title

DP 4. Frog Jump with K Distance | Lecture 3 Follow Up Question

description

Lecture Notes/C++/Java Codes: https://takeuforward.org/data-structure/dynamic-programming-frog-jump-with-k-distances-dp-4/
Problem Link: https://atcoder.jp/contests/dp/tasks/dp_b
Make sure to join our telegram group for discussions: https://linktr.ee/takeUforward
Pre-req for this Series: https://www.youtube.com/watch?v=yVdKa8dnKiE&list=PLgUwDviBIf0rGlzIn_7rsaR2FQ5e6ZOL9
Full Playlist: https://www.youtube.com/watch?v=FfXoiwwnxFw&list=PLgUwDviBIf0qUlt5H_kiKYaNSqJ81PMMY
In this video, we have discussed how to solve the frog jump problem with K distance allowed. I have taught you how you should write a 1D recurrence. This problem is a follow-up to Lecture 3's follow-up question.
If you have not yet checked our SDE sheet, you should definitely do it: https://takeuforward.org/interviews/strivers-sde-sheet-top-coding-interview-problems/
You can also get in touch with me at my social handles: https://linktr.ee/takeUforward

detail

{'title': 'DP 4. Frog Jump with K Distance | Lecture 3 Follow Up Question', 'heatmap': [{'end': 506.697, 'start': 474.737, 'weight': 0.823}, {'end': 635.555, 'start': 532.436, 'weight': 0.74}, {'end': 659.003, 'start': 643.436, 'weight': 0.729}], 'summary': 'Discusses minimum energy for frog jumps, covering the solution to the follow-up question from lecture 3, addressing the minimum energy required for the frog to jump from the 0 index to the n-1 index considering the possibility of jumping to k indexes. it also explains dynamic programming, space complexity, memoization, time complexity, space optimization, and optimizing dynamic programming solutions.', 'chapters': [{'end': 416.738, 'segs': [{'end': 55.706, 'src': 'embed', 'start': 0.209, 'weight': 0, 'content': [{'end': 1.47, 'text': 'hey, everyone, welcome back to the channel.', 'start': 0.209, 'duration': 1.261}, {'end': 3.131, 'text': 'i hope you guys are doing extremely well.', 'start': 1.47, 'duration': 1.661}, {'end': 5.692, 'text': 'so this is going to be a very, very short video.', 'start': 3.131, 'duration': 2.561}, {'end': 11.056, 'text': "why? because in this video i'll be talking about the lecture 3 follow-up question solution.", 'start': 5.692, 'duration': 5.364}, {'end': 11.976, 'text': 'now, in the lecture 3,', 'start': 11.056, 'duration': 0.92}, {'end': 26.532, 'text': 'i did give you a follow-up question where i asked you like the question that we solved was if the frog could jump from the i-th index to i plus 1 and the frog could jump from i to i plus 2.', 'start': 11.976, 'duration': 14.556}, {'end': 35.874, 'text': 'in this scenario, what is the minimum energy that the frog will take in order to jump from 0 index to n minus 1-th index?', 'start': 26.532, 'duration': 9.342}, {'end': 37.615, 'text': 'that was the question in lecture 3.', 'start': 35.874, 'duration': 1.741}, {'end': 41.575, 'text': 'but i did give you a follow up where i said you that what if i give you k?', 'start': 37.615, 'duration': 3.96}, {'end': 43.756, 'text': 'yes, what if i give you k?', 'start': 41.575, 'duration': 2.181}, {'end': 52.382, 'text': 'and i ask you that you are allowed to jump from i plus 1 to i plus 2 plus i plus three dot dot dot dot till i plus kth index,', 'start': 43.756, 'duration': 8.626}, {'end': 55.706, 'text': "you're allowed to jump to all these indexes.", 'start': 52.382, 'duration': 3.324}], 'summary': 'Solving the minimum energy frog jump problem with extended jump range.', 'duration': 55.497, 'max_score': 0.209, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8209.jpg'}, {'end': 120.855, 'src': 'embed', 'start': 97.213, 'weight': 2, 'content': [{'end': 104.275, 'text': 'And then the energy taken will be absolute of A of index minus absolute of index minus 1.', 'start': 97.213, 'duration': 7.062}, {'end': 112.778, 'text': "And if I'm taking the second jump, which is again only possible if index is greater than 1, then I can take the second step or second jump.", 'start': 104.275, 'duration': 8.503}, {'end': 120.855, 'text': 'which is F of index minus two plus absolute of A of index minus absolute of A of index minus two.', 'start': 113.171, 'duration': 7.684}], 'summary': 'Calculating energy taken for jumps based on index and array values.', 'duration': 23.642, 'max_score': 97.213, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB897213.jpg'}, {'end': 354.167, 'src': 'embed', 'start': 327.848, 'weight': 3, 'content': [{'end': 338.35, 'text': 'if this is the jump energy, can I say this? that i can store the minimal of all the jumps, i can store the minimal of all the jumps,', 'start': 327.848, 'duration': 10.502}, {'end': 347.194, 'text': 'which is minimum steps, and whatever the jump energy is, and at the end of the day i can return whatever minimum steps or minimum energy that it took.', 'start': 338.35, 'duration': 8.844}, {'end': 354.167, 'text': 'so instead of writing index minus one, index minus two, i changed it into a for loop.', 'start': 347.194, 'duration': 6.973}], 'summary': 'Algorithm optimized to store minimal jumps and energy using a for loop.', 'duration': 26.319, 'max_score': 327.848, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8327848.jpg'}, {'end': 399.767, 'src': 'embed', 'start': 372.336, 'weight': 5, 'content': [{'end': 376.697, 'text': 'so if you remember, over here we were taking care if index is greater than one.', 'start': 372.336, 'duration': 4.361}, {'end': 379.237, 'text': 'we were taking care if index is greater than two.', 'start': 376.697, 'duration': 2.54}, {'end': 390.881, 'text': 'so we need to make sure that whatever this value is yes, whatever this value is index minus j, like the the after, after subtracting it,', 'start': 379.237, 'duration': 11.644}, {'end': 394.663, 'text': 'should be greater than equal to zero, like it should not go to a negative index.', 'start': 390.881, 'duration': 3.782}, {'end': 399.767, 'text': "and if that's the case, i can make sure that these couple of steps are performed.", 'start': 394.663, 'duration': 5.104}], 'summary': 'Ensure index is greater than or equal to zero for value', 'duration': 27.431, 'max_score': 372.336, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8372336.jpg'}], 'start': 0.209, 'title': 'Minimum energy for frog jumps', 'summary': 'Discusses the solution to the follow-up question from lecture 3, addressing the minimum energy required for the frog to jump from the 0 index to the n-1 index considering the possibility of jumping to k indexes based on the recurrence and steps involved. it also explains the recurrence relationship for index jumps, modifying the code for different jump scenarios, utilizing a for loop to handle multiple jumps, addressing the issue of negative index values, and returning the minimum energy required for the jumps.', 'chapters': [{'end': 120.855, 'start': 0.209, 'title': 'Lecture 3 follow-up question solution', 'summary': 'Discusses the solution to the follow-up question from lecture 3, addressing the minimum energy required for the frog to jump from the 0 index to the n-1 index, considering the possibility of jumping to k indexes, based on the recurrence and steps involved.', 'duration': 120.646, 'highlights': ['The solution addresses the minimum energy required for the frog to jump from the 0 index to the n-1 index, considering the possibility of jumping to k indexes.', 'The recurrence for the problem involves expressing each step in terms of indexes, with the energy taken for each jump being calculated based on the absolute difference of the indexes.', 'The lecture emphasizes the importance of understanding the lecture 3 content to comprehend the follow-up question solution.']}, {'end': 416.738, 'start': 121.336, 'title': 'Recurrence relationship for index jumps', 'summary': 'Discusses the recurrence relationship for index jumps, explaining how to modify the code for different jump scenarios, utilizing a for loop to handle multiple jumps, and addressing the issue of negative index values, ultimately returning the minimum energy required for the jumps.', 'duration': 295.402, 'highlights': ['The chapter discusses how to modify the code for different jump scenarios, including allowing jumps from I plus 1 to I plus 3, I plus 5, and a variable K, where K can vary from question to question.', 'It explains the utilization of a for loop to handle multiple jumps, using the index minus j approach and storing the minimal of all the jumps to calculate the minimum energy required for the jumps.', 'The chapter addresses the issue of negative index values by ensuring that the value after subtracting the index minus j should be greater than or equal to zero, and if not, breaking out of the loop to indicate no further jumps are possible.']}], 'duration': 416.529, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8209.jpg', 'highlights': ['The solution addresses the minimum energy required for the frog to jump from the 0 index to the n-1 index, considering the possibility of jumping to k indexes.', 'The chapter discusses how to modify the code for different jump scenarios, including allowing jumps from I plus 1 to I plus 3, I plus 5, and a variable K.', 'The recurrence for the problem involves expressing each step in terms of indexes, with the energy taken for each jump being calculated based on the absolute difference of the indexes.', 'It explains the utilization of a for loop to handle multiple jumps, using the index minus j approach and storing the minimal of all the jumps to calculate the minimum energy required for the jumps.', 'The lecture emphasizes the importance of understanding the lecture 3 content to comprehend the follow-up question solution.', 'The chapter addresses the issue of negative index values by ensuring that the value after subtracting the index minus j should be greater than or equal to zero, and if not, breaking out of the loop to indicate no further jumps are possible.']}, {'end': 678.307, 'segs': [{'end': 514.763, 'src': 'heatmap', 'start': 474.737, 'weight': 0, 'content': [{'end': 478.32, 'text': 'There are k steps for every junction.', 'start': 474.737, 'duration': 3.583}, {'end': 480.021, 'text': 'For every junction, there are k steps.', 'start': 478.38, 'duration': 1.641}, {'end': 485.305, 'text': 'So, you can say that for if there are n states, and there are k jumps.', 'start': 480.501, 'duration': 4.804}, {'end': 494.137, 'text': 'I can say the time complexity to be this and the space complexity to be bigger of n of recursion stack space plus a bigger of n of memorization.', 'start': 485.666, 'duration': 8.471}, {'end': 498.182, 'text': 'This is what the time complexity and the space complexity of the recursive solution will be.', 'start': 494.457, 'duration': 3.725}, {'end': 503.574, 'text': 'Now, if you remember the tabulation, How did I write the tabulation? It was again very simple.', 'start': 498.622, 'duration': 4.952}, {'end': 506.697, 'text': 'I declared a DP array of size n.', 'start': 504.054, 'duration': 2.643}, {'end': 508.658, 'text': 'I said DP of 0.', 'start': 506.697, 'duration': 1.961}, {'end': 513.402, 'text': 'Yes, which means in order to reach the index 0 from 0 will be 0.', 'start': 508.658, 'duration': 4.744}, {'end': 514.763, 'text': 'That is what I definitely said.', 'start': 513.402, 'duration': 1.361}], 'summary': 'Time complexity is o(n) and space complexity is o(n) for the recursive solution.', 'duration': 41.767, 'max_score': 474.737, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8474737.jpg'}, {'end': 678.307, 'src': 'heatmap', 'start': 643.436, 'weight': 1, 'content': [{'end': 647.978, 'text': "that's how you convert it into probably a multi-dimension dp, like it's.", 'start': 643.436, 'duration': 4.542}, {'end': 650.279, 'text': "it's a basically nested loop dp.", 'start': 647.978, 'duration': 2.301}, {'end': 652.86, 'text': 'so i hope you have understood the follow-up question.', 'start': 650.279, 'duration': 2.581}, {'end': 659.003, 'text': 'now the question arises can we space optimize this?', 'start': 652.86, 'duration': 6.143}, {'end': 662.411, 'text': 'yes, we can, But can we space optimize??', 'start': 659.003, 'duration': 3.408}, {'end': 669.979, 'text': 'Generally, in the previous questions we have seen we have space optimized this from a big O of N to a big O of one.', 'start': 662.492, 'duration': 7.487}, {'end': 672.781, 'text': 'Now, can we do this over here? The answer to that would be no.', 'start': 670.299, 'duration': 2.482}, {'end': 676.705, 'text': 'We can just optimize the space from big O of N to a big O of K.', 'start': 673.122, 'duration': 3.583}, {'end': 678.307, 'text': 'So do you remember?', 'start': 677.606, 'duration': 0.701}], 'summary': 'Nested loop dynamic programming can be space optimized to o(k) from o(n).', 'duration': 145.871, 'max_score': 643.436, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8643436.jpg'}], 'start': 416.738, 'title': 'Dynamic programming and space complexity', 'summary': 'Discusses memoization and time complexity, with a time complexity of o(nk) and space complexity of o(n), recursive solution covering time and space complexity, and space optimization in dynamic programming with a focus on minimizing steps and space optimization from o(n) to o(k).', 'chapters': [{'end': 494.137, 'start': 416.738, 'title': 'Memoization and time complexity', 'summary': 'Discusses the impact of applying memoization, analyzing the time and space complexity in terms of states and jumps, leading to a time complexity of o(nk) and a space complexity of o(n).', 'duration': 77.399, 'highlights': ['The time complexity analysis considers the number of states and the loops per index, resulting in a time complexity of O(nk) where n is the number of states and k is the number of jumps.', 'The space complexity is determined by the recursion stack space and the memory required for memoization, resulting in a space complexity of O(n).']}, {'end': 563.413, 'start': 494.457, 'title': 'Time and space complexity of recursive solution', 'summary': 'Covers the time and space complexity of a recursive solution and explains the tabulation approach using dynamic programming with a focus on minimizing steps and utilizing a for loop.', 'duration': 68.956, 'highlights': ['The tabulation approach involves declaring a DP array of size n and setting initial values, then iterating through the array to calculate and store the minimum steps, providing a clear and simple method for dynamic programming.', 'The lecture emphasizes the use of a for loop to iterate through k steps, demonstrating a practical application of dynamic programming for solving problems with multiple steps and optimizing the solution.']}, {'end': 678.307, 'start': 563.413, 'title': 'Space optimization in dynamic programming', 'summary': 'Discusses dynamic programming for space optimization, explaining the approach and time complexity, and concludes that space can be optimized from o(n) to o(k) but not to o(1).', 'duration': 114.894, 'highlights': ['The time complexity is O(N*K), where N and K are the input parameters, due to nested loop dynamic programming approach.', 'Space optimization can be achieved from O(N) to O(K) but not to O(1) in this scenario, as explained in detail.', 'The chapter explains the approach of using dynamic programming for space optimization and its limitations, providing a comprehensive understanding of the concept.']}], 'duration': 261.569, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8416738.jpg', 'highlights': ['The time complexity analysis considers the number of states and the loops per index, resulting in a time complexity of O(nk) where n is the number of states and k is the number of jumps.', 'The time complexity is O(N*K), where N and K are the input parameters, due to nested loop dynamic programming approach.', 'The space complexity is determined by the recursion stack space and the memory required for memoization, resulting in a space complexity of O(n).', 'The tabulation approach involves declaring a DP array of size n and setting initial values, then iterating through the array to calculate and store the minimum steps, providing a clear and simple method for dynamic programming.', 'The lecture emphasizes the use of a for loop to iterate through k steps, demonstrating a practical application of dynamic programming for solving problems with multiple steps and optimizing the solution.', 'Space optimization can be achieved from O(N) to O(K) but not to O(1) in this scenario, as explained in detail.', 'The chapter explains the approach of using dynamic programming for space optimization and its limitations, providing a comprehensive understanding of the concept.']}, {'end': 1045.138, 'segs': [{'end': 724.345, 'src': 'embed', 'start': 697.226, 'weight': 0, 'content': [{'end': 702.57, 'text': 'we required couple of states that was dp of i minus 1 and the other one was dp of i minus 2..', 'start': 697.226, 'duration': 5.344}, {'end': 705.172, 'text': 'So there were only couple of steps that were required.', 'start': 702.57, 'duration': 2.602}, {'end': 713.798, 'text': 'So what you can do is instead of carrying these guys, you can carry the previous couple of states in couple of variables which was previous two.', 'start': 705.492, 'duration': 8.306}, {'end': 717.24, 'text': 'at previous and you can call this as i.', 'start': 714.438, 'duration': 2.802}, {'end': 719.341, 'text': 'so there will be no dp of i carried.', 'start': 717.24, 'duration': 2.101}, {'end': 721.683, 'text': 'so whenever like assume this is the current i.', 'start': 719.341, 'duration': 2.342}, {'end': 724.345, 'text': 'so in the next step, when i will move here,', 'start': 721.683, 'duration': 2.662}], 'summary': 'Algorithm optimization: reduce steps to compute previous states.', 'duration': 27.119, 'max_score': 697.226, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8697226.jpg'}, {'end': 859.889, 'src': 'embed', 'start': 826.305, 'weight': 3, 'content': [{'end': 828.166, 'text': "This i minus 2 is what you'll require.", 'start': 826.305, 'duration': 1.861}, {'end': 830.508, 'text': "This i minus 3 is what you'll require.", 'start': 828.626, 'duration': 1.882}, {'end': 832.85, 'text': 'This i minus 4 is what you require.', 'start': 830.908, 'duration': 1.942}, {'end': 835.472, 'text': "So you'll just require the four last values.", 'start': 833.15, 'duration': 2.322}, {'end': 838.133, 'text': "Or can I say you'll just require the k last values.", 'start': 835.872, 'duration': 2.261}, {'end': 840.775, 'text': 'values. does that make sense?', 'start': 838.954, 'duration': 1.821}, {'end': 848.881, 'text': "because you'll just require those k last values, because you'll be like dp of i minus 1, dp of i minus 2, dp of i minus 3, dp of i minus 4,", 'start': 840.775, 'duration': 8.106}, {'end': 849.762, 'text': 'nothing beyond that.', 'start': 848.881, 'duration': 0.881}, {'end': 852.904, 'text': "so you'll just be requiring k last values.", 'start': 849.762, 'duration': 3.142}, {'end': 859.889, 'text': 'now if i just, uh, change to the next dot and i say in the next iteration yes, in the next iteration,', 'start': 852.904, 'duration': 6.985}], 'summary': 'To calculate, only the last 4 values are needed for the k-th iteration.', 'duration': 33.584, 'max_score': 826.305, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8826305.jpg'}, {'end': 1009.632, 'src': 'embed', 'start': 982.802, 'weight': 4, 'content': [{'end': 986.263, 'text': 'that is the reason I did not add it in the lecture 3..', 'start': 982.802, 'duration': 3.461}, {'end': 993.206, 'text': 'And yesterday I could not upload the video because I was suffering from very, very high fever and I have a lot of pain over here,', 'start': 986.263, 'duration': 6.943}, {'end': 994.527, 'text': 'like I am burning a lot.', 'start': 993.206, 'duration': 1.321}, {'end': 1001.389, 'text': "i'm still recording this and i will be uploading this, and right after this video there will be lecture four coming up.", 'start': 995.547, 'duration': 5.842}, {'end': 1002.63, 'text': "don't worry about that.", 'start': 1001.389, 'duration': 1.241}, {'end': 1006.231, 'text': 'uh, this is this video is for the yesterday one.', 'start': 1002.63, 'duration': 3.601}, {'end': 1008.032, 'text': 'so yeah, in case you have understood.', 'start': 1006.231, 'duration': 1.801}, {'end': 1009.632, 'text': 'uh, the lecture three follow up.', 'start': 1008.032, 'duration': 1.6}], 'summary': 'Video for lecture 3 delayed due to illness, but lecture 4 coming up.', 'duration': 26.83, 'max_score': 982.802, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8982802.jpg'}], 'start': 678.307, 'title': 'Optimizing dynamic programming', 'summary': 'Discusses optimizing dynamic programming solutions from o(n) to o(1) by using previous states, reducing computational steps and memory usage. it also covers space optimization by utilizing last k values to compute the current value, highlighting that no optimization is needed when k=n.', 'chapters': [{'end': 758.928, 'start': 678.307, 'title': 'Optimizing big o(n) to big o(1)', 'summary': 'Discusses the optimization of a dynamic programming solution from o(n) to o(1), by utilizing only the previous two states for computing any particular i-th state, resulting in significant reduction in computational steps and memory usage.', 'duration': 80.621, 'highlights': ['By carrying the previous two states in a couple of variables, the computational steps and memory usage are significantly reduced, resulting in an O(1) solution.', 'For computing any particular i-th state, only the previous two states (dp of i-1 and dp of i-2) are required, reducing computational complexity and memory usage.', 'The approach of carrying the previous two states in a couple of variables results in a more efficient solution, optimizing the dynamic programming solution.']}, {'end': 1045.138, 'start': 758.928, 'title': 'Optimizing space in dynamic programming', 'summary': "Discusses the optimization of space in dynamic programming by utilizing the last k values to compute the current value, emphasizing that no space optimization is required for k=n, and the explanation for the follow-up question was not included due to time constraints and the lecturer's illness.", 'duration': 286.21, 'highlights': ['The lecturer explains the concept of utilizing the last k values to compute the current value in dynamic programming, demonstrating the reduction of space usage from k elements to maintain the last k values, and emphasizes that no space optimization is necessary for k=n.', "The lecturer shares the reason for not including the explanation for the follow-up question in lecture 3 due to time constraints, and the delay in video upload due to the lecturer's illness.", "The lecturer requests viewers to write 'understood' if they comprehend the content, and urges them to like the video and subscribe to the channel. Additionally, the lecturer promotes the SD sheet in the video description for placement preparation in colleges across India."]}], 'duration': 366.831, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Kmh3rhyEtB8/pics/Kmh3rhyEtB8678307.jpg', 'highlights': ['By carrying the previous two states in a couple of variables, the computational steps and memory usage are significantly reduced, resulting in an O(1) solution.', 'For computing any particular i-th state, only the previous two states (dp of i-1 and dp of i-2) are required, reducing computational complexity and memory usage.', 'The approach of carrying the previous two states in a couple of variables results in a more efficient solution, optimizing the dynamic programming solution.', 'The lecturer explains the concept of utilizing the last k values to compute the current value in dynamic programming, demonstrating the reduction of space usage from k elements to maintain the last k values, and emphasizes that no space optimization is necessary for k=n.', "The lecturer shares the reason for not including the explanation for the follow-up question in lecture 3 due to time constraints, and the delay in video upload due to the lecturer's illness."]}], 'highlights': ['The solution addresses the minimum energy required for the frog to jump from the 0 index to the n-1 index, considering the possibility of jumping to k indexes.', 'The time complexity is O(N*K), where N and K are the input parameters, due to nested loop dynamic programming approach.', 'By carrying the previous two states in a couple of variables, the computational steps and memory usage are significantly reduced, resulting in an O(1) solution.', 'The chapter discusses how to modify the code for different jump scenarios, including allowing jumps from I plus 1 to I plus 3, I plus 5, and a variable K.', 'The recurrence for the problem involves expressing each step in terms of indexes, with the energy taken for each jump being calculated based on the absolute difference of the indexes.']}