Symmetry-Based Search Space Reduction For Grid Maps

06/21/2011
by   Daniel Harabor, et al.
0

In this paper we explore a symmetry-based search space reduction technique which can speed up optimal pathfinding on undirected uniform-cost grid maps by up to 38 times. Our technique decomposes grid maps into a set of empty rectangles, removing from each rectangle all interior nodes and possibly some from along the perimeter. We then add a series of macro-edges between selected pairs of remaining perimeter nodes to facilitate provably optimal traversal through each rectangle. We also develop a novel online pruning technique to further speed up search. Our algorithm is fast, memory efficient and retains the same optimality and completeness guarantees as searching on an unmodified grid map.

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1 Introduction

Pathfinding on uniform-cost undirected grid maps is a problem commonly appearing in the literature: for example in application areas such as robotics [7]artificial intelligence [12] and video games [3, 11]. In such contexts it is often the case that queries sent to the pathfinding system need to be solved as quickly as possible. Traditionally, this requirement is met through the application of hierarchical decomposition techniques that transform the search space into a much smaller approximate representation [2, 11]

. Such methods are very fast, particularly when compared to the classical A* algorithm, but have the disadvantage that solutions found in the abstract state space are often not optimal when mapped back to the original grid. An alternative speedup method is to develop better heuristics to guide the search 

[1, 10, 5]. Though usually fast, optimal and more effective than the popular Manhattan or Octile heuristic (both analogous to Euclidean distance but optimised to 4 and 8-connected grids), they have the disadvantage of requiring significant memory overhead.

In this paper we present Rectangular Symmetry Reduction (RSR): a graph pruning algorithm for undirected uniform-cost grid maps which is fast, memory efficient, optimality preserving and which can, in some cases, eliminate entirely the need to search. The central idea that we will explore involves the identification and elimination of path symmetries from the search space.

To deal with path symmetries RSR makes use of an off-line empty rectangle decomposition [6] that converts an arbitrary undirected uniform-cost grid map into an equivalent one where only nodes from the perimeter of each empty rectangle need to be explored during search. We extend this approach in several directions: (i) we generalise the method from 4-connected grid maps to the 8-connected case where the increase in branching factor makes effective symmetry elimination more challenging; (ii) we develop a new offline pruning technique that reduces the number of nodes which need to be explored during search; (iii) we give a novel online pruning strategy which speeds up node expansion by selectively evaluating either all neighbours associated with a particular node or only a small subset. We prove that in each case both optimality and completeness are preserved.

We perform a thorough empirical analysis, comparing RSR with three similar state-of-the-art graph pruning algorithms on a number of synthetic and realistic benchmarks, including one well known set from the popular roleplaying game Baldur’s Gate II. Compared to Harabor and Botea’s method [6], we both extend the applicability and improve the speed on the subset of instances where both algorithms are applicable. We also compare RSR to the recent Swamps-based pruning method of Pochter et al [8] and the enhanced Portal Heuristic of Goldenberg et al [5]. We show that RSR has complementary strengths compared to both of these methods and identify classes of instances where RSR is clearly the better choice, dominating convincingly across a large number of instances.

2 Related Work

In the presence of symmetry, search algorithms often evaluate many equivalent states and make little real progress toward the goal. The topic of how best to deal with symmetry has received significant attention in other parts of the literature [9] but there are very few works that explicitly identify and deal with symmetry in pathfinding domains such as grid maps. The only work of which we are aware is Empty Rectangular Rooms [6]: a symmetry breaking technique specific to 4-connected uniform-cost grid maps which we refer to as 4ERR. We discuss the main differences between 4ERR and RSR in Sections 1 and 3.

The dead-end heuristic [1] and Swamps-based pathfinding [8] are two closely-related pruning techniques that identify areas in the search space not relevant for reaching the goal. This is a similar yet complementary goal to RSR, which tries to reduce the search effort involved in exploring any given area. Both methods decompose the map into a series of obstacle-free areas. A preliminary online search in the decomposed graph is then used to identify areas that can be ignored during a subsequent search in the original grid.

The gateway heuristic [1] and the portal heuristic [5]

are two similar memory-based techniques which also attempt to speed up optimal pathfinding on grid maps. Both decompose the map into a series of adjacent areas and both pre-compute a database of exact distances between all pairs of nodes that transition from one area to another (called variously “’portals” or “gates”). The main idea is to use this information to improve the accuracy of cost-to-go estimates during search as a way of reducing the number states expanded by A*. Where the portal heuristic differs from other similar works

[1, 10] is in its use of the decomposed graph to further prune the state space. A preliminary search identifies portals relevant to the problem instance at hand and, during a subsequent search in the original grid, any nodes from an area that contains no relevant portals are ignored.

In the algorithm engineering community the problem of quickly computing optimal shortest paths has received significant attention. State of the art methods such as Contraction Hierarchies [4] are based on a combination of Dijkstra’s algorithm together with memory-intensive abstractions. Such algorithms are very fast but they are also highly optimised for road networks in which certain topological properties hold true: for example, the existence of “highway” edges that appear on most shortest paths between arbitrary pairs of nodes. Though mostly orthogonal to RSR, there has been very little work applying these ideas to searching on grid maps. One recent result however [11] suggests they are not as effective when the underlying graph contains a high degree of path symmetry.

3 Rectangular Symmetry Reduction

We begin by making precise the notion of a symmetric relationship between paths in a uniform cost graph:

Definition 1.

Two paths and are symmetric if they share the same start and goal node and one can be derived from the other by interchanging the order of the moves.

When applying Definition 1 to an undirected uniform cost grid 111We say uniform but infact straight moves cost 1 and diagonal moves cost approx. . we notice that each node expanded can often be reached from one or more of its ancestors in the search tree by several symmetric paths of equal length. If the nodes on these alternative paths have an -value smaller than the current node (even an equal -value is often sufficient), A* will needlessly expand them. We address this problem using the high-level strategy in Algorithm 1; which identifies and eliminates path symmetries from the grid.

0:  A grid map
  1. Decompose the grid map into a series of disjoint rectangles that are free of any obstacles. As per [6], the size and placement of the rectangles can vary across a map, depending on the positions of obstacles.

  2. Prune all tiles from the interior of each rectangle and possibly some from the perimeter (border).

  3. Add to each rectangle macro edges between selected pairs of tiles from the perimeter. The cost of each edge is equal to the Octile (or Manhattan, on 4-connected grids) distance between endpoints.

  4. During search, temporarily re-insert tiles back into the map to handle cases where the start or goal is a location which has been previously pruned.

Algorithm 1 Graph reduction based on empty rectangles

Our approach has similarities with 4ERR [6], a symmetry breaking algorithm limited to 4-connected grid maps. The main differences are: (i) we generalise 4ERR to 8-connected grid maps (ii) we give a stronger offline pruning operator to eliminate more nodes from the grid (iii) we give a new online pruning operator that reduces a node’s branching factor and further speeds up search.

The generalisation to uniform-cost 8-connected grids is more challenging than it might look at a first glance. On a 4-connected map no node requires more than one macro-edge (to the closest node on the opposite side of the perimeter) to retain optimality [6]. Thus, it is easy to maintain a low branching factor. As we show in the next section, many more macro-edges are needed to preserve optimality on 8-connected maps. We will identify a set of macro-edges that is necessary and sufficient to ensure that empty rectangles can be crossed optimally. Keeping the branching factor within reasonable limits is a primary motivation for the enhancements reported in the next sections.

4 Optimal Room Traversal with Macro-Edges

After interior nodes are eliminated, macro-edges between selected pairs of perimeter nodes have to be added to ensure that rectangles can be traversed optimally. A straightforward approach would be adding a macro-edge between any two nodes on the perimeter of a rectangle. We will call the subgraph resulting from such an operation a perimeter clique.

Although the perimeter clique approach guarantees optimality, it has the disadvantage of creating up a large branching factor and slowing down search (the number of necessary macro edges is quadratic in the number of perimeter nodes). We introduce an alternative strategy, that creates much fewer macro-edges, by defining a dominance relation between macro-edges.

Definition 2.

A macro-edge connecting two arbitrary nodes and in a perimeter clique is non-dominated if all other paths between and in the perimeter clique have a cost strictly larger than the macro-edge at hand.

By starting with a perimeter clique and applying Definition 2 it is easy to see that the set of non-dominated macro-edges are precisely the ones that we identify below. There are three cases to discuss: connections between nodes on the same rectangle side, connections between orthogonal rectangle sides, and connections between opposite rectangle sides. In the discussion that follows note that the length of each added macro-edge is equal to the heuristic distance between its two endpoints – as measured using Octile distance.

The first case is simple: adjacent nodes on the same perimeter side are connected just as in the original grid. In the second case, two nodes on orthogonal sides of the perimeter of a rectangle are connected iff the shortest path between them is a diagonal (45-degree) line; this is illustrated in Figure 1 (left). Notice that in both cases we introduce no more than two macro edges per node. In the third case, we generate for each perimeter node a “fan” of neighbours from the opposite side . Figure 1 (right), illustrates this idea. Starting from a node such as we step to the closest neighbour from the opposite side of and extend the fan by progressing away from the middle in both directions adding each node we encounter. The last node on either side of the fan is placed diagonally, at 45 degrees, from (such as ) or located in the corner of the perimeter (whichever we encounter first). There is no need to add further nodes, such as , as these can be reached optimally from via the path .

Figure 1: (Left) Macro edges between nodes on orthogonal sides of an empty rectangle. (Right) Each node on the perimeter is connected to a set of nodes on the opposite side.

In the rest of this paper, by macro-edge we will mean a non-dominated macro-edge. We show next that the non-dominated macro-edges computed using our strategy are both necessary and sufficient to ensure optimal traversal between any two perimeter nodes.

Proposition 1.

All non-dominated macro-edges are necessary to ensure optimal paths in a perimeter clique.

Proof.

By definition, a non-dominated macro-edge is the only way to travel optimally between its end nodes in a perimeter clique. Therefore, dropping would result in losing path optimality in the perimeter clique. ∎

Lemma 1.

Let be an empty rectangle in an 8-connected grid map. Let and be two perimeter locations. Then, and can be connected optimally through a path that contains only non-dominated macro-edges.

Proof.

Sketch: We split the proof over the 3 cases discussed earlier: 1) and are on the same side of the perimeter; 2) and are on orthogonal sides of the perimeter; and 3) and are on opposite sides of the perimeter.

In the first case we can simply walk along the perimeter from to ; the optimality of this path is immediate. In the second and third case we argue as follows: the two nodes can be connected through an optimal path that has one diagonal macro-edge (at one end of the path) and zero or more straight macro-edges. See again the example of travelling from to in Figure 1 (Right). ∎

Node Insertion: Sometimes a node from the interior of an empty rectangle is required as a start or goal location for an agent. To handle such situations we give an online procedure that temporarily re-inserts nodes back into the map for the duration of a search. It proceeds as follows: If the start and goal are interior nodes in the same room no insertion is necessary; an optimal path is trivially available. On the other hand, if the start and goal are not in the same rectangle, add four “fans” (collections) of macro edges. Each fan connects the start (goal) node to a set of nodes on one side of the rectangle’s perimeter. Fans are built as shown earlier.

Given the simple geometry of rectangles, it is possible to identify in constant time the set of nodes which the start or goal must be connected to. Further, these neighbours could be generated on the fly.

Lemma 2.

Let be an empty rectangle in an 8-connected grid map. For any nodes , , with a re-inserted interior node and a node on the perimeter, it is always possible to find an optimal length path which mentions no interior nodes except for .

Proof.

Our re-insertion procedure connects the start or goal to a set of nodes on each side of . The procedure in each case is the same as the one given when connecting two nodes on opposite sides of . To prove optimality we can simply run the argument given for Step 3 of Lemma 1 for each node on the perimeter of , in each case substituting for the newly inserted node. ∎

We claim that eliminating symmetries as outlined earlier preserves the completeness and the solution optimality:

Theorem 1.

For every optimal path on an original grid, there exists an optimal path on the modified graph with the property that and have the same cost.

Proof.

Consider an optimal path on the original map and a rectangle that is crossed by . Let and be the two perimeter points along . According to Lemma 1, there is a way to connect and optimally in the modified graph. Thus, we can replace the original segment in with the cost-wise equivalent segment that corresponds to the modified graph. The case when (or ) is the start or goal node is addressed similarly using Lemma 2. By performing such a path segment replacement for all rectangles intersected by , we obtain a path that satisfies the desired properties. ∎

5 Reducing The Branching Factor Further

Consider an empty rectangle of width and height where . After adding all non-dominated macro edges, each node from the perimeter will have between to neighbours from the opposite side of the rectangle, up to 2 neighbours from the same side of the rectangle and up to 5 other neighbours from adjacent rectangles. Such a high branching factor is undesirable as individual node expansion operations take longer.

In this section we study two branching factor reduction methods. The first is an offline technique that prunes nodes from the perimeter of each rectangle. The second is an online pruning strategy which we apply during individual node expansion operations. We discuss both methods in the context of 8-connected grid maps however they are equally applicable to 4-connected maps.

Perimeter Reduction: We observe that in many cases there are nodes on the perimeter of an empty rectangle which have no neighbours from any adjacent rectangle. These nodes represent intermediate locations between entry and exit points that lead into and out of each empty rectangle. To speed up search we propose pruning from the perimeter of each rectangle all such nodes. To preserve optimality, we will connect the neighbours of each pruned node directly to each other. The weight of each new edge is set appropriately to the octile distance between the two neighbours. Figure 2 (Left) shows an example. As we will see this optimisation can have a dramatic effect on the average performance of A* on certain types of maps.

Lemma 3.

Perimeter reduction preserves path optimality.

Proof.

Sketch: Each time we prune a node from the perimeter we add a new edge between all its neighbours with weight equal to the distance between each pair of neighbours. Thus, if a path exists between a pair of nodes before the application of perimeter reduction it is guaranteed to exist afterward. Further, the length of this path is unchanged. ∎

Figure 2: (Left) From each empty rectangle we prune all (dark grey) nodes which have no neighbours in any adjacent rectangle. Remaining nodes are then connected directly. (Right) Assume that is the parent of . When is expanded, there is no need to generate its secondary neighbors. These can be reached directly from on a shorter or equal-length path.

Online Node Pruning: Given a perimeter node , let us partition its macro neighbors (connected to by macro-edges) on the perimeter into primary neighbours and secondary neighbours. Secondary neighbours are those which are located on the opposite side of the perimeter to as compared to (excluding any corner nodes). Primary neighbours are all the rest.

When expanding an arbitrary node from the perimeter of a rectangle we observe that it is not necessary to consider any secondary neighbours if both the node and its predecessor belong to the same rectangle. Figure 2 (Right) shows an example of such a situation; any path to a secondary neighbour is strictly dominated by an alternative path through the predecessor. We apply this observation as follows: During node expansion, determine which rectangle the parent of the current node belongs to. If the current node has no parent or the parent belongs to a different rectangle, then process (i.e., generate) all primary and secondary neighbours. Otherwise, process only primary neighbours.

Lemma 4.

Online node pruning preserves path optimality.

Proof.

Sketch: Let be a node on the perimeter of a rectangle. Assume that its parent belongs to the same rectangle. Let be a secondary successor of . Recall that and are on opposite sides of the rectangle. We argue below that passing through cannot possibly improve the best path between and . Therefore, there is no need to consider macro-edges when and belong to the same rectangle.

There are 4 cases when a node and its parent belong to the same rectangle. In case 1, is a re-inserted node from the interior of the rectangle. Obviously, the path segment is suboptimal, as we zigzag from to on one side of the rectangle and then to on the opposite side of the rectangle. In cases 2, 3, and 4, and are on opposite sides, on orthogonal sides or on the same side of the rectangle. As in case 1, it is possible to check in each case that taking a detour through does not improve the shortest path from to . ∎

6 Memory Requirements and Dynamic Environments

Memory Requirements: In the most straightforward implementation, RSR requires storing the id of the parent rectangle for each node in the original grid. This equates to an memory overhead, where is the set of nodes in the underlying graph. Due to the simple geometric nature of empty rectangles, the set of macro edges associated with each perimeter node can be computed on-the-fly in constant time.
Dynamic Environments: In many application areas, most notably video games, the assumption of a static environment is sometimes unreasonable. For example: obstacles may appear on the grid or existing obstacles may be destroyed as the game progresses. In such cases the underlying graph representing the world must be updated. If a new obstacle appears, or an existing one is destroyed, we can simply invalidate the affected rectangles and recompute new ones. The repair operation must be very fast as most such applications run in real time. As we will show, RSR appears particularly well suited for this task.

7 Experimental Setup

We evaluate the performance of RSR on three benchmarks taken from the freely available pathfinding library Hierarchical Open Graph (HOG)222http://www.googlecode.com/p/hog2: Adaptive Depth is a set of 12 maps of size 100100 in which approximately of each map is divided into rectangular rooms of varying size and a large open area interspersed with large randomly placed obstacles. Baldur’s Gate is a set of 120 maps taken from BioWare’s popular roleplaying game Baldur’s Gate II: Shadows of Amn. Often appearing as a standard benchmark in the literature [1, 6, 8] these maps range in size from 5050 to 320320 and have a distinctive 45-degree orientation. Rooms is a set of 300 maps of size 256256 which are divided into symmetric rows of small rectangular areas (), connected by randomly placed entrances. This benchmark has previously appeared in [10, 8, 5]. As discussed later, we also use a variant of each benchmark where every map is scaled up by a factor of 3. In effect, our input data contains 864 maps in total, with sizes up to .

Since our work is applicable to both 4 and 8 connected grid maps we used two copies each map: one in which diagonal transitions are allowed and another in which they are not. For each map we generated 100 valid problem instances, checking that every instance could be solved both with and without the use of diagonal transitions. Our test machine had a 2.93GHz Intel Core 2 Duo processor, 4GB RAM and ran OSX 10.6.2. Our implementation of A* is based on one provided in HOG, which we adapted to facilitate our online node pruning enhancement.

8 Results

To evaluate RSR we use a generic implementation of A* and discuss performance in terms of search time speedup. That is, the relative improvement to the average time A* needs to solve an instance when running on a pruned vs. unpruned grid. For example, a speedup of 2.0 is twice as fast (higher is better). Note that on approximately 2% of all instances the start and goal are located in the same rectangle and RSR computes the optimal solution without search. We exclude these instances from our results on the basis that they are outliers, even though RSR solves them in constant time.

Figure 3: Average A* speedup on each of our three benchmarks. Results are given in terms of relative improvement to A* search time (i.e. speedup).

Pre-processing Times: Table 1 presents a summary of average pre-processing times for each of our three (non-scaled) benchmarks. We also give the average number of nodes and edges as an indication of map size. We notice that RSR takes very little time to pre-process all input maps. We did not encounter any that took longer than a second, and most required significantly less than that. An interesting implication from this result is that RSR appears well suited to pathfinding in dynamic environments as outlined in Section 6.

Benchmark Avg. Nodes Avg. Edges Preproc RSR Preproc Swamps
Adaptive Depth 8765 32773 0.10 5.06
Baldur’s Gate 4507 16557 0.65 3.15
Rooms 51437 166417 0.39 16.9
Table 1: Input map size and average pre-processing times (seconds).

Comparison to 4ERR: We now compare the performance RSR against the 4ERR [6] graph pruning algorithm. As 4ERR works only on 4-connected grids, here we restrict our attention to this type of maps. To assess the individual impact of both perimeter reduction (PR) and online node pruning (OP) we also develop and compare two variant algorithms: 4ERR+PR and 4ERR+OP.

Figure 3 (A to C) presents our main result. Note that RSR shows a convincing speed improvement over 4ERR and all its variants across all input maps. This allows us to conclude that that RSR is the better choice on 4-connected maps. When analysing the impact of each enhancement, we note that 4ERR+PR yields the biggest improvement on all three benchmarks, speeding up A* by up to 20 times. 4ERR+OP compares well with 4ERR+PR on both Adaptive Depth and Baldur’s Gate but is of little benefit on Rooms where perimeter pruning has already reduced the branching factor.

The large performance variation from one benchmark to another can be attributed to how effectively we can decompose the map. A good decomposition forms large rectangles with few perimeter nodes after pruning. This is the case for Rooms. A poor decomposition builds small rectangles with many transitionary perimeter nodes that cannot be pruned. This is the case for Baldur’s Gate.

Comparison to Swamps: Next, we compare and contrast the performance of RSR with the Swamps algorithm [8]. To evaluate Swamps we used the authors’ source code, including their own implementation of A*. We then ran all experiments using their recommended running parameters: a swamp seed radius of 6 and “no change limit” of 2. Figure 3 (D to F) gives search time speedup results for both RSR and Swamps running on the 8-connected variants of our three benchmark problem sets. On Adaptive Depth and Rooms, where the terrain can be naturally decomposed into rectangles, RSR achieves higher speedups and is shown consistently better than Swamps. On Baldur’s Gate, where this is not the case, Swamps-based pruning is more effective.

Next, we scaled every map in each benchmark by a factor of 3 and randomly generated a new set of 100 problem instances per map. Scaling has the effect of producing larger open areas and allows us to measure the impact of this variable on search time speedup. We present our findings in Figure 3 (G to I). We observe that while the maximum speedup achieved by both algorithms has increased, the gain for Swamps is very small while RSR shows dramatic improvement. Infact, if we limit our attention to problems of similar length to those seen on the original maps we notice that the performance of Swamps actually decreases.

The observed performance characteristics are not unexpected: Swamps prune out areas that can be avoided without introducing a detour while rectangle-based symmetry reduction allows for a faster exploration of areas that need to be searched. Since it appears that the two algorithms are orthogonal, a natural extension of this work would be to combine the two: first, apply 4(or 8)ERR+PR (as appropriate) to a grid in order to eliminate as many interior nodes as possible; then, apply a Swamps-based decomposition to the resultant graph.

Algorithm Extra Memory Baldur’s Gate Rooms
PH-e 3.16 11.9
PH-e 3.07 17.54
RSR 2.8 18.2
Table 2: Avg. A* search time speedup: RSR vs PH-e. RSR figures are across all maps on each benchmark. PH-e figures are for a small subset selected by its authors (1 of 120 from Baldur’s Gate and 5 of 300 from Rooms).

Comparison to Portal Heuristic: We now compare RSR with PH-e – the enhanced variant of the recent Portal Heuristic algorithm [5]. Although we did not have access to a working implementation of this method we will discuss its performance vs. RSR based on published results obtained by the original authors. As in [5] we focus on the 4-connected variants of the Baldur’s Gate and Rooms benchmarks. Table 2 summarises the main result.

PH-e performs well when it can decompose the map into areas of similar size with few transitionary nodes connecting them. RSR performs well when it can decompose the map into large rectangles with few perimeter nodes. On Rooms, both decomposition approaches are highly effective. On Baldur’s Gate both are comparatively less effective. Notice however that PH-e requires up to 7 times more memory than RSR to achieve similar results. As with Swamps, we believe PH-e is entirely orthogonal to RSR and the two can be easily combined. For example, PH-e could be used to more accurately guide search on a map pruned by RSR. Alternatively, symmetry elimination could be used to speed up pathfinding between successive pairs of portals during PH-e’s refinement phase.

9 Conclusion

We introduce RSR, a new search space reduction algorithm applicable to pathfinding on uniform cost grid maps. RSR is fast, memory efficient, optimality preserving and can, in some cases, eliminate entirely the need to search. When running on a grid pruned by RSR, A* is up to 38 times faster than otherwise.

We compare RSR with a range of search space reduction algorithms from the literature. Compared to 4ERR [6], on which it is based, RSR is shown significantly faster on the set of instances where both methods can be applied (i.e. 4-connected maps). Next, we compare RSR to Swamps-based pruning [8] and show that the two algorithms have complementary strengths. We find that Swamps are more useful on maps with small open areas while RSR becomes more effective as larger open areas are available on a map. We also identify a broad range of instances where RSR dominates convincingly and is clearly the better choice. Finally, we compare RSR to the enhanced Portal Heuristic [5]. We show that our method exhibits similar or improved performance but requires up to 7 times less memory. As with Swamps, we find that the two ideas are complementary and could be easily combined.

Future work includes reducing the branching factor in RSR further through the development of better map decompositions and stronger online node pruning strategies. Another interesting topic is combining RSR with Swamps or the Portal Heuristic.

10 Acknowledgements

We would like to thank Alban Grastien and Patrik Haslum for providing feedback on early drafts of this work. We also thank Ariel Felner and Meir Goldenberg for providing us with assistance in comparing RSR with their enhanced Portal Heuristic algorithm.

NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

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