Trade-offs for fully dynamic transitive closure on DAGs: breaking through the O ( n ² barrier

Abstract

We present an algorithm for directed acyclic graphs that breaks through the O(n 2 ) barrier on the single-operation complexity of fully dynamic transitive closure, where n is the number of edges in the graph. We can answer queries in O(n ) worst-case time and perform updates in O(n ω(1, ,1)− + n 1+ ) worst-case time, for any ∈ [0, 1], where ω(1, , 1) is the exponent of the multiplication of an n × n matrix by an n × n matrix. The current best bounds on ω(1, , 1) imply an O(n 0.575 ) query time and an O(n 1.575 ) update time in the worst case. Our subquadratic algorithm is randomized, and has one-sided error. As an application of this result, we show how to solve single-source reachability in O(n 1.575 ) time per update and constant time per query.

Mantaining Dynamic Matrices for Fully Dynamic Transitive Closure

“…In the special case of deletions only, our algorithm achieves O(n) amortized time for deletions and O(1) time for queries: this generalizes to directed graphs the bounds of [12], and improves over [9]. In [5], we show how to break through the O(n 2 ) barrier in the case of directed acyclic graphs.…”

“…similarly for G 1 , H 1 , E 2 , F 2 , G 2 , and then for E, F , G, H 10. end Init * sets the initial value of X (line 2) and initializes the different portions of Data Structure 4 in accordance with the topological order τ of the graph of dependencies as explained in the previous subsection (lines [3][4][5][6][7][8][9]. Set * Before describing our implementation of Set * , we first define a useful shortcut for performing simultaneous SetRow and SetCol operations with the same i on more than one variable in a polynomial P :…”

Dynamic Connectivity for Axis-Parallel Rectangles

“…According to Corollary 4, 6 ). Noting that the number of intersections of the rectangles' boundaries with the q-grid is X = O(nq), we also get an alternative bound M…”

“…For this reason we can skip the creation of O(rn/q) component vertices. Thus the number of vertices of H is reduced to O(r 6 ). This implies that the amortized cost of an update isÕ(r 6 + n/r), which is asymptotically minimized for r = n 1/7 and yields an update time of O(n 6/7 ) = O(n 0.858 ).…”

Dynamic shortest paths and transitive closure: Algorithmic techniques and data structures

“…In [9] the authors show how to trade off query times for updates on directed acyclic graphs: each query can be answered in time O(n ε ) and each update can be performed in time O(n ω(1,ε,1)−ε + n 1+ε ), for any ε ∈ [0, 1], where ω(1, ε, 1) is the exponent of the multiplication of an n × n ε matrix by an n ε × n matrix. Balancing the two terms in the update bound yields that ε must satisfy the equation…”