The knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items to include in the collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. The knapsack problem has been studied for more than a century, with early works dating back to 1897. The subset sum problem is a special case of the decision and 0-1 problems where for each kind of item, the weight equals the value: w i = v i {\displaystyle w_{i}=v_{i}} . In the field of cryptography, the term knapsack problem is often used to refer specifically to the subset sum problem. The subset sum problem is one of Karp's 21 NP-complete problems. == Applications == Knapsack problems appear in real-world decision-making processes in a wide variety of fields, such as finding the least wasteful way to cut raw materials, selection of investments and portfolios, selection of assets for asset-backed securitization, and generating keys for the Merkle–Hellman and other knapsack cryptosystems. One early application of knapsack algorithms was in the construction and scoring of tests in which the test-takers have a choice as to which questions they answer. For small examples, it is a fairly simple process to provide the test-takers with such a choice. For example, if an exam contains 12 questions each worth 10 points, the test-taker need only answer 10 questions to achieve a maximum possible score of 100 points. However, on tests with a heterogeneous distribution of point values, it is more difficult to provide choices. Feuerman and Weiss proposed a system in which students are given a heterogeneous test with a total of 125 possible points. The students are asked to answer all of the questions to the best of their abilities. Of the possible subsets of problems whose total point values add up to 100, a knapsack algorithm would determine which subset gives each student the highest possible score. A 1999 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems related to the field of combinatorial algorithms and algorithm engineering, the knapsack problem was the 19th most popular and the third most needed after suffix trees and the bin packing problem. == Definition == The most common problem being solved is the 0-1 knapsack problem, which restricts the number x i {\displaystyle x_{i}} of copies of each kind of item to zero or one. Given a set of n {\displaystyle n} items numbered from 1 up to n {\displaystyle n} , each with a weight w i {\displaystyle w_{i}} and a value v i {\displaystyle v_{i}} , along with a maximum weight capacity W {\displaystyle W} , maximize ∑ i = 1 n v i x i {\displaystyle \sum _{i=1}^{n}v_{i}x_{i}} subject to ∑ i = 1 n w i x i ≤ W {\displaystyle \sum _{i=1}^{n}w_{i}x_{i}\leq W} and x i ∈ { 0 , 1 } {\displaystyle x_{i}\in \{0,1\}} . Here x i {\displaystyle x_{i}} represents the number of instances of item i {\displaystyle i} to include in the knapsack. Informally, the problem is to maximize the sum of the values of the items in the knapsack so that the sum of the weights is less than or equal to the knapsack's capacity. The bounded knapsack problem (BKP) removes the restriction that there is only one of each item, but restricts the number x i {\displaystyle x_{i}} of copies of each kind of item to a maximum non-negative integer value c {\displaystyle c} : maximize ∑ i = 1 n v i x i {\displaystyle \sum _{i=1}^{n}v_{i}x_{i}} subject to ∑ i = 1 n w i x i ≤ W {\displaystyle \sum _{i=1}^{n}w_{i}x_{i}\leq W} and x i ∈ { 0 , 1 , 2 , … , c } . {\displaystyle x_{i}\in \{0,1,2,\dots ,c\}.} The unbounded knapsack problem (UKP) places no upper bound on the number of copies of each kind of item and can be formulated as above except that the only restriction on x i {\displaystyle x_{i}} is that it is a non-negative integer. maximize ∑ i = 1 n v i x i {\displaystyle \sum _{i=1}^{n}v_{i}x_{i}} subject to ∑ i = 1 n w i x i ≤ W {\displaystyle \sum _{i=1}^{n}w_{i}x_{i}\leq W} and x i ∈ N . {\displaystyle x_{i}\in \mathbb {N} .} One example of the unbounded knapsack problem is given using the figure shown at the beginning of this article and the text "if any number of each book is available" in the caption of that figure. == Computational complexity == The knapsack problem is interesting from the perspective of computer science for many reasons: The decision problem form of the knapsack problem (Can a value of at least V be achieved without exceeding the weight W?) is NP-complete, thus there is no known algorithm that is both correct and fast (polynomial-time) in all cases. There is no known polynomial algorithm which can tell, given a solution, whether it is optimal (which would mean that there is no solution with a larger V). This problem is co-NP-complete. There is a pseudo-polynomial time algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine, described below. Many cases that arise in practice, and "random instances" from some distributions, can nonetheless be solved exactly. There is a link between the "decision" and "optimization" problems in that if there exists a polynomial algorithm that solves the "decision" problem, then one can find the maximum value for the optimization problem in polynomial time by applying this algorithm iteratively while increasing the value of k. On the other hand, if an algorithm finds the optimal value of the optimization problem in polynomial time, then the decision problem can be solved in polynomial time by comparing the value of the solution output by this algorithm with the value of k. Thus, both versions of the problem are of similar difficulty. One theme in research literature is to identify what the "hard" instances of the knapsack problem look like, or viewed another way, to identify what properties of instances in practice might make them more amenable than their worst-case NP-complete behaviour suggests. The goal in finding these "hard" instances is for their use in public-key cryptography systems, such as the Merkle–Hellman knapsack cryptosystem. More generally, better understanding of the structure of the space of instances of an optimization problem helps to advance the study of the particular problem and can improve algorithm selection. Furthermore, notable is the fact that the hardness of the knapsack problem depends on the form of the input. If the weights and profits are given as integers, it is weakly NP-complete, while it is strongly NP-complete if the weights and profits are given as rational numbers. However, in the case of rational weights and profits it still admits a fully polynomial-time approximation scheme. === Unit-cost models === The NP-hardness of the Knapsack problem relates to computational models in which the size of integers matters (such as the Turing machine). In contrast, decision trees count each decision as a single step. Dobkin and Lipton show an 1 2 n 2 {\displaystyle {1 \over 2}n^{2}} lower bound on linear decision trees for the knapsack problem, that is, trees where decision nodes test the sign of affine functions. This was generalized to algebraic decision trees by Steele and Yao. If the elements in the problem are real numbers or rationals, the decision-tree lower bound extends to the real random-access machine model with an instruction set that includes addition, subtraction and multiplication of real numbers, as well as comparison and either division or remaindering ("floor"). This model covers more algorithms than the algebraic decision-tree model, as it encompasses algorithms that use indexing into tables. However, in this model all program steps are counted, not just decisions. An upper bound for a decision-tree model was given by Meyer auf der Heide who showed that for every n there exists an O(n4)-deep linear decision tree that solves the subset-sum problem with n items. Note that this does not imply any upper bound for an algorithm that should solve the problem for any given n. == Solving == Several algorithms are available to solve knapsack problems, based on the dynamic programming approach, the branch and bound approach or hybridizations of both approaches. === Dynamic programming in-advance algorithm === The unbounded knapsack problem (UKP) places no restriction on the number of copies of each kind of item. Besides, here we assume that x i > 0 {\displaystyle x_{i}>0} m [ w ′ ] = max ( ∑ i = 1 n v i x i ) {\displaystyle m[w']=\max \left(\sum _{i=1}^{n}v_{i}x_{i}\right)} subject to ∑
Open-source software security
Open-source software security is the measure of assurance or guarantee in the freedom from danger and risk inherent to an open-source software system. == Implementation debate == === Benefits === Proprietary software forces the user to accept the level of security that the software vendor is willing to deliver and to accept the rate that patches and updates are released. It is assumed that any compiler that is used creates code that can be trusted, but it has been demonstrated by Ken Thompson that a compiler can be subverted using a compiler backdoor to create faulty executables that are unwittingly produced by a well-intentioned developer. With access to the source code for the compiler, the developer has at least the ability to discover if there is any mal-intention. Kerckhoffs' principle is based on the idea that an enemy can steal a secure military system and not be able to compromise the information. His ideas were the basis for many modern security practices, and followed that security through obscurity is a bad practice. === Drawbacks === Simply making source code available does not guarantee review. An example of this occurring is when Marcus Ranum, an expert on security system design and implementation, released his first public firewall toolkit. At one time, there were over 2,000 sites using his toolkit, but only 10 people gave him any feedback or patches. Having a large amount of eyes reviewing code can "lull a user into a false sense of security". Having many users look at source code does not guarantee that security flaws will be found and fixed. == Metrics and models == There are a variety of models and metrics to measure the security of a system. These are a few methods that can be used to measure the security of software systems. === Number of days between vulnerabilities === It is argued that a system is most vulnerable after a potential vulnerability is discovered, but before a patch is created. By measuring the number of days between the vulnerability and when the vulnerability is fixed, a basis can be determined on the security of the system. There are a few caveats to such an approach: not every vulnerability is equally bad, and fixing a lot of bugs quickly might not be better than only finding a few and taking a little bit longer to fix them, taking into account the operating system, or the effectiveness of the fix. === Poisson process === The Poisson process can be used to measure the rates at which different people find security flaws between open and closed source software. The process can be broken down by the number of volunteers Nv and paid reviewers Np. The rates at which volunteers find a flaw is measured by λv and the rate that paid reviewers find a flaw is measured by λp. The expected time that a volunteer group is expected to find a flaw is 1/(Nv λv) and the expected time that a paid group is expected to find a flaw is 1/(Np λp). === Morningstar model === By comparing a large variety of open source and closed source projects a star system could be used to analyze the security of the project similar to how Morningstar, Inc. rates mutual funds. With a large enough data set, statistics could be used to measure the overall effectiveness of one group over the other. An example of such as system is as follows: 1 Star: Many security vulnerabilities. 2 Stars: Reliability issues. 3 Stars: Follows best security practices. 4 Stars: Documented secure development process. 5 Stars: Passed independent security review. === Coverity scan === Coverity in collaboration with Stanford University has established a new baseline for open-source quality and security. The development is being completed through a contract with the Department of Homeland Security. They are utilizing innovations in automated defect detection to identify critical types of bugs found in software. The level of quality and security is measured in rungs. Rungs do not have a definitive meaning, and can change as Coverity releases new tools. Rungs are based on the progress of fixing issues found by the Coverity Analysis results and the degree of collaboration with Coverity. They start with Rung 0 and currently go up to Rung 2. Rung 0 The project has been analyzed by Coverity's Scan infrastructure, but no representatives from the open-source software have come forward for the results. Rung 1 At rung 1, there is collaboration between Coverity and the development team. The software is analyzed with a subset of the scanning features to prevent the development team from being overwhelmed. Rung 2 There are 11 projects that have been analyzed and upgraded to the status of Rung 2 by reaching zero defects in the first year of the scan. These projects include: AMANDA, ntp, OpenPAM, OpenVPN, Overdose, Perl, PHP, Postfix, Python, Samba, and Tcl.
AI Code Generators Reviews: What Actually Works in 2026
Trying to pick the best AI code generator? An AI code generator is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI code generator slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.
Stefano Soatto
Stefano Soatto is professor of computer science at the University of California, Los Angeles (UCLA), in Los Angeles, CA, where he is also professor of electrical engineering and founding director of the UCLA Vision Lab. He is also Vice President of applied science for Amazon Web Services' (AWS) AI division. == Academic biography == Soatto obtained his D. Eng. in electrical engineering, cum laude, from the University of Padua in 1992, was an EAP Fellow at the University of California, Berkeley in 1990–1991, and received his Ph.D. in control and dynamical systems from the California Institute of Technology in 1996 with dissertation "A Geometric Approach to Dynamic Vision". In 1996–97 he was a postdoctoral scholar at Harvard University, and subsequently held positions as assistant and associate professor of electrical engineering and biomedical engineering at Washington University in St. Louis, and of mathematics and computer science at the University of Udine, Italy. He has been at UCLA since 2000. He is also Vice President of applied science for Amazon Web Services' (AWS) AI division. == Research == Soatto's research focuses on computer vision, machine learning and robotics. He co-developed optimal algorithms for structure from motion (SFM, or visual SLAM, simultaneous localization and mapping, in robotics; Best Paper Award at CVPR 1998), characterized its ambiguities (David Marr Prize at ICCV 1999), also characterized the identifiability and observability of visual-inertial sensor fusion (Best Paper Award at ICRA 2015). His research focus is the development of representations, that are functions of the data that capture their informative content and discard irrelevant variability in the data (a generalized form of 'noise' or 'clutter'). Soatto's lab first to demonstrate real-time SFM and augmented reality (AR) on commodity hardware in live demos at CVPR 2000, ICCV 2001, and ECCV 2002. He also co-led the UCLA-Golem Team in the second DARPA Grand Challenge for autonomous vehicles, with Emilio Frazzoli (co-founder of NuTonomy), and Amnon Shashua (co-founder of Mobileye). == Recognition == Soatto was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2013 for contributions to dynamic visual processes. He received the David Marr Prize in Computer Vision in 1999. He was named to the 2022 class of ACM Fellows, "for contributions to the foundations and applications of visual geometry and visual representations learning".
Sumio Watanabe
Sumio Watanabe (渡辺 澄夫, Watanabe Sumio; born 1959) is a Japanese mathematician and engineer working in probability theory, applied algebraic geometry and Bayesian statistics. He is currently a professor at Tokyo Institute of Technology in the Department of Computational Intelligence and Systems Science. He is the author of the text, Algebraic Geometry and Statistical Learning Theory, which proposes a generalization of Fisher's regular statistical theory to singular statistical models. == Books == Mathematical Theory of Bayesian Statistics, CRC Press, 2018, ISBN 9781482238068 Algebraic Geometry and Statistical Learning Theory, Cambridge University Press, 2009.
Alerts.in.ua
alerts.in.ua is an online service that visualizes information about air alerts and other threats on the map of Ukraine. == History == The idea of the site appeared in the first weeks of the 2022 Russian invasion of Ukraine, during the development of other projects related to alerting the population about alarms. So, on March 2, 2022, the "Lviv Siren" bot was created, which reported on air alarms in Lviv on Twitter. Later, the idea arose to monitor alarms all over Ukraine and display them on a map. However, the lack of a single official source reporting alarms made this task much more difficult. On March 15, 2022, the Ajax Systems company announced the creation of the official Telegram channel "Air Alarm". This channel receives signals from the "Air Alarm" application and instantly publishes messages about the start and end of alarms in different regions of Ukraine. This immediately solved the problem with the source of information and gave impetus to the further implementation of the project. On March 22, 2022, the first version of the "Air Alarm Map" website was published, located on the war.ukrzen.in.ua domain. The map quickly gained popularity in social networks. It, like several other similar projects, began to be widely distributed by the mass media: Suspilne, Novyi Kanal, UNIAN, DW, Fakty ICTV, Vikna TV, Ukrainian Radio, STB, Espresso, dev.ua, itc.ua and state bodies: Center for Countering Disinformation at the National Security and Defense Council of Ukraine, Verkhovna Rada of Ukraine, Khmelnytska OVA, etc. On April 8, 2022, the site moved to the alerts.in.ua domain, where it is still available today. On August 25, 2022, the service began monitoring local official channels in addition to the main "Air Alarm". On September 11, 2022, the English version of the site was published. On March 22, 2023, its own Android application was published. The project is actively developing and has its own community. == Description == The main part of the site is a map of Ukraine, on which the regions where an air alert or other threats have been declared are highlighted in real time. As of October 16, 2022, 5 types of threats are supported: Air alarm. The threat of artillery fire. The threat of street fighting. Chemical threat. Nuclear threat. Additionally, based on media reports, information is published about other dangerous events, such as explosions, demining, etc. On the site, you can view the history of announced alarms with links to sources. Alarm statistics for different time periods are also available. For developers, there is an API that allows you to develop your own services based on information about declared alarms. The site is available in Ukrainian, English, Polish and Japanese. == Use == The map is used by: To monitor the situation in the country and the region. To illustrate the alarms announced in the mass media: TSN, Ukrainian truth, Channel 24, Suspilne, RBC Ukraine, Gromadske, Glavkom. As a map of alarms in mobile applications, there is Alarm and AirAlert. As an API for its services, including alternative alarm maps, Telegram, Viber channels, Discord bots, IoT projects, etc. == Statistics == 89.5% of users use the map from a mobile phone, 10% from a PC and 1% from a tablet. Top 6 countries by visit: Ukraine, United States, Poland, Germany, Great Britain and Japan . == Alternative projects == eMap was created by the developer Vadym Klymenko. AlarmMap is an online from the Ukrainian office of Agroprep. The official map of air alarms was developed by Ajax Systems together with the developer Artem Lemeshev, Stfalcon with the support of the Ministry of Statistics.
AI Bug Finders Reviews: What Actually Works in 2026
Looking for the best AI bug finder? An AI bug finder is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI bug finder slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.