithms for Parity Games

An ordered tree is (n, h)-Universal if any ordered tree embeds into it as long as it has height at most h at most n leaves. Example of a (4, 2)-Universal tree

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SLIDE 9

Quasipolynomial Algorithms for Parity Games

An ordered tree is (n, h)-Universal if any ordered tree embeds into it as long as it has height at most h at most n leaves. Example of a (4, 2)-Universal tree

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SLIDE 10

Quasipolynomial Algorithms for Parity Games

Parity Game Universal Tree Solution Algorithm

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SLIDE 11

Quasipolynomial Algorithms for Parity Games

Parity Game Universal Tree Algorithm

Attractor decomposition [Czerwinski, Daviaud, Fijalkow, Jurdzinski, Lazic, Parys ’19] [Jurdzinski, Lazic ’17] [Parys ’19] [Lehtinen, Schewe, Wojtczak ’19] [Jurdzinski, Morvan ’20] [Calude, Jain, Khoussainov, Li, Stephan ’17]

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SLIDE 12

Quasipolynomial Algorithms for Parity Games

Register Games: [Lehtinen ’18], [Lehtinen, Boker ’19] n vertices, d priorities G n · dk vertices, 2k priorities Gk Runtime - O

nk · dk2

k Exponential algorithm

Corollary [Lehtinen ’18]

There is an O(nlog n · dlog2 n) algorithm to solve parity games.

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SLIDE 13

A result

Theorem:

Register Number = Strahler Number

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SLIDE 14

Strahler Number

Definition

The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor.

2 1 2 1 1 1

str(u) =

max{str(v) | v is a child of u}

if maximum is unique, max{str(v) | v is a child of u} + 1

therwise
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SLIDE 15

Strahler Universal trees

Definition

An ordered tree T is k-Strahler (n, h)-Universal if any ordered tree with at most n vertices, height at most h Strahler number no more than k can be embedded in it.

Size

There are Strahler universal trees of size O(poly(n) · hk)

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SLIDE 16

Algorithms for Parity Games

Parity Game Strahler Universal Tree Algorithm Solution

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SLIDE 17

A polynomial time algorithm

Theorem

Given k, the Strahler number of a Parity game, we can find the winning sets for Audrey and Steven in time poly(n) · d

k and quasi-linear space.

Corollary

Solving parity games is polynomial if k · lg d

= O(log n).

(Previously known for k = O(1) and d = O(lg n))

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