Edge sampling and graph parameter estimation via vertex neighborhood accesses
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Edge sampling and graph parameter estimation via vertex neighborhood accesses. / Tetek, Jakub; Thorup, Mikkel.
STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. ed. / Stefano Leonardi; Anupam Gupta. Association for Computing Machinery, Inc., 2022. p. 1116-1129.Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review
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TY - GEN
T1 - Edge sampling and graph parameter estimation via vertex neighborhood accesses
AU - Tetek, Jakub
AU - Thorup, Mikkel
N1 - Publisher Copyright: © 2022 ACM.
PY - 2022
Y1 - 2022
N2 - In this paper, we consider the problems from the area of sublinear-time algorithms of edge sampling, edge counting, and triangle counting. Part of our contribution is that we consider three different settings, differing in the way in which one may access the neighborhood of a given vertex. In previous work, people have considered indexed neighbor access, with a query returning the i-th neighbor of a given vertex. Full neighborhood access model, which has a query that returns the entire neighborhood at a unit cost, has recently been considered in the applied community. Between these, we propose hash-ordered neighbor access, inspired by coordinated sampling, where we have a global fully random hash function, and can access neighbors in order of their hash values, paying a constant for each accessed neighbor. For edge sampling and counting, our new lower bounds are in the most powerful full neighborhood access model. We provide matching upper bounds in the weaker hash-ordered neighbor access model. Our new faster algorithms can be provably implemented efficiently on massive graphs in external memory and with the current APIs for, e.g., Twitter or Wikipedia. For triangle counting, we provide a separation: a better upper bound with full neighborhood access than the known lower bounds with indexed neighbor access. The technical core of our paper is our edge-sampling algorithm on which the other results depend. We now describe our results on the classic problems of edge and triangle counting. We give an algorithm that uses hash-ordered neighbor access to approximately count edges in time Õ(n/"m + 1/"2) (compare to the state of the art without hash-ordered neighbor access of Õ(n/"2 m) by Eden, Ron, and Seshadhri [ICALP 2017]). We present an ω(n/"m) lower bound for "≥m/n in the full neighborhood access model. This improves the lower bound of ω(n/s"m) by Goldreich and Ron [Rand. Struct. Alg. 2008]) and it matches our new upper bound for "≥ m/n. We also show an algorithm that uses the more standard assumption of pair queries ("are the vertices u and v adjacent?"), with time complexity of Õ(n/"m + 1/"4). This matches our lower bound for "≥ m1/6/n1/3. Finally, we focus on triangle counting. For this, we use the full power of the full neighbor access. In the indexed neighbor model, an algorithm that makes Õ(n/"10/3 T1/3 + min(m,m3/2/"3 T)) queries for T being the number of triangles, is known and this is known to be the best possible up to the dependency on "(Eden, Levi, Ron, and Seshadhri [FOCS 2015]). We improve this significantly to Õ(min(n,n/"T1/3 + n m/"2 T)) full neighbor accesses, thus showing that the full neighbor access is fundamentally stronger for triangle counting than the weaker indexed neighbor model. We also give a lower bound, showing that this is the best possible with full neighborhood access, in terms of n,m,T.
AB - In this paper, we consider the problems from the area of sublinear-time algorithms of edge sampling, edge counting, and triangle counting. Part of our contribution is that we consider three different settings, differing in the way in which one may access the neighborhood of a given vertex. In previous work, people have considered indexed neighbor access, with a query returning the i-th neighbor of a given vertex. Full neighborhood access model, which has a query that returns the entire neighborhood at a unit cost, has recently been considered in the applied community. Between these, we propose hash-ordered neighbor access, inspired by coordinated sampling, where we have a global fully random hash function, and can access neighbors in order of their hash values, paying a constant for each accessed neighbor. For edge sampling and counting, our new lower bounds are in the most powerful full neighborhood access model. We provide matching upper bounds in the weaker hash-ordered neighbor access model. Our new faster algorithms can be provably implemented efficiently on massive graphs in external memory and with the current APIs for, e.g., Twitter or Wikipedia. For triangle counting, we provide a separation: a better upper bound with full neighborhood access than the known lower bounds with indexed neighbor access. The technical core of our paper is our edge-sampling algorithm on which the other results depend. We now describe our results on the classic problems of edge and triangle counting. We give an algorithm that uses hash-ordered neighbor access to approximately count edges in time Õ(n/"m + 1/"2) (compare to the state of the art without hash-ordered neighbor access of Õ(n/"2 m) by Eden, Ron, and Seshadhri [ICALP 2017]). We present an ω(n/"m) lower bound for "≥m/n in the full neighborhood access model. This improves the lower bound of ω(n/s"m) by Goldreich and Ron [Rand. Struct. Alg. 2008]) and it matches our new upper bound for "≥ m/n. We also show an algorithm that uses the more standard assumption of pair queries ("are the vertices u and v adjacent?"), with time complexity of Õ(n/"m + 1/"4). This matches our lower bound for "≥ m1/6/n1/3. Finally, we focus on triangle counting. For this, we use the full power of the full neighbor access. In the indexed neighbor model, an algorithm that makes Õ(n/"10/3 T1/3 + min(m,m3/2/"3 T)) queries for T being the number of triangles, is known and this is known to be the best possible up to the dependency on "(Eden, Levi, Ron, and Seshadhri [FOCS 2015]). We improve this significantly to Õ(min(n,n/"T1/3 + n m/"2 T)) full neighbor accesses, thus showing that the full neighbor access is fundamentally stronger for triangle counting than the weaker indexed neighbor model. We also give a lower bound, showing that this is the best possible with full neighborhood access, in terms of n,m,T.
KW - Edge counting
KW - Edge sampling
KW - Sublinear-time algorithms
KW - Triangle counting
UR - http://www.scopus.com/inward/record.url?scp=85132757056&partnerID=8YFLogxK
U2 - 10.1145/3519935.3520059
DO - 10.1145/3519935.3520059
M3 - Article in proceedings
AN - SCOPUS:85132757056
SP - 1116
EP - 1129
BT - STOC 2022 - Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
A2 - Leonardi, Stefano
A2 - Gupta, Anupam
PB - Association for Computing Machinery, Inc.
T2 - 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022
Y2 - 20 June 2022 through 24 June 2022
ER -
ID: 316818149