1 Search-Space Characterization for Real-time Heuristic Search Daniel Huntley and Vadim Bulitko Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G 2E8, Canada fdhuntley j [email protected] Abstract—Recent real-time heuristic search algorithms have selection and parameterization. Finally, we want to provide a demonstrated outstanding performance in video-game pathfind- way for other researchers to characterize benchmark search ing. However, their applications have been thus far limited to that problems for the development of future real-time heuristic domain. We proceed with the aim of facilitating wider applica- tions of real-time search by fostering a greater understanding of search algorithms. We address these goals through an analysis the performance of recent algorithms. We first introduce eight of real-time heuristic search performance, correlating it to algorithm-independent complexity measures for search spaces search space features. and correlate their values with algorithm performance. The complexity measures are statistically shown to be significant predictors of algorithm performance across a set of commercial B. Contributions video-game maps. We then extend this analysis to a wider variety This paper makes the following contributions. First, we of search spaces in the first application of database-driven real- present a set of complexity measures which are useful for time search to domains outside of video-game pathfinding. In characterizing search space complexity as it pertains to the doing so, we gain insight into algorithm performance and possible enhancement as well as into search space complexity. performance of modern real-time search algorithms. Some of these complexity measures are original to this work, and some Index Terms—real-time heuristic search, search-space com- plexity, video-game pathfinding have been adapted from the literature. We then empirically link the values of the collected com- plexity measures to the performance of modern database- I. INTRODUCTION driven real-time search algorithms. We begin with an exam- Heuristic search is a mainstay of artificial intelligence ination of algorithm performance in pathfinding on a varied research. A demand for quickly generated solutions to search set of video-game maps. This has served as the traditional problems gave rise to the sub-field of real-time heuristic test bed for subgoaling real-time search [1]–[3]. This exami- search. Real-time heuristic search algorithms make decisions nation demonstrates a statistical correlation between solution in constant time, independent of the number of states in the suboptimality and complexity measure values. It also shows problem being solved. that machine learning can be used to predict performance and Recent database-driven real-time heuristic search algo- facilitate algorithm parameterization. rithms have demonstrated excellent performance in video- We continue with an examination of algorithm performance game pathfinding [3]. However, despite being formulated for beyond video-game pathfinding. This study is performed using general search, most of these algorithms have not been applied mazes and road maps. To our knowledge, this is the first time to a broader selection of problems. In preliminary experimen- that these algorithms have been applied to these domains. arXiv:1308.3309v1 [cs.AI] 15 Aug 2013 tation, we found that many of the algorithms yielded mixed These additional search spaces represent an incremental step or poor results in other search spaces, such as combinatorial towards more general spaces, and introduce several of the search puzzles. Motivated by this mixed performance, we seek challenges that must be addressed if contemporary real-time to establish a way of empirically characterizing search spaces algorithms are to be successfully adapted for broader domains based on their suitability for different real-time heuristic search such as planning. algorithms. This would assist algorithm selection, and provide insight for the development of future algorithms. II. PROBLEM FORMULATION In this section we discuss the goals of our research. We then In this section we provide definitions for the terminology describe our specific contributions, and outline the layout of that will be used throughout the remainder of this paper. the remainder of this paper. We formally define heuristic search problems, and present our methodology for measuring the effectiveness of heuristic A. Motivation search algorithms. There are three goals motivating our research. First, we seek a greater understanding of where database-driven real- A. Heuristic Search Problems time heuristic search is and is not effective. Second, we wish We define a search space as follows: S is a finite set of to demonstrate that this knowledge can facilitate algorithm vertices or states; E is a finite set of transitions, or weighted 2 edges, on S × S. Edge weights are also known as edge costs neighbour state, and a new planning step begins. Because an and are all positive. Search space size is defined as the number entire solution usually cannot be constructed within a single of states jSj. In a search space, a search problem is defined as planning step, real-time heuristic search algorithms frequently a pair of states: sstart 2 S is the start state and sgoal 2 S is the make suboptimal moves, thereby increasing solution subopti- goal state. We only consider search problems with a single mality. Thus a major line of research is improving the quality goal state, since most video-game pathfinding is performed of planning and, subsequently, the resulting moves. with a single goal state. Recent real-time search algorithms attempt to do so by aug- The agent’s goal is to find a path (i.e., a sequence of edges) menting their online planning with the offline pre-computation between the start and the goal state. Path cost is the sum of of a search-space-specific database. These databases are used its edge costs and the agent attempts to find a path with the to provide one or several intermediate goals, or subgoals, for lowest possible cost (i.e., an optimal path between the start use during search. When an appropriate subgoal is found, and the goal states). search is directed towards that state rather than the original The search space comes with a heuristic function h which goal. This approach tends to improve search performance as any agent can use to guide its search. For a pair of arguments a heuristic function is typically more accurate for a near-by s1; s2 2 S the heuristic function estimates the cost of a shortest subgoal than it is for a distant original goal. ∗ path between s1 and s2. An optimal heuristic function h gives We refer to the class of algorithms that are not real-time as the cost of an optimal path between two states. A heuristic conventional heuristic search methods. Popular search methods ∗ function h is admissible if 8s 2 S [h(s; sgoal) ≤ h (s; sgoal)]. in this class include both optimal algorithms such as A* [5] A heuristic search algorithm that is guaranteed to find a and IDA* [6], and suboptimal algorithms such as HPA* [7] or solution if one exists is called complete. An algorithm that PRA* [8]. We do not focus on conventional heuristic search, is guaranteed to find an optimal solution is called an optimal but discuss some of them briefly. algorithm. Learning algorithms are those which make updates to their heuristic function during execution. A. LRTA*: The Foundation Learning real-time A* (LRTA*) [9] is the first real-time B. Search Performance heuristic search algorithm we discuss. LRTA* serves as the We will use two major metrics to assess the performance foundation for three of the subsequent real-time search algo- of a heuristic search algorithm in this paper: solution sub- rithms discussed in this section. optimality and pre-computation time. These measures are collectively referred to as search performance. We define Algorithm 1 LRTA*(sstart; sgoal; d) solution suboptimality as the ratio of the cost of a solution 1: s sstart to the cost of the optimal solution. For example, if a given 2: while s 6= sgoal do solution has cost 5 and the optimal solution has cost 4, then 3: expand LSS of s 0 0 0 the solution suboptimality is 1:25. 4: find a frontier s minimizing g(s; s ) + h(s ; sgoal) 0 0 Most of the algorithms in this paper construct a database 5: h(s; sgoal) g(s; s ) + h(s ; sgoal) for the given search space. Thus we measure database con- 6: change s one step towards s0 struction time or pre-computation time. Any such preparatory 7: end while behaviour performed by a search agent is referred to as offline being , whereas any work done during active solving Pseudo-code for LRTA* is given as Algorithm 1. The agent online of specific search problems is referred to as being . begins by initializing its current location s to the start state The comparative online and offline time consumption of the sstart (line 1). Next, a set of successor states up to d moves algorithms presented here has been previously studied by away is generated surrounding s (line 3). Once the local search Bulitko et al [2] and Lawrence et al. [3] space (LSS) is expanded, its frontier (i.e., border) states are considered and the most promising state s0 is selected in III. REAL-TIME SEARCH ALGORITHMS line 4. The chosen state is that which minimizes the function 0 0 0 Real-time search is a subclass of agent-centered search [4]. g(s; s ) + h(s ; sgoal), where g(s; s ) is the cost of an optimal In real-time search, the agent occupies a single state on the path from s to s0 within the LSS. To deter state revisitation, the 0 0 map at all time.