Fully-dynamic minimum spanning forest with improved worst-case update time

TY - GEN

T1 - Fully-dynamic minimum spanning forest with improved worst-case update time

AU - Wulff-Nilsen, Christian

N1 - Conference code: 49

PY - 2017

Y1 - 2017

N2 - We give a Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted undirected dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions. Each update is supported in O(n1/2-c) worst-case time wh.p. where c > 0 is some constant, and this bound also holds in expectation. This is the first data structure achieving an improvement over the O(√n) deterministic worst-case update time of Eppstein et al., a bound that has been standing for 25 years. In fact, it was previously not even known how to maintain a spanning forest of an unweighted graph in worst-case time polynomially faster than Θ(√n). Our result is achieved by first giving a reduction from fully-dynamic to decremental minimum spanning forest preserving worst-case update time up to logarithmic factors. Then decremental minimum spanning forest is solved using several novel techniques, one of which involves keeping track of low-conductance cuts in a dynamic graph. An immediate corollary of our result is the first Las Vegas data structure for fully-dynamic connectivity where each update is handled in worst-case time polynomially faster than Θ(√n) w.h.p.; this data structure has O(1) worst-case query time.

AB - We give a Las Vegas data structure which maintains a minimum spanning forest in an n-vertex edge-weighted undirected dynamic graph undergoing updates consisting of any mixture of edge insertions and deletions. Each update is supported in O(n1/2-c) worst-case time wh.p. where c > 0 is some constant, and this bound also holds in expectation. This is the first data structure achieving an improvement over the O(√n) deterministic worst-case update time of Eppstein et al., a bound that has been standing for 25 years. In fact, it was previously not even known how to maintain a spanning forest of an unweighted graph in worst-case time polynomially faster than Θ(√n). Our result is achieved by first giving a reduction from fully-dynamic to decremental minimum spanning forest preserving worst-case update time up to logarithmic factors. Then decremental minimum spanning forest is solved using several novel techniques, one of which involves keeping track of low-conductance cuts in a dynamic graph. An immediate corollary of our result is the first Las Vegas data structure for fully-dynamic connectivity where each update is handled in worst-case time polynomially faster than Θ(√n) w.h.p.; this data structure has O(1) worst-case query time.

KW - Dynamic graph connectivity

KW - Dynamic minimum spanning forest

UR - http://www.scopus.com/inward/record.url?scp=85024397829&partnerID=8YFLogxK

U2 - 10.1145/3055399.3055415

DO - 10.1145/3055399.3055415

M3 - Article in proceedings

AN - SCOPUS:85024397829

SP - 1130

EP - 1143

BT - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

PB - Association for Computing Machinery

T2 - 49th Annual ACM SIGACT Symposium on Theory of Computing

Y2 - 19 June 2017 through 23 June 2017

ER -