This paper presents a new deterministic algorithm for the single-source shortest paths (SSSP) problem on real non-negative edge-weighted directed graphs, with running time O(m√{log n} + √{mnlog nlog log n}), which is O(m√{log nlog log n}) for sparse graphs. This improves the recent breakthrough result of O(m log^{2/3} n) time for directed SSSP algorithm [Duan, Mao, Mao, Shu, Yin 2025].

In undirected graphs with real non-negative weights, we give a new randomized algorithm for the single-source shortest path (SSSP) problem with running time $O(m \sqrt{\log n \cdot \log \log n})$ in the comparison-addition model. This is the first algorithm to break the $O(m+n \log n)$ time bound for real-weighted sparse graphs by Dijkstra’s algorithm with Fibonacci heaps. Previous undirected nonnegative SSSP algorithms give time bound of $O(m \alpha(m, n)+ \min \{n \log n, n \log \log r\})$ in comparison-addition model, where $\alpha$ is the inverse-Ackermann function and r is the ratio of the maximum-to-minimum edge weight [Pettie & Ramachandran 2005], and linear time for integer edge weights in RAM model [Thorup 1999]. Note that there is a proposed complexity lower bound of $\Omega(m+\min \{n \log n, n \log \log r\})$ for hierarchy-based algorithms for undirected real-weighted SSSP [Pettie & Ramachandran 2005], but our algorithm does not obey the properties required for that lower bound. As a non-hierarchybased approach, our algorithm shows great advantage with much simpler structure, and is much easier to implement.

We consider the energy complexity of the leader election problem in the single-hop radio network model, where each device v has a unique identifier ID ( v ) ∈{ 1, 2, ⋖ , N } . Energy is a scarce resource for small battery-powered devices. For such devices, most of the energy is often spent on communication, not on computation. To approximate the actual energy cost, the energy complexity of an algorithm is defined as the maximum over all devices of the number of time slots where the device transmits or listens. Much progress has been made in understanding the energy complexity of leader election in radio networks, but very little is known about the tradeoff between time and energy. Chang et al. [STOC 2017] showed that the optimal deterministic energy complexity of leader election is Θ (log log N ) if each device can simultaneously transmit and listen but still leaving the problem of determining the optimal time complexity under any given energy constraint. Time–energy tradeoff: For any k ≥ log log N , we show that a leader among at most n devices can be elected deterministically in O ( k ċ n 1+ε ) + O ( k ċ N 1/k ) time and O ( k ) energy if each device can simultaneously transmit and listen, where ε > 0 is any small constant. This improves upon the previous O ( N )-time O (log log N )-energy algorithm by Chang et al. [STOC 2017]. We provide lower bounds to show that the time–energy tradeoff of our algorithm is near-optimal. Dense instances: For the dense instances where the number of devices is n = Θ ( N ), we design a deterministic leader election algorithm using only O (1) energy. This improves upon the O (log* N )-energy algorithm by Jurdziński, Kutyłowski, and Zatopiański [PODC 2002] and the O (α ( N ))-energy algorithm by Chang et al. [STOC 2017]. More specifically, we show that the optimal deterministic energy complexity of leader election is \(\Theta (\max \lbrace 1, \log \tfrac{N}{n}\rbrace)\) if each device cannot simultaneously transmit and listen, and it is \(Θ (\max \lbrace 1, \log \log \tfrac{N}{n}\rbrace)\) if each device can simultaneously transmit and listen.

We consider the energy complexity of the leader election problem in the single-hop radio network model, where each device ν has a unique identifier ID(ν) ∈ {1, 2, ..., N}. Energy is a scarce resource for small battery-powered devices. For such devices, most of the energy is often spent on communication, not on computation. To approximate the actual energy cost, the energy complexity of an algorithm is defined as the maximum over all devices of the number of time slots where the device transmits or listens.