A simpler and more efficient algorithm for the next-to-shortest path problem

Given an undirected graph $G=(V,E)$ with positive edge lengths and two vertices $s$ and $t$, the next-to-shortest path problem is to find an $st$-path which length is minimum amongst all $st$-paths strictly longer than the shortest path length. In this paper we show that the problem can be solved in linear time if the distances from $s$ and $t$ to all other vertices are given. Particularly our new algorithm runs in $O(|V|\log |V|+|E|)$ time for general graphs, which improves the previous result of $O(|V|^2)$ time for sparse graphs, and takes only linear time for unweighted graphs, planar graphs, and graphs with positive integer edge lengths.Comment: Partial result appeared in COCOA201

Canadians Should Travel Randomly

We study online algorithms for the Canadian Traveller Problem (CTP) introduced by Papadimitriou and Yannakakis in 1991. In this problem, a traveller knows the entire road network in advance, and wishes to travel as quickly as possible from a source vertex s to a destination vertex t, but discovers online that some roads are blocked (e.g., by snow) once reaching them. It is PSPACE-complete to achieve a bounded competitive ratio for this problem. Furthermore, if at most k roads can be blocked, then the optimal competitive ratio for a deterministic online algorithm is 2k + 1, while the only randomized result known is a lower bound of k + 1. In this paper, we show for the first time that a polynomial time randomized algorithm can beat the best deterministic algorithms, surpassing the 2k + 1 lower bound by an o(1) factor. Moreover, we prove the randomized algorithm achieving a competitive ratio of (1 + [√2 over 2])k + 1 in pseudo-polynomial time. The proposed techniques can also be applied to implicitly represent multiple near-shortest s-t paths.NSC Grant 102-2221-E-007-075-MY3Japan Society for the Promotion of Science (KAKENHI 23240002

Compact procedural implementation in DSP software synthesis through recursive graph decomposition

Abstract. Synthesis of digital signal processing (DSP) software from dataflowbased formal models is an effective approach for tackling the complexity of modern DSP applications. In this paper, an efficient method is proposed for applying subroutine call instantiation of module functionality when synthesizing embedded software from a dataflow specification. The technique is based on a novel recursive decomposition of subgraphs in a cluster hierarchy that is optimized for low buffer size. Applying this technique, one can achieve significantly lower buffer sizes than what is available for minimum code size inlined schedules, which have been the emphasis of prior software synthesis work. Furthermore, it is guaranteed that the number of procedure calls in the synthesized program is polynomially bounded in the size of the input dataflow graph, even though the number of module invocations may increase exponentially. This recursive decomposition approach provides an efficient means for integrating subroutinebased module instantiation into the design space of DSP software synthesis. The experimental results demonstrate a significant improvement in buffer cost, especiall