The parking-lot attendant at the trendy P’SPACE Club has a tough job. Whenever someone leaves the nightspot, she must retrieve the patron’s car from the crammed lot, often having to move other vehicles out of the way to clear a path to the exit. She has to do it quickly to earn a generous tip, but being efficient can be a real challenge. The attendant’s quandary is an example of what computer scientists and engineers describe as a motion-planning problem. Such challenges can arise when a robot needs to shift bulky crates from place to place within a crowded warehouse or find its way through an obstacle-strewn maze. Motion-planning predicaments also come up in seemingly simple puzzles in which a would-be solver slides blocks along given paths to achieve a desired configuration.
In recent years, sliding-block puzzles have served as proving grounds for novel motion-planning strategies. They have also attracted the attention of computer scientists interested in the fundamental limits of computation—for example, in determining the relative running time or amount of memory a computer would require to solve various types of demanding problems in their most difficult form (SN: 5/6/00, p. 296: Changes of Mathematical State).
When they confront a puzzle, researchers work out what resources a computer would need to decide whether the challenge is solvable. Where solutions exist, the investigators try to determine how many there are or identify the one with the fewest steps.
One sliding-block puzzle that has played a starring role in several recent research efforts is Rush Hour, a commercial product first distributed in the United States in 1996. A player starts with rectangular blocks shaped like cars and trucks, each in a given location on a square tray, and figures out a sequence of moves that frees a target vehicle from the traffic jam and gets it to the exit.
A study by Gary W. Flake and Eric B. Baum of the NEC Research Institute in Princeton, N.J., now establishes that Rush Hour is indeed a formidable puzzle. From a computational standpoint, their analysis puts Rush Hour on the same level of difficulty as such demanding games as Othello (SN: 8/16/97, p. 100) but below that of chess and Go.
“We find these results to be surprising considering the simplicity of the game,” Flake and Baum remarked in the January Theoretical Computer Science.
The findings suggest that “unjamming a bunch of cars can be a challenging task,” Flake notes. Flake and Baum subtitled their Rush Hour research report: “Why you should generously tip parking-lot attendants.”
Even in the earliest days of computers, researchers couldn’t resist programming them to play games and solve puzzles. It was an entertaining way to show off one’s programming prowess, test a computer, try out problem-solving strategies, and evaluate rival approaches for organizing data or performing searches (SN: 8/2/97, p. 76: https://www.sciencenews.org/sn_arc97/8_2_97/bob1.htm).
A few years ago, when Baum developed a novel computer program that could learn how to solve specific problems, he turned to a puzzle called Blocks World to test his scheme. The subject of dozens of research papers during the past few decades, Blocks World requires sorting and combining several stacks of colored blocks to match a taller stack.
Dubbed Hayek, Baum’s learning system consisted of a collection of modules—each one a little computer program—that, in effect, competed to make contributions to the solution of a given problem. In the evolving system, the modules making the biggest contributions would be the ones most likely to survive. The system would typically find a procedure for solving a big problem by breaking up the lengthy search for a solution into a sequence of achievable interim goals.
Hayek proved surprisingly successful at solving Blocks World puzzles, easily outperforming other machine-learning systems. Working with colleague Igor Durdanovic, Baum then turned to Rush Hour to see whether the same divide-and-conquer strategy would be as successful in another domain.
In a Rush Hour traffic jam, each vehicle on the six-by-six-square tray is one square wide and either two or three squares long. It can travel only backward and forward along the row in which it is initially placed and can’t change its orientation. The player’s goal is to clear a path for a designated car so it can reach the only exit on the grid. “Rush Hour seemed to be a good candidate [for testing Hayek] because of the finite configuration and the simple action space,” Flake notes.
While straightforward to play, Rush Hour turned out to be considerably more challenging to the computer than Blocks World is. “No one knew this at the time,” Flake says.
Flake and Baum had anecdotal evidence that Rush Hour is difficult. In certain Rush Hour puzzles, for example, it takes dozens of moves to free the target car, with some vehicles being forced to move back and forth many times. Such behavior reflects subtle and complicated interactions among the vehicles, even on a six-by-six grid, the researchers say.
In theoretical computer science, the difficulty of problems—or puzzles and games—is assessed in terms of the computational resources required to solve them. Roughly speaking, how much do the calculation time and memory needs increase as a problem gets bigger? For instance, what does it take to solve Rush Hour played on a 12-by-12 grid with more cars and trucks than the standard allocation?
To get a measure of Rush Hour’s computational complexity, Flake and Baum considered a generalized version of the puzzle—one in which the grid could be any number of cells wide and the single exit could be placed at any location on the grid’s perimeter. They focused on the computer resources that would be required to determine whether there’s a legal sequence of moves that permits the target car to exit.
The researchers proved that their generalized Rush Hour belongs to a category of computational problems described as PSPACE. This designation applies to challenges in which a computer can search through all the possibilities to find the answer using a reasonable amount of memory-defined in computer science as “a polynomial amount of space.” Such a search in the case of Rush Hour takes an amount of memory that increases algebraically with grid size, but the time required may increase exponentially.
Even more strikingly, Flake and Baum established that the seemingly simple Rush Hour is as difficult as any other problem that belongs in PSPACE. Rush Hour’s computational complexity is greater than that of Blocks World, whose time to a solution increases algebraically rather than exponentially. However, Rush Hour is less difficult than generalized chess, where both memory and time requirements increase exponentially as the board size gets bigger.
Inspired by the NEC work on Rush Hour, Robert A. Hearn and Erik D. Demaine of the Massachusetts Institute of Technology applied different mathematical formulations to the game and confirmed Rush Hour’s position in the hierarchy of computational difficulty.
Flake calls that work “a nice and natural extension that allows one to map many other [motion-planning] problems into a framework that is very similar to the one we originally developed. As such, it allows one to [establish] the computational properties of many other problems.”
Hearn and Demaine have already taken advantage of their techniques to show that a variety of sliding-block puzzles belongs in the PSPACE category, including simplified versions of Rush Hour in which all blocks are one-by-two rectangles, like dominoes, and can slide in any direction.
The researchers presented their findings at the International Colloquium on Automata, Languages, and Programming, held last month in Málaga, Spain.
Hearn and Demaine now hope to apply their techniques to several other motion-planning problems to determine their computational difficulty.
One project examines generalized chess. Given two configurations of chess pieces, each on a board that is a certain number of squares wide, is it possible to use legal moves to get from one configuration to the other?
Another project analyzes Lunar Lockout, a commercial puzzle in which robot-shaped blocks can slide along rows and columns of a grid until they collide with another block. The goal is to bring a particular robot to a specified position.
There’s an intriguing connection between computers and sliding-block puzzles that belong to the PSPACE category. “You can build computers out of them,” Hearn says. Indeed, he can imagine various arrangements of sliding blocks as logic components in a hypothetical computing machine.
A digital computer contains circuitry that enables it to perform various actions, from storing a bit of information to adding digits and sorting data. Fundamental to these operations are electronic gates for handling Boolean logic. For example, a so-called AND logic gate gives 1 as the output if two input signals are both 1, and it gives 0 as the output if the input signals are both 0 or one is 0 and the other is 1.
Flake and Baum demonstrated that certain block configurations in a variation on Rush Hour permit the same sort of logic operations. A sliding-block configuration doesn’t by itself perform a logic operation, however.
“It merely permits it to happen if, and only if, there exists a sequence of car moves that allow it to happen,” they point out.
Hearn and Demaine worked out a simpler, more flexible scheme for representing those operations. Within this framework, they defined sliding-block logic gates equivalent to AND gates and other logic components of conventional computers.
For example, they devised a logic gate from a nine-by-nine grid with two places that vehicles can enter and one exit. The AND operation is successful if both inputs begin a sequence of possible moves that releases a target car from the grid.
“We showed there are just a few different kinds of primitive gates that suffice for building computers,” Hearn says. In general, he adds, “if you’re allowed to use arbitrarily many blocks, you can make a sliding-block puzzle that can . . . solve any given problem that a [computer] can solve.”
Indeed, Flake adds, “if one wanted to build a physically realizable model of computation from nanoscale components, [a sliding-block puzzle] is a pretty good candidate because the cars need move only a very small distance to complete the computation.”
The fact that it’s possible to turn sliding-block puzzles into a model of computation underscores how versatile such puzzles can be. A Blocks World game, for instance, wouldn’t serve that purpose. The computational possibilities also suggest that developing an all-purpose strategy for solving sliding-block puzzles is out of reach. “Because we know that computer programs can do complicated things, we should not expect to find a simple theory of sliding-block puzzles,” Hearn argues.
Sliding-block puzzles have come a long way since puzzle maker Sam Loyd introduced the fiendish “14-15” puzzle to the United States in the 1870s. The puzzle consisted of 15 square tiles numbered from 1 to 15 in a square tray large enough to hold 16 tiles. Tiles 14 and 15 started out interchanged, and the player had to restore all the tiles to numerical order. The puzzle was a sensation, but no one could solve it. The sneaky truth was that Loyd’s puzzle was insoluble—something that contemporaneous mathematicians soon proved.
Inspired by Loyd’s example and seeking to tap into the public’s evident puzzle-solving frenzy, designers all over the world vied to create addictive pastimes of their own. Their sliding-block puzzles often featured differently shaped pieces in rectangular trays. The Donkey Puzzle, for instance, required a player to move a two-by-two block to a specified location in a four-by-five tray jammed with several unit squares and one-by-two rectangular blocks.
Japanese designer Nob Yoshigahara came up with the idea for Rush Hour in the late 1970s. In Japan, it initially appeared as a commercial product called Tokyo Parking Lot, and players were expected to solve tough traffic-jam challenges. The current U.S. edition offers puzzles that range in difficulty from “beginner to expert.”
Although mathematicians had developed a theory for solving any sliding-block puzzle in which the pieces are all uniform squares, they could not do so for puzzles in which the blocks have different shapes. Indeed, there still isn’t a general-purpose way—other than by trial and error—to determine whether it’s possible to go from one particular block arrangement to another given array. Even in individual cases, such as Rush Hour, where mathematicians can demonstrate that there’s a solution, they can’t readily determine the minimum number of moves to reach the desired pattern.
Today’s popular sliding-block puzzles—involving rectangular blocks instead of just squares—pose surprisingly formidable challenges for puzzle enthusiasts.
Moreover, they give scientists a new way of looking at what computers do. A sliding-domino puzzle, says Hearn, “is perhaps the simplest known physical system that exhibits computational properties.”
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