Sparse Suffix Tree Construction in Small Space
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- Posy Ramsey
- 7 years ago
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1 Spare Suffix Tree Contruction in Small Space Joint work with Philip Bille, Inge Li Gørtz, Hjalte Wedel Vildhøj (Technical Univerity of Denmark) Tvi Kopelowitz, (Weizmann Intitute of Science) Johanne Ficher (Karlruhe Intitute of Technology). to appear in ICALP 2013
2 The pare uffix tree (SST) T b a n a n a Suffix Tree a banana na na na na
3 The pare uffix tree (SST) T b a n a n a Suffix a banana na na na na
4 The pare uffix tree (SST) T b a n a n a a banana na na na na Can be built in O(n) time log 2 and O(n) extra pace
5 The pare uffix tree (SST) T b a n a n a a banana na na na na Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
6 The pare uffix tree (SST) n T b a n a n a a n a n a a banana na b n a a na na na Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
7 The pare uffix tree (SST) n T b a n a n a a n a n a a banana na b n a a na na na Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
8 The pare uffix tree (SST) n T b a n a n a a n a n a a banana na b n a a na na na Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
9 The pare uffix tree (SST) n T b a n a n a a n a n a a banana na b n a a na na na Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
10 The pare uffix tree (SST) n T b a n a n a a n a n a O(b) a banana na b n a a na na na Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
11 The pare uffix tree (SST) n T b a n a n a a n a n a O(b) a banana na b n a a na na na Beating the naive bound ha been open ince 1960 O(n log 2 b) time (Monte-Carlo) O((n + b 2 ) log 2 b) time with high probability (La-Vega) both in O(b) pace
12 n T b a n a n a The pare uffix array (SSA)
13 The pare uffix array (SSA) n 1 b a n a n a T b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
14 The pare uffix array (SSA) n 1 b a n a n a T b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
15 The pare uffix array (SSA) n 1 b a n a n a T b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
16 The pare uffix array (SSA) n 1 b a n a n a T b a n a n a 2 a n a n a Sort the uffixe lexicographically 3 n a n a 4 a n a 5 n a 6 a 7
17 The pare uffix array (SSA) n T b a n a n a Sort the uffixe lexicographically 2 a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7
18 The pare uffix array (SSA) n T b a n a n a Sort the uffixe lexicographically n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7
19 The pare uffix array (SSA) n T b a n a n a Sort the uffixe lexicographically n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace
20 The pare uffix array (SSA) n T b a n a n a Sort the uffixe lexicographically n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
21 The pare uffix array (SSA) n T b a n a n a n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
22 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
23 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
24 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
25 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array Spare Suffix Array b 2 a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Can be built in O(n) time log 2 and O(n) extra pace What if we only care about a few of the uffixe?
26 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array Spare Suffix Array b 2 a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Converion between SSA and SST i imple and take O(n log b) time
27 n T b a n a n a b a n The pare uffix array (SSA) a n n a a a n Suffix Array Spare Suffix Array 2 6 b 5 2 a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 Converion between SSA and SST i imple and take O(n log b) time... o we will focu on building the pare uffix array
28 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array Spare Suffix Array b 2 a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 O(n log 2 b) time (Monte-Carlo) O((n + b 2 ) log 2 b) time with high probability (La-Vega) both in O(b) pace
29 LCE - a fundamental tool for pattern matching T a b c b a b n a b c a b a b a For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch
30 LCE - a fundamental tool for pattern matching T a b c b a b i n a b c a b a b a For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch j
31 LCE - a fundamental tool for pattern matching T a b c b a b i n a b c a b a b a For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch j
32 LCE - a fundamental tool for pattern matching T a b c b a b i n a b c a b a b a j For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch
33 LCE - a fundamental tool for pattern matching T a b c b a b i n a b c a b a b a j For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch LCE data tructure are typically baed on the uffix array or uffix tree.
34 LCE - a fundamental tool for pattern matching T a b c b a b i n a b c a b a b a j For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch LCE data tructure are typically baed on the uffix array or uffix tree. We do the oppoite - we ue batched LCE querie to contruct the pare uffix array
35 LCE - a fundamental tool for pattern matching T a b c b a b i n a b c a b a b a j For any (i, j), the longet common extenion i the larget l uch that T [i... i + l 1] = T [j... j + l 1] it the furthet you can go before hitting a mimatch LCE data tructure are typically baed on the uffix array or uffix tree. We do the oppoite - we ue batched LCE querie to contruct the pare uffix array Thee LCE querie will be anwered uing Karp-Rabin fingerprint to enure that the pace remain mall
36 Karp-Rabin fingerprint of tring S a b a c c b a b c b φ(s) = S 1 k=0 S[k]rk mod p Here p = Θ(n 4 ) i a prime and 1 r < p i a random integer with high probability, S 1 = S 2 iff φ(s 1 ) = φ(s 2 )
37 Karp-Rabin fingerprint of tring S a b a c c b a b c b φ(s) = S 1 k=0 S[k]rk mod p Here p = Θ(n 4 ) i a prime and 1 r < p i a random integer with high probability, S 1 = S 2 iff φ(s 1 ) = φ(s 2 ) Oberve that φ(s) fit in an O(log n) bit word
38 Karp-Rabin fingerprint of tring S a b a c c b a b c b φ(s) = S 1 k=0 S[k]rk mod p Here p = Θ(n 4 ) i a prime and 1 r < p i a random integer with high probability, S 1 = S 2 iff φ(s 1 ) = φ(s 2 ) Oberve that φ(s) fit in an O(log n) bit word Given φ(s[0, l]) and φ(s[0, r]) we can compute φ(s[l + 1, r]) in O(1) time
39 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1]
40 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] T
41 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] T
42 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] T
43 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] T We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
44 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] 1 T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
45 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
46 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
47 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
48 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
49 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
50 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
51 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
52 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
53 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
54 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint
55 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint In each pa we tore (at mot) 4b prefix fingerprint
56 Simple, Monte-Carlo batched LCE querie Input : a tring, T of length n and b pair, (i, j) Output : for each pair (i, j) output the larget l.t. T [i... i + l 1] = T [j... j + l 1] prefix fingerprint T b We find the larget l for each pair by binary earch (in parallel) comparion are performed uing fingerprint In each pa we tore (at mot) 4b prefix fingerprint thi take O(n log b) time, O(b) pace and i correct whp.
57 Building the pare uffix array uing batched LCE T b a n a n a
58 Building the pare uffix array uing batched LCE T b a n a n a 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
59 Building the pare uffix array uing batched LCE T b a n a n a 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
60 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
61 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
62 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
63 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 2 < 4 becaue n < 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7
64 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 2 < 4 becaue n < 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion
65 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 3 n a n a 4 a n a 5 n a 6 a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion Pick a random pivot and compare each other uffix to it - Thi partition the uffixe in O(n log b) time and O(b) pace
66 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 4 a n a 6 a 3 n a n a 5 n a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion Pick a random pivot and compare each other uffix to it - Thi partition the uffixe in O(n log b) time and O(b) pace
67 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 4 a n a 6 a 3 n a n a 5 n a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion Pick a random pivot and compare each other uffix to it - Thi partition the uffixe in O(n log b) time and O(b) pace - Recure on each partition (the batch till contain b LCE)
68 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 4 a n a 6 a 3 n a n a 5 n a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion Pick a random pivot and compare each other uffix to it - Thi partition the uffixe in O(n log b) time and O(b) pace - Recure on each partition (the batch till contain b LCE)
69 Building the pare uffix array uing batched LCE T b a n a n a The LCE of two uffixe give u their order 1 b a n a n a 2 a n a n a 4 a n a 6 a 3 n a n a 5 n a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion The depth of the recurion i O(log b) whp. o... The total time i O(n log 2 b) and the pace i O(b)
70 Building the pare uffix array uing batched LCE 1 b a n a n a n a a T b a n a n a 2 a n 4 a n a 6 a 3 n a n a 5 n a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion The depth of the recurion i O(log b) whp. o... The total time i O(n log 2 b) and the pace i O(b)
71 Building the pare uffix array uing batched LCE T b a n a n a Thi algorithm i Monte-Carlo and La-Vega. It can be made Monte-Carlo only by aborting the quickort early 1 b a n a n a 2 a n a n a 4 a n a 6 a 3 n a n a 5 n a 7 We perform randomied quickort on the b uffixe uing batched LCE for uffix comparion The depth of the recurion i O(log b) whp. o... The total time i O(n log 2 b) and the pace i O(b)
72 The pare uffix array (SSA) n T b a n a n a a n a n a b n a a n Suffix Array Spare Suffix Array b 2 a n a n a 4 a n a 6 a 1 b a n a n a 3 n a n a 5 n a 7 O(n log 2 b) time (Monte-Carlo) O((n + b 2 ) log 2 b) time with high probability (La-Vega) both in O(b) pace
73 Verifying the pare uffix array T b a n a n a n Suffix Array How can we tell if thi uffix array i correct?
74 Verifying the pare uffix array T b a n a n a n Suffix Array How can we tell if thi uffix array i correct? Check that 2 < 4, 4 < 6, 6 < 1, 1 < 3...
75 Verifying the pare uffix array T b a n a n a n Suffix Array How can we tell if thi uffix array i correct? Check that 2 < 4, 4 < 6, 6 < 1, 1 < 3... Thi uffice becaue lexicographical ordering i tranitive
76 Verifying the pare uffix array T b a n a n a n Suffix Array How can we tell if thi uffix array i correct? Check that 2 < 4, 4 < 6, 6 < 1, 1 < 3... Thi uffice becaue lexicographical ordering i tranitive We could check 2 < 4 uing an LCE query if we verified it
77 Verifying the pare uffix array T b a n a n a n Suffix Array How can we tell if thi uffix array i correct? Check that 2 < 4, 4 < 6, 6 < 1, 1 < 3... Thi uffice becaue lexicographical ordering i tranitive We could check 2 < 4 uing an LCE query if we verified it So it uffice to verify a batch of b LCE querie
78 Verifying batched LCE querie Input : a tring, T of length n and b triple, (i, j, l) Output : for each triple (i, j, l) check that T [i + l] T [j + l] and T [i... i + l 1] = T [j... j + l 1] T b
79 Verifying batched LCE querie Input : a tring, T of length n and b triple, (i, j, l) Output : for each triple (i, j, l) check that T [i + l] T [j + l] and T [i... i + l 1] = T [j... j + l 1] eay! O(n) time T b
80 Verifying batched LCE querie Input : a tring, T of length n and b triple, (i, j, l) Output : for each triple (i, j, l) check that T [i... i + l 1] = T [j... j + l 1] T b
81 Verifying batched LCE querie Input : a tring, T of length n and b triple, (i, j, l) Output : for each triple (i, j, l) check that T [i... i + l 1] = T [j... j + l 1] T b We proceed in round in decending order... in the k-th round every l = 2 k
82 Verifying batched LCE querie Input : a tring, T of length n and b triple, (i, j, l) Output : for each triple (i, j, l) check that T [i... i + l 1] = T [j... j + l 1] T b We proceed in round in decending order... in the k-th round every l = 2 k Let focu on a ingle round
83 A firt example T 1 2 3
84 A firt example T 1 2 3
85 A firt example T 1 2 3
86 A firt example T 1 2 3
87 A firt example T 1 2 3
88 A firt example T If yellow (1) and blue (2) match then the right half of green (3) matche
89 A firt example T If yellow (1) and blue (2) match then the right half of green (3) matche Thi i a lock-tepped cycle
90 A econd example T 1 2 3
91 A econd example T 1 2 3
92 A econd example T 1 2 3
93 A econd example T 1 2 3
94 A econd example T 1 2 3
95 A econd example T If yellow (1),blue (2) and green (3) match then the right 3 /4 of green (3) i periodic
96 A econd example T If yellow (1),blue (2) and green (3) match then the right 3 /4 of green (3) i periodic
97 A econd example T If yellow (1),blue (2) and green (3) match then the right 3 /4 of green (3) i periodic Thi i an unlocked cycle
98 A econd example T 1 2 3
99 A econd example T offet Thee trick only work when the offet are mall
100 A econd example T l offet Thee trick only work when the offet are mall in particular when they um to at mot l/4
101 The overall idea T l We build a graph containing V 9 n/(l log b) node...
102 The overall idea T l We build a graph containing V 9 n/(l log b) node...
103 The overall idea T l We build a graph containing V 9 n/(l log b) node...
104 The overall idea T l l/(9 log b) We build a graph containing V 9 n/(l log b) node...
105 The overall idea T l l/(9 log b) We build a graph containing V 9 n/(l log b) node...
106 The overall idea T l > l/(9 log b) We build a graph containing V 9 n/(l log b) node...
107 The overall idea T l > l/(9 log b) We build a graph containing V 9 n/(l log b) node...
108 The overall idea T l We build a graph containing V 9 n/(l log b) node...
109 The overall idea T l We build a graph containing V 9 n/(l log b) node...
110 The overall idea T l We build a graph containing V 9 n/(l log b) node...
111 The overall idea T l We build a graph containing V 9 n/(l log b) node...
112 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query
113 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query
114 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query We can apply one of the two trick to any hort cycle
115 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query We can apply one of the two trick to any hort cycle (length at mot 2 log b + 1)
116 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query We can apply one of the two trick to any hort cycle (length at mot 2 log b + 1) Thi break the cycle (becaue we delete an edge)
117 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query We can apply one of the two trick to any hort cycle (length at mot 2 log b + 1) Thi break the cycle (becaue we delete an edge) Fact If every node ha degree at leat three there i a hort cycle
118 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query We can apply one of the two trick to any hort cycle (length at mot 2 log b + 1) Thi break the cycle (becaue we delete an edge) Fact If every node ha degree at leat three there i a hort cycle (low degree node are eaily handled)
119 The overall idea T l We build a graph containing V 9 n/(l log b) node... and one edge per LCE query Fact If every node ha degree at leat three there i a hort cycle We can find a hort cycle in the graph via a BFS in O( E ) = O(b) time Thi give the additive O(b 2 log b) term All other tep take O(n log b) time over all round (and ue O(b) pace)
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