A Practical Succinct Data Structure for Tree-Like Graphs Johannes - - PowerPoint PPT Presentation

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Mar 04, 2024 17 likes •592 views

A Practical Succinct Data Structure for Tree-Like Graphs Johannes Fischer TU Dortmund, Germany joint work with Daniel Peters (PTB Berlin) leading adj(x,y) neighbours(x) deg(x) terms [bits] n 2 lg arbitrary O(1) O(1) O(#in) or

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A Practical Succinct Data Structure for Tree-Like Graphs

Johannes Fischer TU Dortmund, Germany joint work with Daniel Peters (PTB Berlin)

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leading terms [bits] adj(x,y) neighbours(x) deg(x) arbitrary O(1)

O(#in) or O(#out)

O(1)

lg n2 m

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leading terms [bits] adj(x,y) neighbours(x) deg(x) arbitrary O(1)

O(#in) or O(#out)

O(1) x-page 2m + xn O(x) O(#neighbours) O(1) planar 2(m+n) O(1) triangular 2m+n separable O(n)

lg n2 m

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leading terms [bits] adj(x,y) neighbours(x) deg(x) arbitrary O(1)

O(#in) or O(#out)

O(1) x-page 2m + xn O(x) O(#neighbours) O(1) planar 2(m+n) O(1) triangular 2m+n separable O(n)

lg n2 m

and not planar!

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What if …

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What if …

… the graph is neither

planar/separable/x-page/…

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What if …

… the graph is neither

planar/separable/x-page/…

…but still “easy“, in

particular not dense?

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What if …

… the graph is neither

planar/separable/x-page/…

…but still “easy“, in

particular not dense?

⇒ adjacency array ??

(m lg n + n lg m bits)

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leading terms [bits] adj(x,y) neighbours(x) deg(x) arbitrary O(1)

O(#in) or O(#out)

O(1) k-page 2m + kn O(k) O(#neighbours) O(1) planar 2(m+n) O(1) triangular 2m+n separable O(n) tree-like (lg 3)(m+n) +k lg n O(1)

O(#in) and O(#out)

#out: O(1)

lg n2 m

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leading terms [bits] adj(x,y) neighbours(x) deg(x) arbitrary O(1)

O(#in) or O(#out)

O(1) k-page 2m + kn O(k) O(#neighbours) O(1) planar 2(m+n) O(1) triangular 2m+n separable O(n) tree-like (lg 3)(m+n) +k lg n O(1)

O(#in) and O(#out)

#out: O(1)

lg n2 m

k: #non-tree edges

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1 3 2 4 5 6 7 8 9

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1 3 2 4 5 6 7 8 9

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1 3 2 4 5 6 7 8 9 3 4 2 4

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1 3 2 4 5 6 7 8 9 3 4 2 4

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1 3 2 4 5 6 7 8 9 3 4 2 4

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1 3 2 4 5 6 7 8 9 3 4 2 4

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1 3 2 4 5 6 7 8 9 3 4 2 4

10 1110

1 1 2 3 4

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1 3 2 4 5 6 7 8 9 3 4 2 4

10 1110 20110110

1 1 2 3 4 2 3 5 6 3 7 8 4

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1 3 2 4 5 6 7 8 9 3 4 2 4

10 1110 20110110 1022000

1 1 2 3 4 2 3 5 6 3 7 8 4 9 5 4 2 6 7 8

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1 3 2 4 5 6 7 8 9 3 4 2 4

10 1110 20110110 1022000 20

1 1 2 3 4 2 3 5 6 3 7 8 4 9 5 4 2 4 6 7 8 9

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1 1 2 3 4 2 3 5 6 3 7 8 4 9 5 4 2 4 6 7 8 9

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3 4 2 4

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3 4 2 4

n+m trits

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3 4 2 4

n+m trits k log n bits

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Listing Successors

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1 3 2 4 5 6 7 8 9 3

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1 3 2 4 5 6 7 8 9 3

3 7 8

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1 3 2 4 5 6 7 8 9 3

3 5 6 7 8

Listing Successors

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1 3 2 4 5 6 7 8 9 3

6 5

5 6 7 8

Listing Successors

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4 2 7 8

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4 2

2 4

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needs rank2

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Listing Predecessors

2 7 8

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4 1

2 7 8

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4 2 6

7 8

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needs selectα

⇒ could be represen- ted as wavelet tree

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Practical Results

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Size

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Time

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Realistic Test: BFS

WebGraph: Boldi/Vigna 2004 WebGraph: Claude/Navarro 2010

Graph Space [MByte] Time for BFT [sec] GLOUDS WebGraph GLOUDS WebGraph amazon-2008 8.551 13.654 0.95 0.86 cnr-2000 3.069 2.765 0.36 0.63 in-2004 15.813 11.231 1.60 1.59 uk-2002 324.108 187.494 32.04 22.96 enwiki-2013 208.896 322.257 5.58 12.57 dblp-2010 0.992 1.752 0.36 0.54 dblp-2011 5.348 8.501 1.18 1.26 hollywood-2011 202.991 18.532 2.59 14.37

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Realistic Test: BFS

WebGraph: Boldi/Vigna 2004 WebGraph: Claude/Navarro 2010

Graph

m n sizeGLOUDS sizeWebGraph

amazon-2008 7.015 0.626 cnr-2000 9.879 1.109 in-2004 12.232 1.408 uk-2002 16.096 1.728 enwiki-2013 24.093 0.648 dblp-2010 4.952 0.566 dblp-2011 6.800 0.628 hollywood-2011 105.003 10.95

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A Practical Succinct Data Structure for Tree-Like Graphs

Listing Successors

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Listing Predecessors

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Listing Predecessors

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Listing Predecessors

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Listing Predecessors

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Practical Results

Size

Time

Realistic Test: BFS

Realistic Test: BFS

T h a t ’ s i t !