## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, December 14, 2023, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Ulysse Schaller

## Balanced Bidirectional Breadth-First-Search on Scale-Free Networks

To find a shortest path between two nodes s and t in a given graph, a classical approach is to start a Breadth-First Search (BFS) from s and run it until the search discovers t. Alternatively, one can start two Breadth First Searches, one from s and one from t, and alternate their layer expansions until they meet. This bidirectional BFS can be balanced by always expanding a layer on the side that has seen fewer vertices so far. This usually results in significant speedups in real-world networks, and it has been shown that this indeed yields sublinear running time on scale-free graph models such as Chung-Lu graphs and GIRGs. We improve this layer-balanced bidirectional BFS approach by using a finer balancing technique. Instead of comparing the size of the two BFS trees after each layer expansion, we perform this comparison after each vertex expansion. This gives rise to two algorithms that run faster than the layer-balanced bidirectional BFS on scale-free networks with power-law exponent 2 < tau < 3. The first one is an approximate shortest-path algorithm that outputs a path of length at most 1 longer than the shortest path in time n^{(tau-2)/(tau-1)+o(1)}. The second one is an exact shortest-path algorithm running in time n^{1/2+o(1)}. These runtime bounds hold with high probability when s and t are chosen uniformly at random among the n vertices of the graph. We also develop an edge-balanced bidirectional BFS algorithm that works under adversarial conditions. This approximate shortest-path algorithm runs in time n^{1/2+o(1)} with high probability when the adversary is allowed to choose s and t based on their (expected) degree. This is joint work with Sacha Cerf, Benjamin Dayan, Umberto De Ambroggio, Marc Kaufmann, and Johannes Lengler.

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