Academic literature on the topic 'Runway Rectangle Escape Problem'

A large body of work has demonstrated that parameterized artificial neural networks (ANNs) can efficiently describe ground states of numerous interesting quantum many-body Hamiltonians. However, the standard variational algorithms used to update or train the ANN parameters can get trapped in local minima, especially for frustrated systems and even if the representation is sufficiently expressive. We propose a parallel tempering method that facilitates escape from such local minima. This methods involves training multiple ANNs independently, with each simulation governed by a Hamiltonian with a different "driver" strength, in analogy to quantum parallel tempering, and it incorporates an update step into the training that allows for the exchange of neighboring ANN configurations. We study instances from two classes of Hamiltonians to demonstrate the utility of our approach using Restricted Boltzmann Machines as our parameterized ANN. The first instance is based on a permutation-invariant Hamiltonian whose landscape stymies the standard training algorithm by drawing it increasingly to a false local minimum. The second instance is four hydrogen atoms arranged in a rectangle, which is an instance of the second quantized electronic structure Hamiltonian discretized using Gaussian basis functions. We study this problem in a minimal basis set, which exhibits false minima that can trap the standard variational algorithm despite the problem’s small size. We show that augmenting the training with quantum parallel tempering becomes useful to finding good approximations to the ground states of these problem instances.

0. IntroductionIf we follow the line of much literature surrounding airports and urban mobility, the emphasis often falls on the fact that these spaces are designed to handle the mega-scale and super-human pace of mass transit. Airports have rightly been associated with velocity, as zones of rapid movement managed by enormous processing systems that guide bodies and things in transit (Pascoe; Pearman; Koolhaas; Gordon; Fuller & Harley). Yet this emphasis tends to ignore the spectrum of tempos and flows that are at play in airport terminals — from stillness to the much exalted hyper-rapidity of mobilized publics in the go-go world of commercial aviation.In this photo essay I'd like to pull a different thread and ask whether it's possible to think of aeromobility in terms of “uneven, differential mobility” (Bissell 280). What would it mean to consider waiting and stillness as forms of bodily engagement operating over a number of different scales and temporalities of movement and anticipation, without privileging speed over stillness? Instead of thinking mobility and stillness as diametrically opposed, can we instead conceive of them as occupying a number of different spatio-temporal registers in a dynamic range of mobility? The following is a provisional "visual ethnography" constructed from photographs of air terminal light boxes I have taken over the last five years (in Amsterdam, London, Chicago, Frankfurt, and Miami). Arranged into a "taxonomy of differentiality", each of these images comes from a slightly different angle, mode or directionality. Each view of these still images displayed in billboard-scale light-emitting devices suggests that there are multiple dimensions of visuality and bodily experience at play in these image-objects. The airport is characterized by an abundance of what appears to be empty space. This may be due to the sheer scale of mass transport, but it also arises from a system of active and non-active zones located throughout contemporary terminals. This photo series emphasises the "emptiness" of these overlooked left-over spaces that result from demands of circulation and construction.1. We Move the WorldTo many travellers, airport gate lounges and their surrounding facilities are loaded with a variety of contradictory associations and affects. Their open warehouse banality and hard industrial sterility tune our bodies to the vast technical and commercial systems that are imbricated through almost every aspect of contemporary everyday life.Here at the departure gate the traveller's body comes to a moment's rest. They are granted a short respite from the anxious routines of check in, body scans, security, information processing, passport scanning, itineraries, boarding procedures and wayfaring the terminal. The landside processing system deposits them at this penultimate point before final propulsion into the invisible airways that pipe them into their destination. We hear the broadcasting of boarding times, check-in times, name's of people that break them away from stillness, forcing people to move, to re-arrange themselves, or to hurry up. Along the way the passenger encounters a variety of techno-spatial experiences that sit at odds with the overriding discourse of velocity, speed and efficiency that lie at the centre of our social understanding of air travel. The airline's phantasmagorical projections of itself as guarantor and enabler of mass mobilities coincides uncomfortably with the passenger's own wish-fulfilment of escape and freedom.In this we can agree with the designer Bruce Mau when he suggests that these projection systems, comprised of "openings of every sort — in schedules, in urban space, on clothes, in events, on objects, in sightlines — are all inscribed with the logic of the market” (Mau 7). The advertising slogans and images everywhere communicate the dual concept that the aviation industry can deliver the world to us on time while simultaneously porting us to any part of the world still willing to accept Diners, VISA or American Express. At each point along the way these openings exhort us to stop, to wait in line, to sit still or to be patient. The weird geographies depicted by the light boxes appear like interpenetrating holes in space and time. These travel portals are strangely still, and only activated by the impending promise of movement.Be still and relax. Your destination is on its way. 2. Attentive AttentionAlongside the panoramic widescreen windows that frame the choreography of the tarmac and flight paths outside, appear luminous advertising light boxes. Snapped tightly to grid and locked into strategic sightlines and thoroughfares, these wall pieces are filled with a rotating menu of contemporary airport haiku and ersatz Swiss graphic design.Mechanically conditioned air pumped out of massive tubes creates the atmosphere for a very particular amalgam of daylight, tungsten, and fluorescent light waves. Low-oxygen-emitting indoor plants are no match for the diesel-powered plant rooms that maintain the constant flow of air to every nook and cranny of this massive processing machine. As Rem Koolhaas puts it, "air conditioning has launched the endless building. If architecture separates buildings, air conditioning unites them" (Koolhaas). In Koolhaas's lingo, these are complex "junkspaces" unifying, colliding and coalescing a number of different circulatory systems, temporalities and mobilities.Gillian Fuller reminds us there is a lot of stopping and going and stopping in the global circulatory system typified by air-terminal-space.From the packing of clothes in fixed containers to strapping your belt – tight and low – stillness and all its requisite activities, technologies and behaviours are fundamental to the ‘flow’ architectures that organize the motion of the globalizing multitudes of today (Fuller, "Store" 63). It is precisely this functional stillness organised around the protocols of store and forward that typifies digital systems, the packet switching of network cultures and the junkspace of airports alike.In these zones of transparency where everything is on view, the illuminated windows so proudly brought to us by J C Decaux flash forward to some idealized moment in the future. In this anticipatory moment, the passenger's every fantasy of in-flight service is attended to. The ultimate in attentiveness (think dimmed lights, soft pillows and comfy blankets), this still image is captured from an improbable future suspended behind the plywood and steel seating available in the moment —more reminiscent of park benches in public parks than the silver-service imagined for the discerning traveller.3. We Know ChicagoSelf-motion is itself a demonstration against the earth-binding weight of gravity. If we climb or fly, our defiance is greater (Appleyard 180).The commercial universe of phones, cameras, computer network software, financial instruments, and an array of fancy new gadgets floating in the middle of semi-forgotten transit spaces constitutes a singular interconnected commercial organism. The immense singularity of these claims to knowledge and power loom solemnly before us asserting their rights in the Esperanto of "exclusive rollover minutes", "nationwide long distance", "no roaming charges" and insider local knowledge. The connective tissue that joins one part of the terminal to a commercial centre in downtown Chicago is peeled away, revealing techno-veins and tendrils reaching to the sky. It's a graphic view that offers none of the spectacular openness and flights of fancy associated with the transit lounges located on the departure piers and satellites. Along these circulatory ribbons we experience the still photography and the designer's arrangement of type to attract the eye and lure the body. The blobby diagonals of the telco's logo blend seamlessly with the skyscraper's ribbons of steel, structural exoskeleton and wireless telecommunication cloud.In this plastinated anatomy, the various layers of commercially available techno-space stretch out before the traveller. Here we have no access to the two-way vistas made possible by the gigantic transparent tube structures of the contemporary air terminal. Waiting within the less travelled zones of the circulatory system we find ourselves suspended within the animating system itself. In these arteries and capillaries the flow is spread out and comes close to a halt in the figure of the graphic logo. We know Chicago is connected to us.In the digital logic of packet switching and network effects, there is no reason to privilege the go over the stop, the moving over the waiting. These light box portals do not mirror our bodies, almost at a complete standstill now. Instead they echo the commercial product world that they seek to transfuse us into. What emerges is a new kind of relational aesthetics that speaks to the complex corporeal, temporal, and architectural dimensions of stillness and movement in transit zones: like "a game, whose forms, patterns and functions develop and evolve according to periods and social contexts” (Bourriaud 11). 4. Machine in the CaféIs there a possible line of investigation suggested by the fact that sound waves become visible on the fuselage of jet planes just before they break the sound barrier? Does this suggest that the various human senses are translatable one into the other at various intensities (McLuhan 180)?Here, the technological imaginary contrasts itself with the techno alfresco dining area enclosed safely behind plate glass. Inside the cafes and bars, the best businesses in the world roll out their biggest guns to demonstrate the power, speed and scale of their network coverage (Remmele). The glass windows and light boxes "have the power to arrest a crowd around a commodity, corralling them in chic bars overlooking the runway as they wait for their call, but also guiding them where to go next" (Fuller, "Welcome" 164). The big bulbous plane sits plump in its hangar — no sound barriers broken here. It reassures us that our vehicle is somewhere there in the network, resting at its STOP before its GO. Peeking through the glass wall and sharing a meal with us, this interpenetrative transparency simultaneously joins and separates two planar dimensions — machinic perfection on one hand, organic growth and death on the other (Rowe and Slutsky; Fuller, "Welcome").Bruce Mau is typical in suggesting that the commanding problem of the twentieth century was speed, represented by the infamous image of a US Navy Hornet fighter breaking the sound barrier in a puff of smoke and cloud. It has worked its way into every aspect of the design experience, manufacturing, computation and transport.But speed masks more than it reveals. The most pressing problem facing designers and citizens alike is growth — from the unsustainable logic of infinite growth in GDP to the relentless application of Moore's Law to the digital networks and devices that define contemporary society in the first world. The shift of emphasis from speed to growth as a time-based event with breaking points and moments of rupture has generated new possibilities. "Growth is nonlinear and unpredictable ... Few of us are ready to admit that growth is constantly shadowed by its constitutive opposite, that is equal partners with death” (Mau 497).If speed in part represents a flight from death (Virilio), growth invokes its biological necessity. In his classic study of the persistence of the pastoral imagination in technological America, The Machine in the Garden, Leo Marx charted the urge to idealize rural environments at the advent of an urban industrialised America. The very idea of "the flight from the city" can be understood as a response to the onslaught of technological society and it's deathly shadow. Against the murderous capacity of technological society stood the pastoral ideal, "incorporated in a powerful metaphor of contradiction — a way of ordering meaning and value that clarifies our situation today" (Marx 4). 5. Windows at 35,000 FeetIf waiting and stillness are active forms of bodily engagement, we need to consider the different layers of motion and anticipation embedded in the apprehension of these luminous black-box windows. In The Virtual Window, Anne Friedberg notes that the Old Norse derivation of the word window “emphasizes the etymological root of the eye, open to the wind. The window aperture provides ventilation for the eye” (103).The virtual windows we are considering here evoke notions of view and shelter, open air and sealed protection, both separation from and connection to the outside. These windows to nowhere allow two distinct visual/spatial dimensions to interface, immediately making the visual field more complex and fragmented. Always simultaneously operating on at least two distinct fields, windows-within-windows provide a specialized mode of spatial and temporal navigation. As Gyorgy Kepes suggested in the 1940s, the transparency of windows "implies more than an optical characteristic; it implies a broader spatial order. Transparency means a simultaneous perception of different spatial locations" (Kepes 77).The first windows in the world were openings in walls, without glass and designed to allow air and light to fill the architectural structure. Shutters were fitted to control air flow, moderate light and to enclose the space completely. It was not until the emergence of glass technologies (especially in Holland, home of plate glass for the display of commercial products) that shielding and protection also allowed for unhindered views (by way of transparent glass). This gives rise to the thesis that windows are part of a longstanding architectural/technological system that moderates the dual functions of transparency and separation. With windows, multi-dimensional planes and temporalities can exist in the same time and space — hence a singular point of experience is layered with many other dimensions. Transparency and luminosity "ceases to be that which is perfectly clear and becomes instead that which is clearly ambiguous" (Rowe and Slutsky 45). The light box air-portals necessitate a constant fluctuation and remediation that is at once multi-planar, transparent and "hard to read". They are informatic.From holes in the wall to power lunch at 35,000 feet, windows shape the manner in which light, information, sights, smells, temperature and so on are modulated in society. "By allowing the outside in and the inside out, [they] enable cosmos and construction to innocently, transparently, converge" (Fuller, "Welcome" 163). Laptop, phone, PDA and light box point to the differential mobilities within a matrix that traverses multiple modes of transparency and separation, rest and flight, stillness and speed.6. Can You Feel It?Increasingly the whole world has come to smell alike: gasoline, detergents, plumbing, and junk foods coalesce into the catholic smog of our age (Illich 47).In these forlorn corners of mobile consumption, the dynamic of circulation simultaneously slows and opens out. The surfaces of inscription implore us to see them at precisely the moment we feel unseen, unguided and off-camera. Can you see it, can you feel it, can you imagine the unimaginable, all available to us on demand? Expectation and anticipation give us something to look forward to, but we're not sure we want what's on offer.Air travel radicalizes the separation of the air traveller from ground at one instance and from the atmosphere at another. Air, light, temperature and smell are all screened out or technologically created by the terminal plant and infrastructure. The closer the traveller moves towards stillness, the greater the engagement with senses that may have been ignored by the primacy of the visual in so much of this circulatory space. Smell, hunger, tiredness, cold and hardness cannot be screened out.In this sense, the airplanes we board are terminal extensions, flying air-conditioned towers or groundscrapers jet-propelled into highways of the air. Floating above the horizon, immersed in a set of logistically ordained trajectories and pressurized bubbles, we look out the window and don't see much at all. Whatever we do see, it's probably on the screen in front of us which disconnects us from one space-time-velocity at the same time that it plugs us into another set of relations. As Koolhaas says, junkspace is "held together not by structure, but by skin, like a bubble" (Koolhaas). In these distended bubbles, the traveler momentarily occupies an uncommon transit space where stillness is privileged and velocity is minimized. The traveler's body itself is "engaged in and enacting a whole kaleidoscope of different everyday practices and forms" during the course of this less-harried navigation (Bissell 282).7. Elevator MusicsThe imaginary wheel of the kaleidoscope spins to reveal a waiting body-double occupying the projected territory of what appears to be a fashionable Miami. She's just beyond our reach, but beside her lies a portal to another dimension of the terminal's vascular system.Elevators and the networks of shafts and vents that house them, are to our buildings like veins and arteries to the body — conduits that permeate and structure the spaces of our lives while still remaining separate from the fixity of the happenings around them (Garfinkel 175). The terminal space contains a number of apparent cul-de-sacs and escape routes. Though there's no background music piped in here, another soundtrack can be heard. The Muzak corporation may douse the interior of the elevator with its own proprietary aural cologne, but at this juncture the soundscape is more "open". This functional shifting of sound from figure to ground encourages peripheral hearing, providing "an illusion of distended time", sonically separated from the continuous hum of "generators, ventilation systems and low-frequency electrical lighting" (Lanza 43).There is another dimension to this acoustic realm: “The mobile ecouteur contracts the flows of information that are supposed to keep bodies usefully and efficiently moving around ... and that turn them into functions of information flows — the speedy courier, the networking executive on a mobile phone, the scanning eyes of the consumer” (Munster 18).An elevator is a grave says an old inspector's maxim, and according to others, a mechanism to cross from one world to another. Even the quintessential near death experience with its movement down a long illuminated tunnel, Garfinkel reminds us, “is not unlike the sensation of movement we experience, or imagine, in a long swift elevator ride” (Garfinkel 191).8. States of SuspensionThe suspended figure on the screen occupies an impossible pose in an impossible space: half falling, half resting, an anti-angel for today's weary air traveller. But it's the same impossible space revealed by the airport and bundled up in the experience of flight. After all, the dimension this figures exists in — witness the amount of activity in his suspension — is almost like a black hole with the surrounding universe collapsing into it. The figure is crammed into the light box uncomfortably like passengers in the plane, and yet occupies a position that does not exist in the Cartesian universe.We return to the glossy language of advertising, its promise of the external world of places and products delivered to us by the image and the network of travel. (Remmele) Here we can go beyond Virilio's vanishing point, that radical reversibility where inside and outside coincide. Since everybody has already reached their destination, for Virilio it has become completely pointless to leave: "the inertia that undermines your corporeity also undermines the GLOBAL and the LOCAL; but also, just as much, the MOBILE and the IMMOBILE” (Virilio 123; emphasis in original).In this clinical corner of stainless steel, glass bricks and exit signs hangs an animated suspension that articulates the convergence of a multitude of differentials in one image. Fallen into the weirdest geometry in the world, it's as if the passenger exists in a non-place free of all traces. Flows and conglomerates follow one another, accumulating in the edges, awaiting their moment to be sent off on another trajectory, occupying so many spatio-temporal registers in a dynamic range of mobility.ReferencesAppleyard, Donald. "Motion, Sequence and the City." The Nature and Art of Motion. Ed. Gyorgy Kepes. New York: George Braziller, 1965. Adey, Peter. "If Mobility Is Everything Then It Is Nothing: Towards a Relational Politics of (Im)mobilities." Mobilities 1.1 (2006): 75–95. Bissell, David. “Animating Suspension: Waiting for Mobilities.” Mobilities 2.2 (2007): 277-298.Bourriaud, Nicolas. Relational Aesthetics. Trans. Simon Pleasance and Fronza Woods. Paris: Les Presses du Reel, 2002. Classen, Constance. “The Deodorized City: Battling Urban Stench in the Nineteenth Century.” Sense of the City: An Alternate Approach to Urbanism. Ed. Mirko Zardini. Baden: Lars Muller Publishers, 2005. 292-322. Friedberg, Anne. The Virtual Window: From Alberti to Microsoft. Cambridge: MIT P, 2006. Fuller, Gillian, and Ross Harley. Aviopolis: A Book about Airports. London: Black Dog Publishing, 2005. Fuller, Gillian. "Welcome to Windows: Motion Aesthetics at the Airport." Ed. Mark Salter. Politics at the Airport. Minnesota: U of Minnesota P, 2008. –––. "Store Forward: Architectures of a Future Tense". Ed. John Urry, Saolo Cwerner, Sven Kesselring. Air Time Spaces: Theory and Method in Aeromobilities Research. London: Routledge, 2008. 63-75.Garfinkel, Susan. “Elevator Stories: Vertical Imagination and the Spaces of Possibility.” Up Down Across: Elevators, Escalators, and Moving Sidewalks. Ed. Alisa Goetz. London: Merrell, 2003. 173-196. Gordon, Alastair. Naked Airport: A Cultural History of the World's Most Revolutionary Structure. New York: Metropolitan, 2004.Illich, Ivan. H2O and the Waters of Forgetfulness: Reflections on the Historicity of Stuff. Dallas: Dallas Institute of Humanities and Culture, 1985. Kepes, Gyorgy. Language of Vision. New York: Dover Publications, 1995 (1944). Koolhass, Rem. "Junkspace." Content. 6 Mar. 2009 ‹http://www.btgjapan.org/catalysts/rem.html›.Lanza, Joseph. "The Sound of Cottage Cheese (Why Background Music Is the Real World Beat!)." Performing Arts Journal 13.3 (Sep. 1991): 42-53. McLuhan, Marshall. “Is It Natural That One Medium Should Appropriate and Exploit Another.” McLuhan: Hot and Cool. Ed. Gerald Emanuel Stearn. Middlesex: Penguin, 1967. 172-182. Marx, Leo. The Machine in the Garden: Technology and the Pastoral Ideal in America. London: Oxford U P, 1964. Mau, Bruce. Life Style. Ed. Kyo Maclear with Bart Testa. London: Phaidon, 2000. Munster, Anna. Materializing New Media: Embodiment in Information Aesthetics. New England: Dartmouth, 2006. Pascoe, David. Airspaces. London: Reaktion, 2001. Pearman, Hugh. Airports: A Century of Architecture. New York: Abrams, 2004. Remmele, Mathias. “An Invitation to Fly: Poster Art in the Service of Civilian Air Travel.” Airworld: Design and Architecture for Air Travel. Ed. Alexander von Vegesack and Jochen Eisenbrand. Weil am Rhein: Vitra Design Museum, 2004. 230-262. Rowe, Colin, and Robert Slutsky. Transparency: Literal and Phenomenal. Perspecta 8 (1963): 45-54. Virilio, Paul. City of Panic. Trans. Julie Rose. Oxford: Berg, 2005.

We study a host of geometric optimization problems that are NP-hard and design polynomial time approximation algorithms for them. More precisely, we are given a family of geometric objects and a point set, mostly in the plane, and study different variants and generalizations of Packing and Covering problems. Our objects of study are mostly family of non-piercing regions in the plane. We call a set of simple and connected regions to be non-piercing if for any pair of intersecting regions, A and B both A\B and B\A are connected regions. A set of disks, squares, half-planes are examples of non-piercing regions, whereas, a set of lines, rectangles are examples of piercing objects. For most of the problems we have studied, a simple local search algorithm is enough to yield a PTAS whose analysis of approximation requires a suitable bipartite graph on the local search solution and the optimal solution to have a balanced sub-linear separator. We study a generalization of the standard packing problem, called the Capacitated Region Packing problem and its slight variant, the Shallow Packing problem. We devise a PTAS for both these problems with restrictions on the capacities. For the former problem, the objects are non-piercing whereas for the latter problem the objects can be even more general and only have sub-quadratic union complexity with the capacities at most some constant for both the cases. The non-triviality here is to show that the intersection graph of arrangements with shallow depth, which is not planar, has balanced sub-linear separators. Our results complement the Maximum Independent Set of Rectangles problem as rectangles are both piercing and have quadratic union complexity. We also study the Shallow Point Packing problem and are able to show that local search works here as well for unit capacity and devise a constant factor approximation algorithm using an adaptation Brannaman-Goodrich technique for packing problems. Runaway Rectangle Escape problem is closely related to the above packing problems and is motivated from routing in printed circuit boards. Here we are given a set of axis-parallel rectangles inside a rectangular boundary R and a maximum allowed depth d. The objective is to extend the maximum number of input rectangles to one of the four sides of R such that the maximum depth of a point is at most d after extension. We show that local search gives a (2 + )-approximation for d = O(1). When the input rectangles are all disjoint then we devise a simple 4(1 + 1=(d 􀀀 1))-approximation algorithm. We also propose a randomized (1 + )-approximation algorithm based on randomized rounding making some density assumptions. Lastly, we show the problem to be NP-hard even when the rectangles are unit squares aligned in a grid. We study the Multi-Cover problem which is a generalization of the Set Cover problem. We give a PTAS for non-piercing regions when the depth of every point is at most constant. We also study different variants of the covering problem like the Unique Coverage, and Prize Collecting Set Cover problem. For Unique Cover we show that local search yields a PTAS for non-piercing regions for bounded depth and degree. For Prize Collecting Set Cover a PTAS works for non-piercing regions if the weight of every region is within a range [1; a], where a is some constant. Lastly, we consider variants of the Art Gallery problems called the Minimum (Horizontal) Sliding Cameras problem, M(H)SC. We are given an orthogonal polygon and we need to deploy mobile guards who can walk along an orthogonal (horizontal) line segment and can guard a point inside the polygon if the perpendicular drawn from the point onto the line segment lies inside the polygon. Our local search algorithm yields a PTAS for the MHSC problem and also the MSC problem when the polygon has no holes. In order to do so we prove an appropriate graph on orthogonal line segments to be planar by proposing a graph drawing scheme.