Matt Guest · YouTube · 7:22
This introductory video segment defines dynamic programming as an optimization technique that breaks problems into overlapping subproblems. It uses the factorial function to illustrate recursive breakdown and solution construction, and the Fibonacci sequence to visualize redundant calculations. The video clarifies that 'programming' in this context means planning, not coding, and establishes that a true dynamic programming problem must exhibit both optimal substructure and overlapping subproblems. This 120-second introductory clip explains two core properties of dynamic programming problems using visuals and simple examples. It first defines optimal substructure through a shortest-path map from San Francisco to San Diego via Los Angeles, summarized by the equation SHORTEST_PATH(SF, SD) = SHORTEST_PATH(SF, LA) + SHORTEST_PATH(LA, SD). It then introduces overlapping subproblems with a brief recursion animation and a Fibonacci recursion tree, highlighting repeated fib(2) and fib(3) calls as redundant work that dynamic programming aims to eliminate. This 120-second clip introduces dynamic programming by first showing a recursion tree with repeated subtrees to explain overlapping subproblems, then stating that two conditions make DP efficient and summarizing them in a Venn diagram as optimal substructures and overlapping subproblems. It next gives a practical test based on recursion and divide-and-conquer, browses LeetCode DP problems to motivate pattern recognition, and begins listing five DP indicators: sequential decision-making, greedy choice property, and state transition, each with named example problems. This video segment covers the last two indicators for identifying dynamic programming problems: 'Path or Array' and 'Counting or Maximization/Minimization'. It explains that problems involving optimal paths, array arrangements, state transitions, counting ways to achieve goals, or optimizing values often use dynamic programming. Examples like Unique Paths, Longest Increasing Subsequence, Coin Change, and Maximum Subarray are provided.
Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.
Generated from the video's visuals and explanation; not verbatim speech.
The video opens by addressing the common frustration learners face with dynamic programming (DP), noting that simply practicing more isn't always effective without a clear definition. It sets the stage for explaining what DP is and how to identify it.
The narrator formally defines dynamic programming as an algorithmic optimization technique. It works by recursively breaking a complex problem into smaller, overlapping subproblems, and then combining their solutions to solve the original problem. To make this concrete, the video uses the calculation of 4! (4 factorial). It shows the recursive breakdown: 4! depends on 3!, which depends on 2!, which depends on 1!. Once the base case (1! = 1) is reached, the solutions are built back up: 2! = , 3! = , and finally 4! = . This perfectly illustrates the 'break down and construct' mechanism.
To further illustrate the concept of 'overlapping subproblems', the video displays a recursion tree for calculating the 7th Fibonacci number, fib(7). The tree visually demonstrates how a naive recursive approach recalculates the same values (like fib(5) and fib(4)) multiple times, leading to exponential time complexity. This redundancy is exactly what dynamic programming aims to eliminate by storing results.
The video then provides historical context, explaining that the term 'dynamic programming' was coined by mathematician Richard Bellman in the 1950s. Crucially, it clarifies a common misconception: 'programming' in this context does not mean writing computer code. Instead, Bellman used it to mean 'planning and decision making'. 'Dynamic' refers to the ever-changing nature of the problems being solved. The core takeaways from this history are optimal decision making and solving subproblems.
Finally, the video outlines the two strict conditions required for a problem to be classified as a dynamic programming problem, visualized using a Venn diagram. The first condition is 'Optimal Substructure': the problem can be divided into subproblems, and the optimal solution to the main problem can be constructed from the optimal solutions of these subproblems. The second condition, introduced just before the clip ends, is 'Overlapping Subproblems': the same subproblems are encountered repeatedly during the recursive breakdown.
The clip opens on a Venn diagram titled `Characteristics of DP Problems`. One circle is labeled `Optimal Substructures`, the other `Overlapping Subproblems`, and their intersection is labeled `Dynamic Programming Problem`. This visual frames the lesson: DP is being introduced through two defining properties rather than through code.
The first property, optimal substructure, is then explained with a map route from San Francisco to San Diego. The narrator treats the trip as one optimization problem and asks how the shortest route is found. Because the route passes through Los Angeles, the example splits the journey into two smaller route problems: San Francisco to Los Angeles, and Los Angeles to San Diego.
The key inference is that if each subroute is optimal, then combining those optimal pieces gives an optimal whole route. The on-screen equation makes this explicit: . This is the clip’s concrete model of optimal substructure: the global optimum is assembled from local optima.
The lesson then shifts to the second property, overlapping subproblems. Before defining it formally through example, the video inserts a short recursion primer. A simple animation shows a generic problem repeatedly producing a `step` and another , continuing until a `BASE CASE` is reached. The narration stresses that this is only a brief introduction to recursion, not a full treatment.
With that background in place, the video uses a Fibonacci recursion tree rooted at 5. The tree expands into smaller calls, and the important visual point is that some subtrees repeat. The narrator identifies these repeated regions as redundant work, specifically calling out repeated calls such as fib(2) and fib(3). This is how the abstract phrase `overlapping subproblems` is made concrete: the same subproblem is solved multiple times in a naive recursive process.
The clip closes this segment by connecting the two ideas. Optimal substructure explains why a problem can be decomposed into subproblems; overlapping subproblems explain why naïve recursion wastes effort by recomputing the same subresults. The narrator states that dynamic programming helps remove that repeated computation, setting up DP as the method that exploits both properties.
The clip opens on a recursion tree whose repeated branches are circled in red. The narration identifies this repetition as overlapping subproblems, meaning the same smaller problem is encountered more than once during recursive decomposition.
It then states the central criterion: if two conditions are satisfied, dynamic programming can solve the problem much more efficiently. At this moment the second condition is explicit in audio, while the first is supplied by the later diagram.
The speaker gives a practical test for recognizing such problems. First, try to solve the problem recursively. Then ask whether divide-and-conquer style decomposition works, and whether reducing the problem to the simplest subproblem can eventually build up to the optimal solution.
From that test, the narration concludes that if this recursive breakdown and rebuild is possible, then recursion applies, and the speaker says you can know the problem is solvable with dynamic programming.
A Venn diagram titled "Characteristics of DP Problems" summarizes the idea visually. One circle is "Optimal Substructures," the other is "Overlapping Subproblems," and their intersection is labeled "Dynamic Programming Problem."
The video then shifts from formal characteristics to pattern recognition. A LeetCode page filtered by the Dynamic Programming tag is scrolled to show that many problems in this category share recognizable structures.
A slide titled "5 DP Indicators" appears. The first indicator is sequential decision-making: a problem involves a sequence of decisions, each affecting future states, and stepwise optimal choices contribute to the overall optimum.
For this first indicator, the clip names House Robber (198) and Decode Ways (91) as examples that exhibit sequential decision-making.
The second indicator is the greedy choice property. The narration explains that when locally optimal choices lead to a globally optimal solution, the problem is often solvable with DP, and explicitly claims that greedy solvability implies DP solvability.
Examples for the greedy choice property are Jump Game II (45) and Best Time to Buy and Sell Stock II (122).
The third indicator is state transition. The speaker says DP problems often move from one state to the next, and the transition depends on previous states; recognizing this dependence is another useful signal.
The examples given for state transition are Climbing Stairs (70) and Minimum Path Sum (64). The fourth indicator is only introduced at the very end of the clip and is not explained within this segment.
The fourth indicator for dynamic programming is when a problem deals with paths or arrays. This includes finding an optimal path, arranging elements in an array, or transitioning from one state to another.
Classic examples of this include the Unique Paths problem and the Longest Increasing Subsequence problem.
The fifth and final indicator is counting or maximization/minimization. If a problem asks you to count the number of ways to reach a goal, or to find the maximum or minimum value, dynamic programming is likely applicable.
Notable examples for this category are the Coin Change problem and the Maximum Subarray problem.
By keeping these five patterns in mind—sequential decision-making, greedy choice property, state transition, path or array, and counting or optimization—you can better identify when to apply dynamic programming and develop your intuition for these types of problems.
Dynamic programming is an algorithmic optimization technique that solves complex problems by recursively breaking them into smaller, overlapping subproblems and combining their solutions to form the solution to the original problem.
The calculation of 4! demonstrates the core DP mechanism: breaking down 4! into 3!, then 2!, then 1! (base case), and building the answer back up (1 -> 2 -> 6 -> 24).
A recursion tree for fib(7) visually shows how naive recursion recalculates the same subproblems (e.g., fib(5), fib(4)) multiple times, highlighting the inefficiency DP addresses.
Coined by Richard Bellman, 'programming' in dynamic programming refers to planning and decision making, not writing computer code.
A problem is a dynamic programming problem if and only if it satisfies two conditions: Optimal Substructure and Overlapping Subproblems.
A problem has optimal substructure if its optimal solution can be constructed efficiently from the optimal solutions of its subproblems.
A problem has overlapping subproblems if a recursive algorithm for it solves the same subproblems repeatedly, rather than generating new ones each time.
The opening Venn diagram defines a dynamic programming problem as the intersection of two properties: it must have optimal substructure and overlapping subproblems. The visual treats these as joint characteristics rather than isolated facts.
A problem has optimal substructure when the optimal solution to the whole problem can be built from optimal solutions to its subproblems. The clip states this definition verbally and then models it with a route decomposition example.
The San Francisco → San Diego route is split at Los Angeles. Once the shortest SF→LA and LA→SD segments are known, they combine to give the shortest SF→SD route in the example. This is the clip’s main intuition for why subproblem optima can compose into a global optimum.
Overlapping subproblems occur when the same smaller problem is encountered multiple times during recursion. The video defines the term by showing repeated subtrees in a Fibonacci call tree rather than by giving a formal theorem.
The clip gives a minimal explanation of recursion: a function calls itself repeatedly until it reaches a base case. The accompanying animation uses a generic problem P and terminal label BASE CASE to visualize that idea.
When computing the fifth Fibonacci number recursively, the tree contains duplicated subcalls such as fib(2) and fib(3). Those repeated branches are the visual signature of overlapping subproblems and the reason naïve recursion does extra work.
Dynamic programming is presented as the technique that removes repeated computation caused by overlapping subproblems. The clip does not yet show memoization or tabulation, but it motivates DP by contrasting redundant recursive work with a more efficient approach.
The recursion-tree example shows that the same subproblem can appear in multiple branches of a recursive solution. The red circles mark repeated subtrees, illustrating why plain recursion may redo work.
The clip says that if two conditions are satisfied, dynamic programming can solve the problem much more efficiently. The later Venn diagram identifies them as optimal substructures and overlapping subproblems.
To decide whether a problem is suitable for DP, the speaker recommends first trying a recursive solution, then checking whether divide-and-conquer decomposition can reduce the problem to simple subproblems that build up to the optimal solution.
A Venn diagram summarizes DP problems as the intersection of "Optimal Substructures" and "Overlapping Subproblems." This is the clip's compact conceptual definition of when DP applies.
After the formal characteristics, the video introduces five practical indicators for recognizing DP problems: sequential decision-making, greedy choice property, state transition, path or array, and counting or maximization/minimization.
The first indicator is a sequence of decisions where each decision impacts future states. If the optimal solution at each step contributes to the overall optimal solution, that suggests a DP problem. Examples: House Robber (198), Decode Ways (91).
The second indicator is that locally optimal choices lead to a globally optimal solution. The speaker states that if a problem can be solved with a greedy algorithm, it can also be solved with dynamic programming. Examples: Jump Game II (45), Best Time to Buy and Sell Stock II (122).
The third indicator is movement from one state to the next where the transition depends on previous states. Recognizing this dependence is presented as a strong sign of a DP problem. Examples: Climbing Stairs (70), Minimum Path Sum (64).
Problems involving finding optimal paths, arranging elements in arrays, or transitioning between states often require dynamic programming.
Problems that ask to count the number of ways to achieve a goal, or to maximize/minimize a value, frequently use dynamic programming.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
The factorial recurrence is shown as , , , and .
The narrator explains that to get factorial(n) you need to get factorial(n-1) first.
n!
Factorial of a nonnegative integer n, defined recursively by with base case (and implicitly , though not shown).
Nonnegative integers
The tree diagram is labeled fib(7), fib(6), fib(5), etc., representing the Fibonacci sequence.
The narrator mentions the origin of the phrase dynamic programming while showing the Fibonacci tree.
fib(n)
The nth Fibonacci number, typically defined by with base cases .
Nonnegative integers
On-screen equation reads `SHORTEST_PATH(SF, SD) = SHORTEST_PATH(SF, LA) + SHORTEST_PATH(LA, SD)`.
The narrator says the whole problem can be summarized with this function.
SHORTEST_PATH(A, B)
A function-like notation for the shortest route between two cities A and B.
Cities in the example map; SF = San Francisco, LA = Los Angeles, SD = San Diego.
Map labels show San Francisco, Los Angeles, and San Diego as endpoints and intermediate city on the route.
The displayed equation uses SF, LA, SD as arguments of SHORTEST_PATH.
SF, LA, SD
Abbreviations for San Francisco, Los Angeles, and San Diego used in the shortest-path example.
City names in the route example.
The narrator refers to calculating the fifth Fibonacci number.
A recursion tree is shown with root node labeled 5 and repeated subtrees labeled 2 and 3.
The narrator explicitly mentions repeated function calls like fib(2) and fib(3).
The video does not display the formal recurrence formula for Fibonacci numbers; only the recursion-tree example is shown.
fib(n)
Recursive computation of the nth Fibonacci number, illustrated by a call tree.
Nonnegative integer indices in the example; visible nodes include 5, 4, 3, 2, 1, 0.
The speaker repeatedly says "dynamic programming" while explaining when a problem can be solved with it.
A Venn diagram labels the intersection of "Optimal Substructures" and "Overlapping Subproblems" as "Dynamic Programming Problem".
DP
Dynamic programming, presented as a method for solving problems that have optimal substructure and overlapping subproblems.
algorithm design and problem-solving
The narrator defines dynamic programming as an algorithmic optimization technique that breaks down a complicated problem into smaller overlapping subproblems in a recursive manner, and uses solutions to the subproblems to construct a solution to the original problem.
The factorial example illustrates breaking down into , then , then .
Dynamic programming is an algorithmic optimization technique that solves complex problems by recursively breaking them into smaller, overlapping subproblems and combining their solutions to form the solution to the original problem.
The problem can be broken down into smaller subproblems.
The subproblems overlap (the same subproblem is solved multiple times).
The solution to the original problem can be constructed from the solutions to the subproblems.
The narrator states the first condition for a DP problem is optimal substructure: if the problem can be divided into subproblems, and its optimal solution can be constructed from optimal solutions of the subproblems.
A Venn diagram shows 'Optimal Substructures' as one of the two intersecting circles defining a 'Dynamic Programming Problem'.
A problem has optimal substructure if its optimal solution can be constructed efficiently from the optimal solutions of its subproblems.
The problem can be divided into subproblems.
The optimal solution of the main problem depends on the optimal solutions of its subproblems.
The narrator introduces the second condition as overlapping subproblems.
A Venn diagram shows 'Overlapping Subproblems' as the other intersecting circle defining a 'Dynamic Programming Problem'.
The video cuts off before fully explaining what overlapping subproblems are.
A problem has overlapping subproblems if a recursive algorithm for it solves the same subproblems repeatedly, rather than generating new ones each time.
The problem can be solved recursively.
The same subproblems appear multiple times in the recursion tree.
A Venn diagram titled `Characteristics of DP Problems` shows `Optimal Substructures` and `Overlapping Subproblems`, with their intersection labeled `Dynamic Programming Problem`.
The narrator introduces the first concept as optimal substructure and later says the second characteristic is overlapping subproblems.
The clip presents dynamic programming as the intersection of two properties: a problem must have optimal substructure and overlapping subproblems. The Venn diagram visually states that both conditions together characterize a DP problem.
Used as an introductory characterization of DP problems.
The clip does not provide a formal proof or exhaustive list beyond these two properties.
The narrator says, `Essentially, a problem exhibits optimal substructure if the optimal solution to the overall problem can be constructed by efficiently combining the optimal solutions of its subproblems.`
The map example divides the San Francisco to San Diego route at Los Angeles into two subroutes.
The equation `SHORTEST_PATH(SF, SD) = SHORTEST_PATH(SF, LA) + SHORTEST_PATH(LA, SD)` is displayed.
Optimal substructure means the best solution to the whole problem can be assembled from best solutions to smaller subproblems. In the route example, once the shortest SF→LA and LA→SD segments are known, they combine to give the shortest SF→SD route.
The example assumes the overall route passes through Los Angeles.
The narrator states that finding the shortest subroutes guarantees the overall shortest route in this example.
The narrator says, `As for the second characteristic, we have overlapping subproblems.` and later explains that repeated work appears in the recursion tree.
A Fibonacci recursion tree shows repeated subtrees for fib(2) and fib(3), highlighted in red.
On-screen text includes `OVERLAPPING SUBPROBLEMS` and references repeated function calls.
Overlapping subproblems occur when the same smaller problem is solved multiple times during a recursive process. The Fibonacci tree makes this concrete by showing repeated calls such as fib(2) and fib(3), which cause redundant computation.
Illustrated using naive recursion on Fibonacci numbers.
The clip defines the term through repetition in a call tree rather than through a formal theorem.
The narrator says recursion is basically when a function calls itself over and over again until it hits a base case.
A simple animation shows `P` branching into `step` and another `P`, repeating downward until `BASE CASE` appears.
The clip gives only a brief informal explanation of recursion and does not define base case formally.
The video uses a simplified animation to explain recursion: a problem P repeatedly calls itself, each time doing a step, until reaching a base case. This prepares the viewer to understand why repeated subcalls can create overlap.
Presented as a brief conceptual introduction, not a full treatment of recursion.
"then it will exhibit overlapping subproblems."
A recursion tree is shown with repeated subtrees circled in red.
The video defines overlapping subproblems through a recursion-tree illustration: different branches of the recursive decomposition revisit the same smaller subproblem. The red circles mark repeated subtrees, visually indicating that the same subproblem is solved more than once.
The problem is being solved recursively.
The recursive decomposition revisits the same subproblem multiple times.
"If these two conditions are satisfied, then dynamic programming can solve the problem much more efficiently."
The Venn diagram names the two conditions as "Optimal Substructures" and "Overlapping Subproblems".
The first condition is not fully spoken inside this clip; it is identified from the later Venn diagram label.
The clip states that if two conditions are satisfied, dynamic programming can solve the problem much more efficiently. The later diagram identifies those two conditions as optimal substructure and overlapping subproblems.
The problem satisfies optimal substructure.
The problem exhibits overlapping subproblems.
"So when dealing with a problem, first try and see if it can be solved recursively."
"To best determine this, try and see if divide and conquer will work for the given problem."
"See if breaking the problem down to the simplest subproblem can eventually build up to the optimal solution."
The speaker gives a practical test: first try to solve the problem recursively, then check whether divide-and-conquer style decomposition works, and whether breaking the problem into the simplest subproblems can eventually build up to the optimal solution. If this can be done, the speaker says you can apply recursion and then know the problem can be solved with dynamic programming.
You are trying to decide whether a problem is suitable for dynamic programming.
Title: "Characteristics of DP Problems".
Left circle: "Optimal Substructures"; right circle: "Overlapping Subproblems"; intersection: "Dynamic Programming Problem".
"All right, so those are the two main characteristics that determine if a problem can be solved with dynamic programming."
A Venn diagram summarizes the two main characteristics that determine whether a problem can be solved with dynamic programming: optimal substructures and overlapping subproblems. Their intersection is labeled as a dynamic programming problem.
Used as a conceptual summary of when DP applies.
"In fact, I found five indicators to look out for."
Slide title: "5 DP Indicators" with list items 1 through 5.
After summarizing the formal characteristics, the speaker introduces five practical indicators for recognizing dynamic programming problems: sequential decision-making, greedy choice property, state transition, path or array, and counting or maximization/minimization.
Used as heuristics for spotting DP problems in practice.
The narrator states: 'A problem is a dynamic programming problem if it satisfies two conditions.'
The Venn diagram visually represents the intersection of 'Optimal Substructures' and 'Overlapping Subproblems' as the 'Dynamic Programming Problem'.
A problem is a dynamic programming problem if and only if it satisfies two conditions: optimal substructure and overlapping subproblems.
The problem can be solved using an algorithmic approach.
For all problems, if it has optimal substructure and overlapping subproblems, then it is a dynamic programming problem.
The narrator says that if we find the shortest routes from San Francisco to Los Angeles and Los Angeles to San Diego, then we are guaranteed to find the overall shortest route from San Francisco to San Diego.
The displayed equation equates the whole shortest path to the sum of two shorter segment paths.
The clip does not prove why this guarantee holds; it asserts it through the example.
If the shortest SF→LA route and the shortest LA→SD route are known, then combining them yields the shortest SF→SD route in the presented example.
The route from San Francisco to San Diego goes through Los Angeles.
Each subroute is chosen to be shortest for its endpoints.
For the specific three-city route example shown in the video.
The narrator says, `This isn't ideal, and dynamic programming will help us cut out all the repeat computations we're doing.`
Repeated subtrees in the Fibonacci recursion tree are highlighted as redundant work.
The clip states the benefit qualitatively and does not provide a complexity analysis.
Dynamic programming helps eliminate repeated computations that arise from overlapping subproblems.
The recursive process contains repeated subproblems, as illustrated by the Fibonacci tree.
In the context of the Fibonacci recursion example shown.
"If these two conditions are satisfied, then dynamic programming can solve the problem much more efficiently."
The clip does not fully restate both conditions in audio here; the diagram later names them.
If the two conditions are satisfied, then dynamic programming can solve the problem much more efficiently.
The problem satisfies the two conditions discussed in the video.
For the problem under discussion, if both conditions hold, then DP is more efficient.
"If you can do this, then you will know if you can apply recursion, at which point you'll know for sure that this problem can be solved with dynamic programming."
This is the speaker's stated implication in the video; the clip does not provide a formal proof.
If you can break the problem down to the simplest subproblem and build up to the optimal solution, then you can apply recursion, and the speaker says you will know for sure that the problem can be solved with dynamic programming.
The problem can be decomposed recursively to the simplest subproblem.
Those subproblems can be built up to the optimal solution.
For the problem being tested, if the recursive decomposition and build-up succeed, then DP applies according to the video.
"In other words, if you can solve a problem using a greedy algorithm, then you can also solve it using dynamic programming."
This is presented as a heuristic claim by the speaker, not proved in the clip.
If a problem can be solved using a greedy algorithm, then it can also be solved using dynamic programming.
The problem is solvable by a greedy algorithm.
For problems exhibiting the greedy choice property as described in the video.
The step-by-step breakdown of is shown: , , , .
Boxes appear sequentially, showing the recursive calls and then the return values building up the final answer: .
Start with the problem of finding 4 factorial.
Definition of factorial.
To find 3!, we need to find 2!.
Recursive definition of factorial.
To find 2!, we need to find 1!.
Recursive definition of factorial.
Base case reached.
Definition of factorial base case.
Substitute 1! back into the equation for 2!.
Arithmetic substitution.
Substitute 2! back into the equation for 3!.
Arithmetic substitution.
Substitute 3! back into the equation for 4!.
Arithmetic substitution.
The recursive breakdown and subsequent build-up correctly computes .
The narrator explains that because the highway goes through Los Angeles, the problem can be divided into two subproblems.
Arrows mark San Francisco, Los Angeles, and San Diego on the map, and the route is visually split at Los Angeles.
The equation `SHORTEST_PATH(SF, SD) = SHORTEST_PATH(SF, LA) + SHORTEST_PATH(LA, SD)` appears after the verbal decomposition.
The derivation is example-based and intuitive rather than formal.
The original problem is stated as a route-finding task on the map.
Directly given by the narration and map example.
The narrator identifies Los Angeles as the intermediate point on the highway route.
Stated in the audio while the map highlights Los Angeles.
The whole journey is decomposed into two smaller route problems.
Explicitly described by the narrator as dividing the problem into two subproblems.
The optimal whole route is written as the sum of the optimal subroutes.
Displayed on screen and summarized verbally as the function form of the example.
The example demonstrates optimal substructure by constructing the global optimum from optimal solutions to subproblems.
A recursion tree rooted at 5 expands into repeated subtrees for 3 and 2.
The narrator says the highlighted areas are repeat work and points out repeated function calls like fib(2) and fib(3).
On-screen text references `FIB(2)` and `FIB(3)` as repeated calls.
The video does not write the recurrence `` explicitly on screen; it is inferred from the standard Fibonacci recursion tree shown.
The tree is introduced as the work needed to compute fib(5).
Narration explicitly says this is the fifth Fibonacci number example.
The tree branches downward into calls labeled 4, 3, 2, 1, and 0.
Visible in the recursion-tree diagram.
The same smaller problems appear more than once in different branches.
The narrator highlights repeated work and names fib(2) and fib(3).
The duplicated subtrees illustrate redundant recomputation in naive recursion.
Directly explained by the narration and visual highlighting.
The Fibonacci example visually establishes what overlapping subproblems mean and why they motivate dynamic programming.
The narration moves from overlapping subproblems to the two-condition test, then to trying recursion and divide-and-conquer, and finally to concluding DP applicability.
A recursion tree with repeated subtrees circled in red supports the explanation.
The video first points to repeated subtrees in a recursion tree and says the problem exhibits overlapping subproblems.
Directly stated in audio and shown by the red-circled repeated branches.
It then states that if the two conditions are satisfied, dynamic programming can solve the problem much more efficiently.
Directly stated in audio.
Next, the speaker proposes a practical test: try to solve the problem recursively and see whether divide-and-conquer works.
Directly stated in audio.
The test is refined by asking whether breaking the problem down to the simplest subproblem can eventually build up to the optimal solution.
Directly stated in audio.
Finally, the speaker concludes that if this can be done, then you can apply recursion and know the problem can be solved with dynamic programming.
Directly stated in audio.
The clip presents an intuitive route from recognizing overlapping subproblems and recursive decomposability to concluding that dynamic programming is applicable.
Venn diagram titled "Characteristics of DP Problems".
"those are the two main characteristics that determine if a problem can be solved with dynamic programming."
The left set is labeled "Optimal Substructures".
Visible in the diagram.
The right set is labeled "Overlapping Subproblems".
Visible in the diagram.
Their intersection is labeled "Dynamic Programming Problem".
Visible in the diagram.
The narration states that these are the two main characteristics determining DP solvability.
Directly stated in audio.
The visual argument summarizes DP problems as exactly those lying in the intersection of optimal substructure and overlapping subproblems.
The example explicitly calculates .
Visual boxes show the recursive calls and return values for factorial(4).
Compute the value of 4! using a recursive approach.
Find the numerical value of 4!.
Apply the recursive formula for .
Definition of factorial.
Apply the recursive formula for .
Definition of factorial.
Apply the recursive formula for .
Definition of factorial.
Reach the base case.
Definition of factorial.
Return from the base case and compute 2!.
Arithmetic.
Return and compute 3!.
Arithmetic.
Return and compute 4!.
Arithmetic.
24
The calculated value 24 matches the standard mathematical result for 4!.
A large tree diagram shows the recursive calls for fib(7), highlighting repeated nodes like fib(5), fib(4), etc.
The video does not explicitly calculate the final value of fib(7), it only shows the tree structure to illustrate overlapping subproblems.
Illustrate the recursive calls made when computing fib(7) without memoization.
(implied by the tree structure)
Base cases (implied)
Show the structure of the recursion tree for fib(7).
Root node of the tree.
Problem statement.
First level of recursion.
Definition of Fibonacci sequence.
Second level of recursion, showing repeated fib(4) and fib(5).
Definition of Fibonacci sequence.
A tree diagram with multiple repeated subproblems (e.g., fib(5) appears twice, fib(4) appears three times).
The visual representation clearly shows the exponential growth and redundancy of naive recursive Fibonacci calculation.
A Google Maps route from San Francisco to San Diego is shown with Los Angeles marked along the way.
The narrator explains the goal is to find the shortest path between the two cities and then decomposes it at Los Angeles.
The equation `SHORTEST_PATH(SF, SD) = SHORTEST_PATH(SF, LA) + SHORTEST_PATH(LA, SD)` is displayed.
No numerical distance is computed in the clip; the map shows `7 h 46 min` and `530 miles`, but these values are not used in the algebraic explanation.
Find the shortest driving route from San Francisco to San Diego using the map example.
A route from San Francisco to San Diego is shown.
The route passes through Los Angeles.
The objective is the shortest path between the endpoints.
Show how the overall shortest route can be built from shortest subroutes.
Los Angeles is singled out as the point where the route can be split.
Stated by the narrator and highlighted on the map.
The original problem becomes two smaller shortest-path problems.
Explicitly described as dividing the problem into two subproblems.
Combine the two optimal subresults to obtain the global optimum.
Displayed formula and narrated conclusion.
The shortest SF→SD route is obtained by adding the shortest SF→LA route and the shortest LA→SD route.
The narrator states that this construction guarantees the overall shortest route in the example.
A recursion tree for the 5th Fibonacci number is shown with repeated subtrees.
The narrator says this is the work required to calculate the fifth Fibonacci number and points out repeated calls like fib(2) and fib(3).
Text on screen references repeated function calls and overlapping subproblems.
The exact numeric Fibonacci value is not stated in the clip.
Use the recursion tree for the 5th Fibonacci number to illustrate repeated computation.
A recursion tree rooted at 5 is shown.
Some subtrees repeat, especially those for 2 and 3.
Demonstrate overlapping subproblems and the inefficiency of naive recursion.
The root call represents the target Fibonacci computation.
Narration identifies the example as the fifth Fibonacci number.
The same subproblems occur multiple times in the tree.
Highlighted visually and named in the audio.
Repeating identical subcalls means extra computation with no new information.
The narrator calls these highlighted areas repeat work.
The Fibonacci tree shows overlapping subproblems through repeated fib(2) and fib(3) calls.
The redundancy is verified by the visual highlighting of duplicate subtrees.
LeetCode page shows topic tag "Dynamic Programming 474" and text "You have solved problems."
"because if you look on LeetCode at various dynamic programming problems, you can actually spot patterns amongst them."
Use a list of LeetCode dynamic programming problems to look for recurring patterns.
A LeetCode page filtered by the Dynamic Programming tag.
The page shows 474 problems in that category.
Identify practical indicators of DP problems from examples.
The video opens the LeetCode Dynamic Programming topic page.
Visible on screen.
It scrolls through many DP-labeled problems such as Climbing Stairs, House Robber, Decode Ways, Jump Game II, Unique Paths, Palindromic Substrings, Coin Change, Longest Increasing Subsequence, Edit Distance, and Word Break.
Visible in the scrolling list.
The speaker uses this browsing as motivation for extracting five indicators.
Directly stated in audio.
The example serves as motivation: browsing many DP problems reveals recurring patterns that can be summarized as indicators.
The verification is the subsequent slide listing the five indicators.
"Some examples on LeetCode that exhibit sequential decision making are the problems House Robber and Decode Ways."
Text appears beside item 1: "HOUSE ROBBER (198)" and "DECODE WAYS (91)".
Name LeetCode problems that illustrate sequential decision-making.
The indicator is sequential decision-making.
Provide concrete examples.
The speaker names House Robber as an example.
Directly stated in audio and shown on slide.
The speaker names Decode Ways as an example.
Directly stated in audio and shown on slide.
House Robber (198) and Decode Ways (91).
Both titles and numbers are visible on the slide.
"Some examples of this are the Jump Game II and Best Time to Buy and Sell Stock II."
Text appears beside item 2: "JUMP GAME II (45)" and "BEST TIME TO BUY AND SELL STOCK II (122)".
Name LeetCode problems that illustrate the greedy choice property indicator.
The indicator is greedy choice property.
Provide concrete examples.
The speaker names Jump Game II as an example.
Directly stated in audio and shown on slide.
The speaker names Best Time to Buy and Sell Stock II as an example.
Directly stated in audio and shown on slide.
Jump Game II (45) and Best Time to Buy and Sell Stock II (122).
Both titles and numbers are visible on the slide.
"Some examples of this are the Climbing Stairs and Minimum Path Sum problems."
Text appears beside item 3: "CLIMBING STAIRS (70)" and "MINIMUM PATH SUM (64)".
Name LeetCode problems that illustrate state transition.
The indicator is state transition.
Provide concrete examples.
The speaker names Climbing Stairs as an example.
Directly stated in audio and shown on slide.
The speaker names Minimum Path Sum as an example.
Directly stated in audio and shown on slide.
Climbing Stairs (70) and Minimum Path Sum (64).
Both titles and numbers are visible on the slide.
Text 'UNIQUE PATHS (62)' and 'LONGEST INCREASING SUBSEQUENCE (300)' appear on screen.
The speaker mentions these as good examples of the path or array indicator.
Identify problems that fit the 'Path or Array' dynamic programming indicator.
Unique Paths (LeetCode 62)
Longest Increasing Subsequence (LeetCode 300)
Provide examples of problems involving optimal paths or array arrangements.
The speaker lists Unique Paths and Longest Increasing Subsequence as examples.
These problems involve finding optimal paths or arranging elements in an array.
Unique Paths (62), Longest Increasing Subsequence (300).
The examples match the definition of the Path or Array indicator.
Text 'COIN CHANGE (322)' and 'MAXIMUM SUBARRAY (53)' appear on screen.
The speaker mentions these as notable examples on LeetCode for the counting or optimization indicator.
Identify problems that fit the 'Counting or Maximization/Minimization' dynamic programming indicator.
Coin Change (LeetCode 322)
Maximum Subarray (LeetCode 53)
Provide examples of problems involving counting ways or optimizing a value.
The speaker lists Coin Change and Maximum Subarray as examples.
These problems require counting the number of ways to achieve a goal or maximizing/minimizing a value.
Coin Change (322), Maximum Subarray (53).
The examples match the definition of the Counting or Optimization indicator.
Four blue-outlined boxes appear sequentially from left to right, representing the recursive calls factorial(4), factorial(3), factorial(2), and factorial(1). Then, numbers appear inside them from right to left, representing the return values 1, 2, 6, and 24.
Box for factorial(4)
Box for factorial(3)
Box for factorial(2)
Box for factorial(1)
Arrows indicating recursive calls
Arrows indicating return values
Boxes appear sequentially from left to right.
Numbers appear sequentially from right to left inside the boxes.
The structure of the boxes remains constant once they appear.
Visually demonstrates the 'break down' phase (left to right) and the 'construct solution' phase (right to left) of dynamic programming/recursion.
A large binary tree diagram is shown with nodes labeled fib(7), fib(6), fib(5), etc. Some nodes are highlighted in teal to show repetition.
Nodes labeled fib(n)
Edges connecting parent to child nodes
Highlighted nodes (e.g., fib(5), fib(4))
None, it is a static diagram.
The tree structure represents the recursive calls.
Illustrates the concept of overlapping subproblems, where the same subproblem (e.g., fib(5)) is computed multiple times in a naive recursive approach.
A Venn diagram with two intersecting circles. The left circle is cyan and labeled 'Optimal Substructures'. The right circle is yellow and labeled 'Overlapping Subproblems'. The intersection is green and labeled 'Dynamic Programming Problem'.
Cyan circle (Optimal Substructures)
Yellow circle (Overlapping Subproblems)
Green intersection (Dynamic Programming Problem)
Red arrows point to the circles as the narrator explains them.
The spatial relationship of the sets.
Visually defines that a Dynamic Programming Problem is the intersection of problems having Optimal Substructure and Overlapping Subproblems.
A Venn diagram titled `Characteristics of DP Problems` shows two overlapping circles labeled `Optimal Substructures` and `Overlapping Subproblems`, with the intersection labeled `Dynamic Programming Problem`.
Title text `Characteristics of DP Problems`.
Left circle `Optimal Substructures`.
Right circle `Overlapping Subproblems`.
Intersection label `Dynamic Programming Problem`.
The diagram remains static while the narrator introduces the first concept.
The intersection continues to represent the conjunction of both properties.
The visual encodes that a DP problem is characterized by having both optimal substructure and overlapping subproblems.
A map of California shows a route from San Francisco to San Diego, with arrows pointing to San Francisco, Los Angeles, and San Diego.
Text overlays appear in sequence: `HOW?`, `DIVIDED INTO 2 SUBPROBLEMS`, `GUARANTEED TO FIND THE OVERALL SHORTEST ROUTE`, and `FROM SF -> SD`.
The equation `SHORTEST_PATH(SF, SD) = SHORTEST_PATH(SF, LA) + SHORTEST_PATH(LA, SD)` appears near the end of the segment.
Google Maps route from San Francisco to San Diego.
Red arrows marking cities.
Overlay text describing subproblems and guarantee.
Shortest-path equation.
The route is first shown as one whole trip.
Los Angeles is highlighted as the split point.
The problem is reframed as two subroutes.
The final equation summarizes the decomposition.
The endpoints remain San Francisco and San Diego throughout the example.
The animation turns a geographic route into an algorithmic decomposition example for optimal substructure.
A simple white-background animation shows `P` branching into `step` and another `P`, repeating downward until `BASE CASE` appears.
Text at the bottom reads `ANIMATION SHOWING HOW RECURSION WORKS`.
The animation is schematic and does not specify a concrete function besides the generic label P.
Node `P`.
Branch label `step`.
Terminal label `BASE CASE`.
The structure grows downward level by level.
Each recursive call produces another `step` and another `P`.
The process terminates at `BASE CASE`.
The pattern of self-reference remains the same at each level.
The animation visualizes recursion as repeated self-calling until a stopping condition is reached.
A recursion tree rooted at 5 expands into nodes labeled 4, 3, 2, 1, and 0.
Red outlines highlight repeated subtrees, and captions mention `FIB(2)` and `FIB(3)`.
The exact edge rule is not written on screen, though the tree matches the standard Fibonacci recursion pattern.
Root node 5.
Internal nodes 4, 3, 2.
Leaf nodes 1 and 0.
Red highlight loops around repeated subtrees.
The full tree is shown first.
Repeated regions are then emphasized visually.
Captions identify fib(2) and fib(3) as examples of repeated calls.
The tree structure itself does not change; only the highlighting and explanatory overlays change.
The duplicated subtrees make overlapping subproblems visible as repeated recursive work.
A binary recursion tree is displayed with nodes labeled by numbers and red circles around repeated subtrees.
The exact recurrence represented by the tree is not stated in the clip.
Root node labeled 5
Internal nodes labeled 4, 3, 2
Leaf nodes labeled 1 and 0
Red circles around repeated subtrees
The narration progresses from identifying overlapping subproblems to explaining how recursion and DP relate.
The red circles remain fixed while subtitles change beneath the tree.
The tree structure itself does not animate during this interval.
Repeated subtrees stay highlighted in red.
The static tree visually demonstrates that recursive decomposition can revisit the same subproblem multiple times, supporting the idea of overlapping subproblems.
A Venn diagram titled "Characteristics of DP Problems" appears.
Left circle labeled "Optimal Substructures"
Right circle labeled "Overlapping Subproblems"
Intersection labeled "Dynamic Programming Problem"
The scene cuts from the recursion tree to the Venn diagram.
No further animation occurs within the diagram interval.
The two-set structure remains fixed.
The intersection label remains visible throughout.
The diagram summarizes the conceptual claim that DP problems are characterized by the intersection of optimal substructure and overlapping subproblems.
A LeetCode page filtered by Dynamic Programming is scrolled vertically.
Topic tag "Dynamic Programming 474"
Problem rows with titles, acceptance rates, and difficulty labels
Sidebar with company counts
The page scrolls downward through many DP problems.
The cursor moves over the topic area before scrolling.
The page remains within the LeetCode interface.
The list stays filtered to Dynamic Programming.
The scrolling example motivates the shift from formal characteristics to practical pattern recognition across many DP problems.
A slide titled "5 DP Indicators" lists five numbered items, with the current item highlighted in red as the narration advances.
Item 4 is introduced only at the very end of the clip and is not explained within the provided duration.
Title "5 DP Indicators"
Numbered list items 1 through 5
Example problem labels appearing beside items 1 through 3
The highlight moves from item 1 to item 2 to item 3 as the narration proceeds.
Example problem names appear beside the corresponding indicator.
The overall list of five indicators remains on screen throughout.
The slide format stays constant.
The slide organizes the practical recognition heuristics for DP problems and ties each indicator to named LeetCode examples.
The fourth item '4. Path or Array' is colored red while the others are white.
List of 5 DP Indicators
Item 4 changes color to red
Items 1, 2, 3, and 5 remain white
Directs viewer attention to the current topic being discussed.
The narrator clarifies: 'The term doesn't refer to programming in the coding sense. However, Bellman used programming to signify planning and decision making.'
Learners often assume 'programming' in 'dynamic programming' refers to writing computer code.
In this context, 'programming' means planning and decision making, as coined by Richard Bellman.
The narrator says, `Now, recursion is a topic that deserves its own video, but for this example we'll touch on it briefly.`
Viewers might assume this clip fully explains recursion.
The video explicitly limits itself to a brief, example-driven introduction to recursion for the purpose of explaining overlapping subproblems.
After saying overlapping subproblems are when problems overlap, the narrator asks, `Obviously, but what is meant by this?`
The Fibonacci tree is then used to define the term concretely.
The phrase `problems overlap` sounds tautological if taken literally.
The clip clarifies the term by showing repeated recursive calls such as fib(2) and fib(3) in a computation tree.
"if you can solve a problem using a greedy algorithm, then you can also solve it using dynamic programming."
The clip presents this as a heuristic relationship and does not discuss counterexamples or limitations.
One might think that if a problem is solvable greedily, it is not a dynamic programming problem.
The video explicitly says the opposite: if a problem can be solved using a greedy algorithm, then it can also be solved using dynamic programming.
The speaker recommends first checking whether the problem can be solved recursively and then concludes that this indicates DP applicability.
The clip does not separately explain memoization versus tabulation.
One might stop at plain recursion after decomposing the problem.
The video uses recursion as a diagnostic step, then argues that once overlapping subproblems and optimal build-up are present, dynamic programming is the more efficient approach.
The Venn diagram shows 'Optimal Substructures' as a necessary component of 'Dynamic Programming Problem'.
The definition of dynamic programming inherently requires the property of optimal substructure.
The Venn diagram shows 'Overlapping Subproblems' as a necessary component of 'Dynamic Programming Problem'.
The definition of dynamic programming inherently requires the property of overlapping subproblems.
The Venn diagram places `Dynamic Programming Problem` in the intersection of `Optimal Substructures` and `Overlapping Subproblems`.
The general DP characterization contains optimal substructure as one of its two required properties.
The narrator introduces overlapping subproblems as the second characteristic.
The Venn diagram and later Fibonacci tree both connect this property to DP.
The DP characterization also contains overlapping subproblems as the second required property.
The shortest-path example is used immediately after introducing optimal substructure.
The equation directly expresses composition of optimal subresults.
The route example applies the definition of optimal substructure to a concrete shortest-path problem.
The generic recursion animation precedes the Fibonacci tree.
The narrator uses recursion to explain where repeated calls come from.
Understanding recursive self-calling is used as background for recognizing repeated subproblems in the Fibonacci tree.
The Fibonacci recursion tree is explicitly presented as the simplest example of overlapping subproblems.
The Fibonacci example applies the abstract notion of overlapping subproblems to a visible recursive computation.
The narrator says dynamic programming will help cut out repeat computations.
The DP framework is presented as the method that addresses the inefficiency caused by overlapping subproblems.
The narration links overlapping subproblems to the claim that DP can solve the problem more efficiently.
The recursion tree highlights repeated subtrees.
Recognizing overlapping subproblems is presented as one of the conditions needed before concluding that dynamic programming is appropriate.
The speaker first mentions "these two conditions" and later says "those are the two main characteristics".
The Venn diagram names the two characteristics explicitly.
The verbal two-condition test and the Venn diagram express the same conceptual criterion for DP problems.
After summarizing the two main characteristics, the speaker says, "But before I end this video, I want to talk about a few patterns that you can look out for."
The scene changes from the Venn diagram to the "5 DP Indicators" slide.
The five indicators are presented as practical patterns derived from looking at many DP problems after stating the formal characteristics.
Item 1 on the "5 DP Indicators" slide is "Sequential Decision-Making".
"The first of which is sequential decision making."
Sequential decision-making is one member of the five-indicator list.
Narrator defines dynamic programming.
Factorial example is used to explain recursion and building up solutions.
Narrator lists the two conditions for a DP problem.
Venn diagram shows the two conditions.
Narrator explains the historical meaning of 'programming' in this context.
The narrator defines optimal substructure in words and illustrates it with the route example.
Los Angeles is highlighted as the intermediate point in the map example.
The shortest-path equation is centered on LA as the split point.
The narrator introduces overlapping subproblems and then explains them through repeated calls.
The Fibonacci tree shows duplicated subtrees.
Repeated fib(2) and fib(3) subtrees are highlighted in the recursion tree.
A simple P → step → P animation ends at BASE CASE.
"If these two conditions are satisfied..."
Venn diagram labels the two conditions.
The speaker describes trying recursion and divide-and-conquer as a test.
"then it will exhibit overlapping subproblems."
Repeated subtrees are circled in red.
Covered · Introduction to the struggles of learning dynamic programming and the video's goal to explain the basics.
Covered · Definition of dynamic programming illustrated with the recursive calculation of 4!.
Covered · Visual demonstration of overlapping subproblems using the Fibonacci recursion tree.
Covered · Historical context of the term 'dynamic programming' coined by Richard Bellman, clarifying the meaning of 'programming'.
Covered · Explanation of the two key characteristics of dynamic programming problems: optimal substructure and overlapping subproblems, shown via a Venn diagram.
Covered · Opening Venn diagram introduces the two DP characteristics and begins the optimal-substructure discussion.
Covered · Map example explains optimal substructure through shortest-path decomposition and displays the equation.
Covered · Transition to the second DP characteristic using a Fibonacci-style recursion tree.
Covered · Brief generic animation explains recursion and its base case.
Covered · Fibonacci recursion tree is used to show repeated subproblems and motivate dynamic programming; the final seconds continue the same example and transition toward a broader statement about recursion.
Covered · Recursion-tree explanation of overlapping subproblems and the practical recursion test for DP.
Covered · Venn diagram summarizing the two main characteristics of DP problems.
Covered · LeetCode browsing used as motivation for recognizing patterns among DP problems.
Covered · Introduction of the five indicators and explanation of sequential decision-making with examples.
Covered · Explanation of the greedy choice property and its relation to DP solvability.
Covered · Explanation of state transition with examples; the fourth indicator is only named at the end and not developed within this clip.
Covered · Discussion of the fourth indicator: Path or Array.
Covered · Discussion of the fifth indicator: Counting or Maximization/Minimization.
Covered · Summary of the five indicators and outro/call to action.
Reviewed English material v1: card/concept_card "Definition of Dynamic Programming" starting at 0:50 — Dynamic programming is an algorithmic optimization technique that solves complex problems by recursively breaking them into smaller, overlapping subproblems and combining their solutions to form the solution to the original problem.
Reviewed English material v1: example/example "Shortest path from San Francisco to San Diego" from 2:11 to 3:02 — Find the shortest driving route from San Francisco to San Diego using the map example. Show how the overall shortest route can be built from shortest subroutes. Los Angeles is singled out as the point where the route can be split.; The original problem becomes two smaller shortest-path problems.; Combine the two optimal subresults to obtain the global optimum.
The Fibonacci recursion tree demonstrates repeated computation by showing that the same subproblems are solved multiple times. For example, when computing the fifth Fibonacci number, the tree contains duplicated subcalls such as fib(2) and fib(3).
Conditions: The problem is being solved recursively.; The recursive decomposition revisits the same subproblem multiple times.
Overlapping subproblems occur when the same smaller problem is solved multiple times during a recursive process. The Fibonacci tree makes this concrete by showing repeated calls such as fib(2) and fib(3), which cause redundant computation.
Conditions: The problem can be solved recursively.; The same subproblems appear multiple times in the recursion tree.
The San Francisco to San Diego route can be split at Los Angeles because the problem exhibits optimal substructure. Once the shortest SF→LA and LA→SD segments are known, they combine to give the shortest SF→SD route.
Conditions: The example assumes the overall route passes through Los Angeles.; The narrator states that finding the shortest subroutes guarantees the overall shortest route in this example.
Richard Bellman used the term 'programming' to signify planning and decision making, not writing computer code. The term 'dynamic' refers to the ever-changing nature of the problems being solved.
Conditions: The context is the historical origin of the term 'dynamic programming'.
To determine if a problem can be solved with dynamic programming, first try to solve the problem recursively. Then check whether divide-and-conquer style decomposition works, and whether breaking the problem down to the simplest subproblem can eventually build up to the optimal solution.
Conditions: You are trying to decide whether a problem is suitable for dynamic programming.
The factorial example illustrates the core mechanism of dynamic programming by showing how a complex problem is recursively broken down into smaller subproblems and then built back up. To compute , it depends on , which depends on , which depends on .
Conditions: The problem can be broken down into smaller subproblems.; The solution to the original problem can be constructed from the solutions to the subproblems.
Optimal substructure means that the optimal solution to the overall problem can be constructed by efficiently combining the optimal solutions of its subproblems. In the route example, once the shortest SF→LA and LA→SD segments are known, they combine to give the shortest SF→SD route.
Conditions: The problem can be divided into subproblems.; The optimal solution of the main problem depends on the optimal solutions of its subproblems.
A problem is a dynamic programming problem if it satisfies two conditions: optimal substructure and overlapping subproblems. Optimal substructure means the optimal solution to the overall problem can be constructed from the optimal solutions of its subproblems.
Conditions: The problem can be solved using an algorithmic approach.; The problem can be divided into subproblems.
YouTube2:45Explanation of Shortest path models, complementing the current application.
Shortest path models
YouTube4:21Explanation of Shortest path models, complementing the current application.
Shortest path models
YouTube5:25Explanation of Shortest path models, complementing the current application.
Shortest path models