On Backtracking in Real-time Heuristic Search

On Backtracking in Real-time Heuristic Search Valeriy K. Bulitko Centre for Science, Athabasca University, 1 University Drive Athabasca, Alberta, T9S 3A3, CANADA valeriyb@athabascau.ca Vadim Bulitko Department of Computing Science, University of Alberta Edmonton, Alberta, T6G 2E8, CANADA bulitko@ualberta.ca September 13, 2021 Abstract Real-time heuristic search algorithms are suitable for situated agents that need to make their decisions in constant time. Since the original work by Korf nearly two decades ago, numerous extensions have been suggested. One of the most intriguing extensions is the idea of backtracking wherein the agent decides to return to a previously visited state as opposed to moving forward greedily. This idea has been empirically shown to have a significant impact on various performance measures. The studies have been carried out in particular empirical testbeds with specific real-time search algorithms that use backtracking. Consequently, the extent to which the trends observed are characteristic of backtracking in general is unclear. In this paper, we present the first entirely theoretical study of backtracking in real-time heuristic search. In particular, we present upper bounds on the solution cost exponential and linear in a parameter regulating the amount of backtracking. The results hold for a wide class of real-time heuristic search algorithms that includes many existing algorithms as a small subclass. Keywords: real-time heuristic search, agent-centered search. 1 Introduction In this paper we study the problem of agent-centered real-time heuristic search (Koenig, 2001). The dis- tinctive property of such search is that an agent must repeatedly plan and execute actions within a constant time interval that is independent of the size of the problem being solved. This restriction severely limits the range of applicable algorithms. For instance, static search algorithms (e.g., A* of Hart, Nilsson, & Raphael, 1968), re-planning algorithms (e.g., D* of Stenz, 1995), anytime algorithms (e.g., ARA* of Likhachev, Gordon, & Thrun, 2004) and anytime re-planning algorithms (e.g., AD* of Likhachev, Ferguson, Gordon, Stentz, & Thrun, 2005) cannot guarantee a constant bound on planning time per action. LRTA* provides such guarantees by planning only a few actions at a time and updating its heuristic function, but the solution quality can be poor (Korf, 1990; Ishida, 1992). As a motivating application, consider navigation in gridworld maps in commercial computer games. In such games, an agent can be tasked to go to any location on the map from its current location. The agent must react quickly to the user’s command regardless of the map’s size and complexity. Consequently, game 1 arXiv:0912.3228v1 [cs.AI] 16 Dec 2009 companies impose a time-per-action limit on their pathfinding algorithms. As an example, Bioware Corp., a major game company, limits planning time to 1-3 ms for all pathfinding units (and there can be many units planning simultaneously). The original real-time search algorithms, RTA* and LRTA*, form a local search space (LSS) around the agent’s current state. Then they greedily take an action toward the most promising state on the frontier of the LSS. A large number of subsequent real-time heuristic search algorithms have followed this canon (e.g., Russell & Wefald, 1991; Furcy & Koenig, 2000; Shimbo & Ishida, 2003; Koenig, 2004; Hern´andez & Meseguer, 2005a, 2005b; Koenig & Likhachev, 2006; Rayner, Davison, Bulitko, Anderson, & Lu, 2007; Bulitko, Sturtevant, Lu, & Yau, 2007; Bulitko, Luˇstrek, Schaeffer, Bj¨ornsson, & Sigmundarson, 2008). Arguably, the most radical departure was an introduction of the so-called backtracking moves by Shue and Zamani (1993a, 1993b), Shue, Li, and Zamani (2001). Their impact on performance of real-time heuristic search and, in particular the cost of the solution the agent finds, has been studied mostly empirically (Shue & Zamani, 1993a, 1993b; Shue et al., 2001; Bulitko, 2004; Bulitko & Lee, 2006; Sigmundarson & Bj¨ornsson, 2006). As a result, it is unclear to what extent the reported findings and trends are specific to the particular algorithms and/or to the testbed environments used. The contribution of this paper is an entirely theoretical investigation of effects of backtracking on real- time search performance. We describe a theoretical framework that generalizes a broad class of existing real- time search algorithms. We show that in the worst case, solution cost can be exponential in the parameter controlling the amount of backtracking. We then identify a special case that affords linear solution cost. Because we consider real-time heuristic search on general graphs, the results of our study are domain- independent and, thus, broadly applicable. The rest of the paper is organized as follows. We first informally review the pioneering LRTA* algo- rithm and introduce the notion of backtracking in Section 2. The search problem and performance metrics are fo

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