The traditional definition for the operation of exponentiation to a natural power (or a positive integer) had introduced approximately as follows:
Exponentiation is an arithmetic operation originally defined as the result of multiple multiplications a number by itself.
But the more precise formulation is still different:
Raising a number X to an integer power N is an arithmetic operation defined as the result of multiple [N by mod times] multiplications or divisions one by number X.
Surprisingly, there are strict mathematical methods that literally allow to hear visual geometric forms and, conversely, to see the beauty of musical harmonies...
The intuition is to employ a depth-first search (DFS) approach through the tree.
During each step in the traversal, we perform the following key calculations:
1. Determine the path that ends at the current node.
2. Compute two different subtree paths that traverse the current node.
3. Maintain an array that keeps track of the cases where paths contain either 0 or 1 prime number.
This method allows us to efficiently count the valid paths in the tree while considering the presence of prime numbers.
Time complexity: O(max(nums) * log(max(nums)) + n * log(n)). Accounting for computing prime scores, using the stack to compute next greater elements, and sorting the tuples.
Space complexity: O(max(nums) + n). Considering the space required for arrays and the stack used for computation.
2809. Min Time to Make Array Sum: Efficient Swift solution, using dynamic programming, for minimizing time to reach a sum in arrays A and B. Time: O(n²), Space: O(n).
2813. Max Elegance of K-Length Subseq: Swift code for elegantly selecting unique k-length subsequences with profit and categories. Solution uses sorting and iteration. Time: O(nlogn), Space: O(n).
Hard++, Extra Category.
Amazon.
Overflow checks have been taken into consideration. The maximum time to move a box is at most 4 * 1000 (four steps to move the box, each taking 1000 time). With at most 1e4 boxes, the total time is at most 4e7, ensuring the solution is safe.
Given two positive integers low and high represented as strings, find the count of stepping numbers in the inclusive range [low, high].
A stepping number is an integer such that all of its adjacent digits have an absolute difference of exactly 1.
Return an integer denoting the count of stepping numbers in the inclusive range [low, high].
Since the answer may be very large, return it modulo 10^9 + 7.
Hard, Acceptance Level 14.5%.
Dynamic Programming (DP).
Efficiently handles large inputs (10^9 + 7).
Latest and Most Hardest Problem.
Simple Swift Math Solution.
Time complexity: O(N logN).
The time complexity of this solution is dominated by the sorting step, making it O(N logN), where N is the length of the input array usageLimits. The rest of the operations involve simple arithmetic and comparisons, which take linear time. Therefore, the overall time complexity of the function is O(NlogN).
Only 10 lines of code.
LeetCode 2612 (Hard). Minimum Reverse Operations.
The algorithm follows a breadth-first search (BFS) approach to determine the minimum number of reverse operations needed to bring the 1 to each position in the array.
To speed up the algorithm, we mark banned positions with -2 instead of using set lookups. This optimization reduces the constant coefficient and improves the speed of the algorithm, but it may still result in a time limit exceeded (TLE) error.
For each visited position, there are potentially O(k) target positions that can be reached through reverse operations. To avoid the multiplicative cost of iterating over all these potential positions, we update the nextNode2s array. This array initially points forward by 2, but we update it dynamically to point beyond all the target positions considered for each visited position. This optimization helps improve the efficiency of the algorithm and avoids unnecessary computations.
Solution of the day.
LeetCode 956 (Hard). Tallest Billboard.
The code uses dynamic programming to solve the problem. It maintains a dictionary dp, where the keys represent the possible height differences between the two billboards, and the values represent the maximum sum of heights achieved for each height difference.
Hash, salt, SHA-1, SHA-2, std::hash.. To a non-programming person that may come up as some kind of a recipe that just does not seem to add up. In a sense, this is indeed supposed to be a gibberish to any third party and a strong, helpful mechanism for us, programmers.
At the start of writing this article, I had one clear idea to get across the table: to finally unveil this mystery of hashing in C++ for beginners. I, a beginner myself, also wanted to solidify my knowledge in this area; so let’s get started.
The life of Langton's Ant seems sad and lonely, but, as we'll soon discover, he is not ready to put up with such an outrageous situation and is trying his best to escape. American scientist Christopher Langton invented his ant back in 1986. Since then, no one has been able to explain the strange behavior of this mysterious model...
As a Product Manager who has worked on the development of delivery route optimisation software for 10+ years, I see that modern technologies can significantly improve the optimisation process and deliver better solutions. AI, machine learning, and other modern technologies have the potential to revolutionise the way delivery routes are optimised in the future.
With the increasing availability of data and the advancement of AI and machine learning algorithms, it is becoming possible to develop more sophisticated prediction models that can be integrated into optimisation algorithms to make more accurate and informed decisions about route planning and scheduling. Machine learning algorithms can be trained to predict customer demand based on historical sales data and other market trends, allowing businesses to optimise their delivery schedules and routes accordingly. AI can also be used to optimise delivery schedules based on customer preferences and other relevant factors.
Blockchain technology could be used to create a secure, decentralised database of information about deliveries, including information about the products being shipped, the route they are taking, and the status of the delivery. This could help increase transparency and accountability in the delivery process as well as reduce the risk of fraud and theft.
Internet of Things (IoT) devices, such as sensors and GPS trackers, may collect real-time data about delivery vehicles and their surroundings. This data could be analysed and used to optimise delivery routes in real time, as well as to track the location of deliveries and monitor the condition of the products being shipped.
The published material is in the Appendix of my book [1]
Modern civilization finds itself at a crossroads in which to choose the meaning of life. Because of the development of technology, the majority of the world's population may be "superfluous" - not in demand in the production of values. There is another option, where each person is a supreme value, an absolute individual and can be indispensably useful in the technology of the collective mind.
In the eighties of the last century, the task of creating a scientific field of "collective intelligence" was set. Collective intelligence is defined as the ability of the collective to find solutions to problems more effectively than each participant individually. The right collective mind must be...
• The method of phase-magnitude interpolation (PMI)
• Accurate measure of frequency, magnitude and phase of signal harmonics
• Detection of resonances
The Fast Fourier Transform (FFT) algorithm is an important tool for analyzing and processing signals of various nature.
It allows to reconstruct magnitude and phase spectrum of a signal into the frequency domain by magnitude sample into the time domain, while the method is computationally optimized with modest memory consumption.
Although there is not losing of any information about the signal during the conversion process (calculations are reversible up to rounding), the algorithm has some peculiarities, which hinder high-precision analysis and fine processing of results further.
The article presents an effective way to overcome such "inconvenient" features of the algorithm.
