Faster Parallel Batch-Dynamic Algorithms for Low Out-Degree Orientation

TL;DR

This paper presents faster parallel batch-dynamic algorithms for maintaining a low out-degree orientation of an undirected graph and is the first parallel batch-dynamic algorithm to maintain an asymptotically optimal orientation with asymptotically optimal expected work bounds.

Abstract

A low out-degree orientation directs each edge of an undirected graph with the goal of minimizing the maximum out-degree of a vertex. In the parallel batch-dynamic setting, one can insert or delete batches of edges, and the goal is to process the entire batch in parallel with work per edge similar to that of a single sequential update and with span (or depth) for the entire batch that is polylogarithmic. In this paper we present faster parallel batch-dynamic algorithms for maintaining a low out-degree orientation of an undirected graph. All results herein achieve polylogarithmic depth, with high probability (whp); the focus of this paper is on minimizing the work, which varies across results. Our first result is the first parallel batch-dynamic algorithm to maintain an asymptotically optimal orientation with asymptotically optimal expected work bounds, in an amortized sense, improving over the prior best work bounds of Liu et al.~[SPAA~'22] by a logarithmic factor. Our second result is a $O(c \log n)$ orientation algorithm with expected worst-case $O(\sqrt{\log n})$ work per edge update, where $c$ is a known upper-bound on the arboricity of the graph. This matches the best-known sequential worst-case $O(c \log n)$ orientation algorithm given by Berglin and Brodal ~[Algorithmica~'18], albeit in expectation. Our final result is a $O(c + \log n)$-orientation algorithm with $O(\log^2 n)$ expected worst-case work per edge update. This algorithm significantly improves upon the recent result of Ghaffari and Koo~[SPAA~'25], which maintains a $O(c)$-orientation with $O(\log^9 n)$ worst-case work per edge whp.

Figures (5)

• Figure 1: Example graph with high span for Brodal-Fagerberg's Algorithm. Let the cutoff for flipping be $4c$. Note that the black edges can all be inserted in any order without the algorithm performing any flips because the out-degree bound is not violated. When the green edge $(v_1,a)$ is inserted, $v_1$ flips, then $v_2$ in the next step (now having $4c$), so on and so forth. Thus there will be $\Omega(t)$ iterations of the algorithm, and $t$ can be as high as $\Theta(\frac{n}{c})$. Note that similar examples arise even if the edges are not inserted in an arbitrary orientation direction.
• Figure 2: Example graph orientation and skyline. A skyline is a subset of edges coming from high-degree vertices where the amount of edges taken depends on the vertex's out-degree. On the left, we show an orientation, where we have colored the edges in an example skyline of size 9 green. On the right we show a bar graph of the vertex out-degrees. The height of the green portion of each column equals the number of out-edges from this vertex placed in the skyline. Looking at the bar graph, note that the bottom of the green bars is (approximately) level.
• Figure 3: Example of counter movements. Here, $T=10$ and $H=3$. Note that the total amount taken from counters above $T$, which is 11, is less than the amount added to counters, which is 10. Note that all counters end up with weight under $T+H=13$, and that no counter below $T$ loses weight.
• Figure 4: Example of finding a skyline of size $10$, where $c'=3$. Note that we take all edges above $6c'$ but that we made an arbitrary choice for the edges between $5c'$ and $6c'$.
• Figure 5: New and old vertices from strata perspective of a skyline with rounding 10. Note that the complete skyline for this graph is not shown, only a partial view. The black boxes denote out-degree being added in this strata from vertices existing at higher strata. The orange boxes denote out-degree being added from vertices who first exist in this strata.

Theorems & Definitions (112)

• Theorem 2.0
• Theorem 2.0
• Theorem 2.0
• Lemma 2.0
• Theorem 2.0
• Theorem 2.0
• Lemma 3.0
• Lemma 3.1
• Lemma 3.3
• proof