New Proof Dramatically Compresses Space Needed for Computation
Once upon a time computers filled entire rooms, reading numbers from spinning tapes and churning them through wires to do chains of basic arithmetic. Today they slip into our pockets, performing in a tiny fraction of a second what used to take hours. But even as chips shrink and gain speed, theorists are flipping the question from how much computation space we can pack into a machine to how little is enough to get the job done.
This inquiry lies at the heart of computational complexity, a measure of the limits of what problems can be solved and at what cost in time and space. For nearly 50 years theorists believed that if solving a problem takes t steps, it should also need roughly t bits of memory—the 0s and 1s that a machine uses to record information. (Technically, that equation was t/ log(t), but for the numbers involved log(t) is typically negligibly small.) If a task involves 100 steps, for instance, you'd expect to need at least 100 bits, enough to diligently log each step. Using fewer bits was thought to require more steps—like alphabetizing your books by swapping them one by one on the shelf instead of pulling them all out and reshelving them. But in a surprising finding described this week at the ACM Symposium on Theory of Computing in Prague, Massachusetts Institute of Technology computer scientist Ryan Williams found that any problem solvable in time t needs only about √ t bits of memory: a 100-step computation could be compressed and solved with something on the order of 10 bits. 'This result shows the prior intuition is completely false,' Williams says. 'I thought there must be something wrong [with the proof] because this is extremely unexpected.'
The breakthrough relies on a 'reduction,' a means of transforming one problem into another that may seem unrelated but is mathematically equivalent. With reductions, packing a suitcase maps onto determining a monthly budget: the size of your suitcase represents your total budget, pieces of clothing correspond to potential expenses, and carefully deciding which clothes can fit is like allocating your budget. Solving one problem would then directly solve the other. This idea is at the core of Williams's result: any problem can be transformed into one you can solve by cleverly reusing space, deftly cramming the necessary information into just a square-root number of bits. Thus, the original problem must be solvable with this compact container.
On supporting science journalism
If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
'This progress is unbelievable,' says Mahdi Cheraghchi, a computer scientist at the University of Michigan. 'Before this result, there were problems you could solve in a certain amount of time, but many thought you couldn't do so with such little space.' Williams's finding, he adds, is 'a step in the right direction that we didn't know how to take.'
While computers have continued to shrink, our theoretical understanding of their efficiency has exploded, suggesting that the real constraint is not how much memory we have but how wisely we use it.
Hashtags
Try Our AI Features
Explore what Daily8 AI can do for you:
Comments
No comments yet...
Related Articles
Yahoo
19 hours ago
- Yahoo
Toxic algae blooms are lasting longer in Lake Erie − why that's a worry for people and pets
Lake Erie algal blooms, August 2011, along the southeast Lake Erie shore of Pelee Island, Ontario, Canada, 5 miles north of the international line. | Michigan Sea Grant Gregory J. Dick, University of Michigan Federal scientists released their annual forecast for Lake Erie's harmful algal blooms on June 26, 2025, and they expect a mild to moderate season. However, anyone who comes in contact with the blooms can face health risks, and it's worth remembering that 2014, when toxins from algae blooms contaminated the water supply in Toledo, Ohio, was considered a moderate year, too. The Conversation asked Gregory J. Dick, who leads the Cooperative Institute for Great Lakes Research, a federally funded center at the University of Michigan that studies harmful algal blooms among other Great Lakes issues, why they're such a concern. bulletin_current 1. What causes harmful algal blooms? Harmful algal blooms are dense patches of excessive algae growth that can occur in any type of water body, including ponds, reservoirs, rivers, lakes and oceans. When you see them in freshwater, you're typically seeing cyanobacteria, also known as blue-green algae. These photosynthetic bacteria have inhabited our planet for billions of years. In fact, they were responsible for oxygenating Earth's atmosphere, which enabled plant and animal life as we know it. Algae are natural components of ecosystems, but they cause trouble when they proliferate to high densities, creating what we call blooms. Harmful algal blooms form scums at the water surface and produce toxins that can harm ecosystems, water quality and human health. They have been reported in all 50 U.S. states, all five Great Lakes and nearly every country around the world. Blue-green algae blooms are becoming more common in inland waters. The main sources of harmful algal blooms are excess nutrients in the water, typically phosphorus and nitrogen. Historically, these excess nutrients mainly came from sewage and phosphorus-based detergents used in laundry machines and dishwashers that ended up in waterways. U.S. environmental laws in the early 1970s addressed this by requiring sewage treatment and banning phosphorus detergents, with spectacular success. Today, agriculture is the main source of excess nutrients from chemical fertilizer or manure applied to farm fields to grow crops. Rainstorms wash these nutrients into streams and rivers that deliver them to lakes and coastal areas, where they fertilize algal blooms. In the U.S., most of these nutrients come from industrial-scale corn production, which is largely used as animal feed or to produce ethanol for gasoline. Climate change also exacerbates the problem in two ways. First, cyanobacteria grow faster at higher temperatures. Second, climate-driven increases in precipitation, especially large storms, cause more nutrient runoff that has led to record-setting blooms. 2. What does your team's DNA testing tell us about Lake Erie's harmful algal blooms? Harmful algal blooms contain a mixture of cyanobacterial species that can produce an array of different toxins, many of which are still being discovered. When my colleagues and I recently sequenced DNA from Lake Erie water, we found new types of microcystins, the notorious toxins that were responsible for contaminating Toledo's drinking water supply in 2014. These novel molecules cannot be detected with traditional methods and show some signs of causing toxicity, though further studies are needed to confirm their human health effects. We also found organisms responsible for producing saxitoxin, a potent neurotoxin that is well known for causing paralytic shellfish poisoning on the Pacific Coast of North America and elsewhere. Saxitoxins have been detected at low concentrations in the Great Lakes for some time, but the recent discovery of hot spots of genes that make the toxin makes them an emerging concern. Our research suggests warmer water temperatures could boost its production, which raises concerns that saxitoxin will become more prevalent with climate change. However, the controls on toxin production are complex, and more research is needed to test this hypothesis. Federal monitoring programs are essential for tracking and understanding emerging threats. 3. Should people worry about these blooms? Harmful algal blooms are unsightly and smelly, making them a concern for recreation, property values and businesses. They can disrupt food webs and harm aquatic life, though a recent study suggested that their effects on the Lake Erie food web so far are not substantial. But the biggest impact is from the toxins these algae produce that are harmful to humans and lethal to pets. The toxins can cause acute health problems such as gastrointestinal symptoms, headache, fever and skin irritation. Dogs can die from ingesting lake water with harmful algal blooms. Emerging science suggests that long-term exposure to harmful algal blooms, for example over months or years, can cause or exacerbate chronic respiratory, cardiovascular and gastrointestinal problems and may be linked to liver cancers, kidney disease and neurological issues. In addition to exposure through direct ingestion or skin contact, recent research also indicates that inhaling toxins that get into the air may harm health, raising concerns for coastal residents and boaters, but more research is needed to understand the risks. The Toledo drinking water crisis of 2014 illustrated the vast potential for algal blooms to cause harm in the Great Lakes. Toxins infiltrated the drinking water system and were detected in processed municipal water, resulting in a three-day 'do not drink' advisory. The episode affected residents, hospitals and businesses, and it ultimately cost the city an estimated US$65 million. 4. Blooms seem to be starting earlier in the year and lasting longer – why is that happening? Warmer waters are extending the duration of the blooms. In 2025, NOAA detected these toxins in Lake Erie on April 28, earlier than ever before. The 2022 bloom in Lake Erie persisted into November, which is rare if not unprecedented. Scientific studies of western Lake Erie show that the potential cyanobacterial growth rate has increased by up to 30% and the length of the bloom season has expanded by up to a month from 1995 to 2022, especially in warmer, shallow waters. These results are consistent with our understanding of cyanobacterial physiology: Blooms like it hot – cyanobacteria grow faster at higher temperatures. 5. What can be done to reduce the likelihood of algal blooms in the future? The best and perhaps only hope of reducing the size and occurrence of harmful algal blooms is to reduce the amount of nutrients reaching the Great Lakes. In Lake Erie, where nutrients come primarily from agriculture, that means improving agricultural practices and restoring wetlands to reduce the amount of nutrients flowing off of farm fields and into the lake. Early indications suggest that Ohio's H2Ohio program, which works with farmers to reduce runoff, is making some gains in this regard, but future funding for H2Ohio is uncertain. In places like Lake Superior, where harmful algal blooms appear to be driven by climate change, the solution likely requires halting and reversing the rapid human-driven increase in greenhouse gases in the atmosphere. Gregory J. Dick, Professor of Biology, University of Michigan This article is republished from The Conversation under a Creative Commons license. Read the original article.
Scientific American
a day ago
- Scientific American
New Proof Dramatically Compresses Space Needed for Computation
Once upon a time computers filled entire rooms, reading numbers from spinning tapes and churning them through wires to do chains of basic arithmetic. Today they slip into our pockets, performing in a tiny fraction of a second what used to take hours. But even as chips shrink and gain speed, theorists are flipping the question from how much computation space we can pack into a machine to how little is enough to get the job done. This inquiry lies at the heart of computational complexity, a measure of the limits of what problems can be solved and at what cost in time and space. For nearly 50 years theorists believed that if solving a problem takes t steps, it should also need roughly t bits of memory—the 0s and 1s that a machine uses to record information. (Technically, that equation was t/ log(t), but for the numbers involved log(t) is typically negligibly small.) If a task involves 100 steps, for instance, you'd expect to need at least 100 bits, enough to diligently log each step. Using fewer bits was thought to require more steps—like alphabetizing your books by swapping them one by one on the shelf instead of pulling them all out and reshelving them. But in a surprising finding described this week at the ACM Symposium on Theory of Computing in Prague, Massachusetts Institute of Technology computer scientist Ryan Williams found that any problem solvable in time t needs only about √ t bits of memory: a 100-step computation could be compressed and solved with something on the order of 10 bits. 'This result shows the prior intuition is completely false,' Williams says. 'I thought there must be something wrong [with the proof] because this is extremely unexpected.' The breakthrough relies on a 'reduction,' a means of transforming one problem into another that may seem unrelated but is mathematically equivalent. With reductions, packing a suitcase maps onto determining a monthly budget: the size of your suitcase represents your total budget, pieces of clothing correspond to potential expenses, and carefully deciding which clothes can fit is like allocating your budget. Solving one problem would then directly solve the other. This idea is at the core of Williams's result: any problem can be transformed into one you can solve by cleverly reusing space, deftly cramming the necessary information into just a square-root number of bits. Thus, the original problem must be solvable with this compact container. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. 'This progress is unbelievable,' says Mahdi Cheraghchi, a computer scientist at the University of Michigan. 'Before this result, there were problems you could solve in a certain amount of time, but many thought you couldn't do so with such little space.' Williams's finding, he adds, is 'a step in the right direction that we didn't know how to take.' While computers have continued to shrink, our theoretical understanding of their efficiency has exploded, suggesting that the real constraint is not how much memory we have but how wisely we use it.
Yahoo
3 days ago
- Yahoo
University of Michigan and NASA partner with high school students to track solar storms
From left, David Greene of Skyline High School in Ann Arbor, and Mojtaba Akhavan-Tafti, a research faculty member in climate and space sciences and engineering (CLaSP), help Lillian Cui, Yufei Fu, Hailey Chung, and other students assemble the SunRISE Ground Radio Lab kit at the M-Air outdoor lab on the North Campus of the University of Michigan in Ann Arbor on May 17, 2023. | Photo by Brenda Ahearn/ University of Michigan, College of Engineering, Communications and Marketing In coordination with researchers from NASA and the University of Michigan, high school students across the country are playing a key role in building our understanding of solar disturbances carrying potential threats to satellites and the power grid. Utilizing $500 antenna kits designed by U of M, students at 18 schools across eight states and Puerto Rico have been working to monitor solar radio bursts, an earlier indicator of geomagnetic storms, as part of the SunRISE Ground Radio Lab. Launched in August 2023, the collaboration has already yielded results, with a study published in Earth and Space Science Wednesday analyzing the project's early findings. Mojtaba Akhavan-Tafti, an associate research scientist at U of M who leads the lab, told Michigan Advance that the project began as a program for both undergraduate and graduate students who were tasked with designing and building their own antenna and taking measurements, which they used for scientific analysis. Since the class started in 2021, it's seen a strong turnout of students, Akhavan-Tafti said, starting with five or six students and growing to 21 to 22 students a semester. SUBSCRIBE: GET THE MORNING HEADLINES DELIVERED TO YOUR INBOX 'Later on, NASA came back to us and said, 'Hey, we know that this program has been very productive. A lot of students speak very highly of it. Is there any chance that you could involve high school students in your project, because, you know, they're at the start of their careers, they're trying to figure out what to do with their lives, and maybe they would be very interested in getting involved?'' Akhavan-Tafti said. However, the antennas students were building cost around $25,000 a unit, with shipping and assembly also presenting concerns. So they put the challenge to the college students, who developed an antenna that is simpler to install with a much smaller $500 price tag. With NASA covering the cost of the antenna kits, and another one of its programs, Radio JOVE, supplying them, the university began reaching out to high schools to see if they would be interested in participating in the lab program, later working with NASA's educational programs to partner with schools across the country. 'When we reached out to high schools originally to start the program, we had two requirements: One is that the school needs to be okay with installing the antenna on campus, and two that they need to be…collecting and uploading data on a regular basis, on a weekly basis, to our data repository,' Akhavan-Tafti said. After receiving the antenna, students work through an online training module introducing students to the field of radio astronomy, explaining what the antennas do and how to assemble them and breaking down what the data collected looks like and how it can be used scientifically. Afterwards, an entire class would often work to install the antenna on campus, with the university offering guidance on how to collect and upload data, Akhavan-Tafti said. The goal of the program is to engage students in science, technology, engineering, arts and math, or STEAM activities, with hopes of helping students realize that it's not rocket science to get involved in the space industry, Akhavan-Tafti explained. 'It's a rapidly growing industry that needs people who are inspired and are ready for the challenge. And why not the next generation?' he said. Alongside tracking low-frequency radio bursts from the sun, the program also held two observation campaigns, capturing data during the previous October 2023 annular solar eclipse and the total solar eclipse in April 2024. While many of the high schools have completely automated their data collection process, the students have also been very helpful in characterizing the data, Akhavan-Tafti said. In addition to sifting through data to separate radio signals from other noise, the students have also been slowly working towards automating the categorization of different types of signals, he said. As students monitor these signals from the ground, NASA has also been working to launch a set of six toaster-sized satellites to observe solar radio emissions from orbit. Akhavan-Tafti explained that these bursts of radio signals are tied to phenomena like solar flares and coronal mass ejections, large bubbles of plasma which carry a magnetic field. When these ejections occur, they often take eight days to reach the earth where they can create geomagnetic storms. However, the radio signals associated with these events travel at the speed of light and arrive at the earth much earlier, Akhavan-Tafti explained, acting as a warning system for geomagnetic storms. While these storms are responsible for auroras, they can also disrupt signals, Akhavan-Tafti noted, pointing to $500 million in lost agricultural harvests in the U.S. last year due to automated tractors that rely on GPS going off track. Additionally, geomagnetic storms can accelerate charged particles to very high speeds, potentially harming satellites and creating health and safety concerns for astronauts, Akahavan-Tafti said. Geomagnetic storms can also create radio blackouts and power outages, according to NASA. For students participating at the college level, they gain experience listening to a customer, designing a system to meet their needs, testing it in multiple environments to ensure it is functional and shipping out their prototype, Akhavan-Tafti explained. For high school students, the online training introduces them to radio astronomy, and exposes them to a number of concepts, including the various types of engineering and resources for getting involved in other science, technology, engineering and math based fields, he said. Students also learn how to work together to assemble their antenna and after completing the project, they receive a certificate from NASA which they can show while applying for university, Akhavan-Tafti said. 'Beyond all of that, the data collection and analysis enables them to apply some of the science and math and engineering that they've learned in the classroom to real world problems to contribute to furthering our understanding of our universe,' Akhavan-Tafti said. Additionally as the space industry grows, this field will be where many students find themselves working, Akhavan-Tafti said, noting his focus is to teach students early on in their careers about the opportunities that are out there. 'They could be the next scientists, they could be the next engineers, they could be the next policy makers, or even entrepreneurs who start their own businesses doing space-related activities. … we want the hardest working and smartest kids to start thinking about space and this space needs them,' he said.
