[PPT] - Listing all fixed-length simple cycles in sparse graphs in optimal PowerPoint Presentation

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Listing all fixed-length simple cycles in sparse graphs in

ptimal time

George Manoussakis

Universit´ e Paris Saclay, Universit´ e Paris Sud, LRI-CNRS, France

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Introduction

Input object: a graph G = (V , E) a graph: |V | = n vertices and |E| = m edges.

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Question: find all (induced) subgraphs with some property induced subgraph subgraph

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Example: find all cycles of length four

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Example: find all cycles of length four

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Introduction - k-degenerate graphs

Definition (k-degenerate graphs) Graph k-degenerate if k is smallest integer such that: every induced subgraph has a vertex of degree at most k,

r equivalently
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Introduction - k-degenerate graphs

Definition (k-degenerate graphs) Graph k-degenerate if k is smallest integer such that: every induced subgraph has a vertex of degree at most k,

r equivalently

it has acyclic orientation in which every vertex has out-degree bounded by k

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Introduction - k-degenerate graphs

Definition (k-degenerate graphs) Graph k-degenerate if k is smallest integer such that: every induced subgraph has a vertex of degree at most k,

r equivalently

it has acyclic orientation in which every vertex has out-degree bounded by k Example:

Figure: A 3-degenerate graph and a 3-bounded orientation

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Introduction - k-degenerate graphs

Why degenerate graphs

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Introduction - k-degenerate graphs

Why degenerate graphs generalize many graphs: planar graphs 5-degenerate, trees 1-degenerate, Barab´ asi-Albert graphs etc.

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Introduction - k-degenerate graphs

Why degenerate graphs generalize many graphs: planar graphs 5-degenerate, trees 1-degenerate, Barab´ asi-Albert graphs etc. many real world graphs have low degeneracy Graph n m k citationCiteseer 268,495 1.1M 15 Flickr 1.72M 15.6M 568 uk-2002 18M 261M 943 Twitter 41.7M 1.20B 2490

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Introduction - k-degenerate graphs

Why degenerate graphs generalize many graphs: planar graphs 5-degenerate, trees 1-degenerate, Barab´ asi-Albert graphs etc. many real world graphs have low degeneracy Graph n m k citationCiteseer 268,495 1.1M 15 Flickr 1.72M 15.6M 568 uk-2002 18M 261M 943 Twitter 41.7M 1.20B 2490 theoretical/practical

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Input: graph G, p ∈ N Output: all cycles subgraphs of G with p vertices (without duplicates)

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Input: graph G, p ∈ N Output: all cycles subgraphs of G with p vertices (without duplicates) This talk: O(n⌊p/2⌋k⌈p/2⌉)

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Input: graph G, p ∈ N Output: all cycles subgraphs of G with p vertices (without duplicates) This talk: O(n⌊p/2⌋k⌈p/2⌉) worst case output size optimal: ∀k, p and n ≥ kp, ∃ n-order k-degenerate graph with Ω(n⌊p/2⌋k⌈p/2⌉) p-cycles.

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Input: graph G, p ∈ N Output: all cycles subgraphs of G with p vertices (without duplicates) This talk: O(n⌊p/2⌋k⌈p/2⌉) worst case output size optimal: ∀k, p and n ≥ kp, ∃ n-order k-degenerate graph with Ω(n⌊p/2⌋k⌈p/2⌉) p-cycles. space optimal:

nly requires memory needed to store the graph
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Related work: 1 cycle:

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Related work: 1 cycle: C3 in planar graphs in linear time [Papadimitriou et al. ’81]

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Related work: 1 cycle: C3 in planar graphs in linear time [Papadimitriou et al. ’81] C3 and C4 in planar and degenerate in linear time [Chiba et al. ’85], [Chrobak et al. ’91]

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Related work: 1 cycle: C3 in planar graphs in linear time [Papadimitriou et al. ’81] C3 and C4 in planar and degenerate in linear time [Chiba et al. ’85], [Chrobak et al. ’91] C5 and C6 in planar in O(n log n) [Richardson et al. ’86]

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Related work: 1 cycle: C3 in planar graphs in linear time [Papadimitriou et al. ’81] C3 and C4 in planar and degenerate in linear time [Chiba et al. ’85], [Chrobak et al. ’91] C5 and C6 in planar in O(n log n) [Richardson et al. ’86] Any length general and degenerate graphs [Alon et al. ’97] for induced cycles : [Cai et al. ’06]

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Related work: all cycles fixed length:

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Related work: all cycles fixed length: any length in planar graphs in O(occurences) [Eppstein ’95]

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Related work: all cycles fixed length: any length in planar graphs in O(occurences) [Eppstein ’95] cycles of length 4 and 5 in degenerate graphs in O(occurences) [Kowalik ’03]

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Related work: all cycles fixed length: any length in planar graphs in O(occurences) [Eppstein ’95] cycles of length 4 and 5 in degenerate graphs in O(occurences) [Kowalik ’03] any length in general graphs, randomized in O(f ∗ occurences) with f time to find one occurence [Meeks ’16].

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Related work: all cycles fixed length: any length in planar graphs in O(occurences) [Eppstein ’95] cycles of length 4 and 5 in degenerate graphs in O(occurences) [Kowalik ’03] any length in general graphs, randomized in O(f ∗ occurences) with f time to find one occurence [Meeks ’16]. this talk: deterministic worst case output size optimal for any length in degenerate graphs in O(n⌊p/2⌋k⌈p/2⌉)

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Outline of the algorithm: given a k-degenerate graph Construct an acyclic orientation of the graph with out-degree bounded by k

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Outline of the algorithm: given a k-degenerate graph Construct an acyclic orientation of the graph with out-degree bounded by k Prove that any cycle can be decomposed into small easily computable parts

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Outline of the algorithm: given a k-degenerate graph Construct an acyclic orientation of the graph with out-degree bounded by k Prove that any cycle can be decomposed into small easily computable parts Form all possible cycles by computing all these possible parts and their possible combinations

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Outline of the algorithm: given a k-degenerate graph Construct an acyclic orientation of the graph with out-degree bounded by k Prove that any cycle can be decomposed into small easily computable parts Form all possible cycles by computing all these possible parts and their possible combinations Using properties of the decomposition output each cycle exactly once

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Acyclic orientation: ∃ some vertex with two out-going edges

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Definition (t-path) t-path of size (i, j) : two oriented paths of length i and j starting from the same vertex

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Definition (t-path) t-path of size (i, j) : two oriented paths of length i and j starting from the same vertex Remark in k-degenerate graphs, at most nki+j such paths

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Algorithm: brute force is optimal input: graph G k-degenerate

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all possible acyclic orientations of a p-length simple cycle
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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all possible acyclic orientations of a p-length simple cycle
2. For each orientation find a t-path decomposition t1, t2, ..., tr
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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all possible acyclic orientations of a p-length simple cycle
2. For each orientation find a t-path decomposition t1, t2, ..., tr
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G.
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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all possible acyclic orientations of a p-length simple cycle
2. For each orientation find a t-path decomposition t1, t2, ..., tr
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G.
4. Construct all possible combinations of these t-paths in time

N(|t1|) ∗ N(|t2|) ∗ ... ∗ N(|tr|)

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all possible acyclic orientations of a p-length simple cycle
2. For each orientation find a t-path decomposition t1, t2, ..., tr
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G.
4. Construct all possible combinations of these t-paths in time

N(|t1|) ∗ N(|t2|) ∗ ... ∗ N(|tr|)

5. For each combination: check if it is a cycle + uniqueness
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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle
2. For each orientation find a t-path decomposition t1, t2, ..., tr
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr)

5. For each combination: check if it is a cycle + uniqueness

Complexity?

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr)

5. For each combination: check if it is a cycle + uniqueness

Complexity?

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr)

5. For each combination: check if it is a cycle + uniqueness

Complexity?

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G O(nkp)
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr)

5. For each combination: check if it is a cycle + uniqueness

Complexity?

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G O(nkp)
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr) O(nrkp)

5. For each combination: check if it is a cycle + uniqueness

Complexity?

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G O(nkp)
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr) O(nrkp)

5. For each combination: check if it is a cycle + uniqueness O(p)

Complexity?

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G O(nkp)
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr) O(nrkp)

5. For each combination: check if it is a cycle + uniqueness O(p)

total time: O(nrkp)

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G O(nkp)
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr) O(nrkp)

5. For each combination: check if it is a cycle + uniqueness O(p)

total time: O(nrkp) + r ≤ ⌊p/2⌋ = ⇒ O(n⌊p/2⌋kp)

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Algorithm: brute force is optimal input: graph G k-degenerate

1. Compute all acyclic orientations of a p-length simple cycle O(2p)
2. For each orientation find a t-path decomposition t1, t2, ..., tr O(p)
3. Compute all possible t-paths of size |t1|, |t2|, ..., |tr| in G O(nkp)
4. Construct all possible combinations of these t-paths in time

N(t1) ∗ N(t2) ∗ ... ∗ N(tr) O(nrkp)

5. For each combination: check if it is a cycle + uniqueness O(p)

total time: O(nrkp) + r ≤ ⌊p/2⌋ = ⇒ O(n⌊p/2⌋kp) + fine tuning = ⇒ O(n⌊p/2⌋k⌈p/2⌉)

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Proof of optimality: (for k, p even)

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Proof of optimality: (for k, p even)

Figure: sets Ki of size k/2, sets Li of size h ≥ k

graph is k-degenerate

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Proof of optimality: (for k, p even)

Figure: sets Ki of size k/2, sets Li of size h ≥ k

graph is k-degenerate number of vertices: n = p

2 k 2 + h p 2 =

⇒ h = Θ(n) if n ≥ kp

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Proof of optimality: (for k, p even)

Figure: sets Ki of size k/2, sets Li of size h ≥ k

graph is k-degenerate number of vertices: n = p

2 k 2 + h p 2 =

⇒ h = Θ(n) if n ≥ kp number of cycles: ( k

2)p/2hp/2 = Ω(kp/2np/2)

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Open problems: List p-cycles in k-degenerate graphs in O(occurences) ? enumerating p-cycles takes time O(n⌊p/2⌋k⌈p/2⌉) how much faster is counting

for general graphs: O(nω) [Alon et al. ’97] for cycles length ≤ 7 O(n⌈p/2⌉+3) [Vassilevska et al. ’09] O(n0.45p+O(1)) [Bjorklund et al. ’14]

what if we do not print cycles ? O(nk2) for C4, [Chiba et al. ’85] O(nk3) for C5, [Kowalik ’03]

ptimal complexity is attained assuming adjacency matrix, if only

adjacency list then O(n⌊p/2⌋k⌈p/2⌉ log k)

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