Part 1: The Dialogue of Dynamic Programming
A Socratic Journey between the Philosopher and the Seeker
The Problem
You are given an m x n integer matrix points (with rows indexed 0 to m-1 and columns indexed 0 to n-1).
Starting from row 0, you must select exactly one cell from each row. The score you receive is calculated as:
Score = Sum of all selected cell values - Sum of penalties
Where the penalty between two consecutive selections is:
- If you pick
points[r][c1]in rowr, and then pickpoints[r+1][c2]in rowr+1 - The penalty is
|c1 - c2|(absolute difference between column indices)
Return the maximum score you can achieve.
Example
Input: points = [[1,2,3],
[1,5,1],
[3,1,1]]
Output: 9
Explanation:
- Pick points[0][2] = 3
- Pick points[1][1] = 5 (penalty = |2-1| = 1)
- Pick points[2][0] = 3 (penalty = |1-0| = 1)
- Score = (3 + 5 + 3) - (1 + 1) = 9
In the agora of Athens, a young seeker approaches the philosopher with a scroll containing this very puzzle...
Act I: The Recognition of Interdependence
The Seeker's Opening
SEEKER: Master, I have discovered a principle: "If choices depend on future outcomes, then one must employ Dynamic Programming."
SOCRATES: Ah! You speak of interdependence. Tell me, young one, what led you to this insight?
SEEKER: Observe this matrix of values. I must choose one from each row. At first, I thought: simply pick the largest in each row!
SOCRATES: And did this strategy serve you well?
SEEKER: (hesitates) Not always. When I pick a cell, I pay a price—a penalty based on distance to my next choice. Sometimes a smaller immediate gain leads to a greater final reward.
SOCRATES: So the best choice NOW depends on what comes AFTER?
SEEKER: Yes! That is precisely it.
SOCRATES: Then you have learned the first truth: Greedy wisdom fails when present choices alter future landscapes.
Act II: The Question of Sequence
On the Nature of Order
SEEKER: But Master, I am troubled. The problem states: "Pick one from each row." It says nothing about ORDER. Why then do we proceed row by row—first, second, third?
SOCRATES: An excellent question! Tell me, CAN you skip a row?
SEEKER: No, I must pick from every row.
SOCRATES: Then if you stand at the fifth row, where were you in your previous step?
SEEKER: The fourth row... I see! I cannot have been at the second row without first passing through the third.
SOCRATES: Precisely. But let me probe deeper: Does the problem FORCE you to think sequentially?
SEEKER: (uncertain) The rules say only that penalties are calculated between adjacent rows...
SOCRATES: Ah! So I could MENTALLY consider row five before row three?
SEEKER: I... yes, I suppose so. But...
SOCRATES: But what troubles you?
The Paradox of Physical Truth
SEEKER: The penalties! They are calculated based on the POSITIONS in the matrix, not my mental order!
SOCRATES: Speak more on this.
SEEKER: If I pick column 2 in row 1, and column 5 in row 2, the penalty is the distance between them—regardless of whether I contemplated row 2 before or after row 1.
SOCRATES: So the penalty is written into the very fabric of the problem's geometry?
SEEKER: Yes! Like cities along a road. The distance between Athens and Sparta does not change based on the order I plan my journey.
SOCRATES: Then you have learned the second truth: The rules do not mandate sequence, but MATHEMATICS does. To calculate the value at row five, you need the value at row four. This is not constraint, but dependency.
Act III: The Mystery of State
What Must Be Remembered?
SOCRATES: Now, let us explore a deeper mystery. Tell me, what is this dp[r][c] you speak of?
SEEKER: It is... (struggles) ...the maximum value at position r, c?
SOCRATES: Is it the value OF that cell, or something more?
SEEKER: More, I think. But I cannot articulate it clearly.
SOCRATES: Then let us reason together. Imagine you have journeyed through many rows and arrived at row 3, column 2. What information do you carry?
SEEKER: I carry... points I have accumulated?
SOCRATES: All the points from the beginning of your journey?
SEEKER: Yes! From row 0 to row 3!
SOCRATES: So dp[3][2] is not merely the value at that cell, but the entire accumulated wealth of your journey up to that point?
SEEKER: (eyes widening) Yes! It is the running total!
The Nature of Accumulation
SOCRATES: Then when you write:
dp[r][c] = dp[r-1][prev_c] + points[r][c] - penalty
What does each term represent?
SEEKER: Let me think... dp[r-1][prev_c] is the wealth I had accumulated up to the previous row...
SOCRATES: Continue.
SEEKER: And points[r][c] is what I gain at the current position...
SOCRATES: And the penalty?
SEEKER: What I must pay for the distance I traveled!
SOCRATES: So the formula says: "My total wealth NOW equals my total wealth BEFORE, plus what I gain, minus what I lose"?
SEEKER: Yes! It is a running sum across my entire journey!
SOCRATES: Then you have learned the third truth: DP values are not snapshots, but chronicles. Each carries the weight of all decisions past.
Act IV: The Great Paradox
The Seeker's Confusion
SEEKER: Master, I am deeply troubled. You have led me to two insights that seem to contradict each other.
SOCRATES: Speak your concern.
SEEKER: First, you helped me see that "my choices NOW affect what is optimal LATER." This is why we need Dynamic Programming.
SOCRATES: Yes, I recall.
SEEKER: But then you said, "The past does not matter—only where you ARE matters, not how you ARRIVED."
SOCRATES: And this troubles you?
SEEKER: These seem OPPOSITE! If the future depends on my choices, doesn't that mean the past matters?
SOCRATES: (smiling) Ah! Now we approach the heart of wisdom. Tell me, are these truly opposite, or do they speak of different things?
The Two Rivers of Causation
SOCRATES: Let us examine each carefully. First: "My choice now affects what is optimal later." What does this mean?
SEEKER: If I choose column 0 at row 1, then choosing column 0 at row 2 costs me nothing. But if I choose column 5 at row 1, then choosing column 0 at row 2 costs me five units.
SOCRATES: So your present choice changes the LANDSCAPE of the future?
SEEKER: Exactly!
SOCRATES: This is what I call Forward Causation—the river that flows from present to future. Your current action shapes what comes after.
The Backward Gaze
SOCRATES: Now the second statement: "The past doesn't matter, only current state." Let me pose a question: Suppose you stand at row 5, column 3, with a score of 100 points.
SEEKER: Yes?
SOCRATES: Does it matter whether you arrived there via row 4 column 1, or row 4 column 5?
SEEKER: (thinking) For future decisions... no! The penalty going forward only depends on where I am NOW (column 3), not where I was before.
SOCRATES: So two different histories can lead to the same present, and from that present, the optimal future is the same?
SEEKER: Yes!
SOCRATES: This is what I call Backward Information—how much history must you remember to make optimal future decisions?
The Resolution
SOCRATES: So let us revisit your paradox:
"My choice now affects what is optimal later"
- Meaning: Present shapes future
- Direction: Forward causation (now → later)
- Example: Picking column 0 vs. 5 changes tomorrow's landscape
"The past doesn't matter, only current state"
- Meaning: To decide optimally from here forward, I need only know WHERE I am, not HOW I arrived
- Direction: Backward information (what history must I carry?)
- Example: At row 5 col 3, the path from row 0 can be forgotten
SEEKER: (illuminated) They describe DIFFERENT DIRECTIONS! One looks forward, one looks backward!
SOCRATES: Indeed! And this is the essence of what mathematicians call the Markov Property.
Act V: The Markov Property
The Principle of Sufficient Present
SOCRATES: Tell me, young seeker, if you wished to know the entire history of Athens, how far back would you look?
SEEKER: To the beginning, to Theseus and beyond!
SOCRATES: But if you wished to know what Athens should do TOMORROW, would you need to know every detail of every year past?
SEEKER: (considering) No... I would need to know our current state—our strength, our alliances, our resources...
SOCRATES: Exactly! The present STATE contains all relevant information from the past, compressed into a single moment. The HOW of arrival matters not; only the WHERE of being.
SEEKER: So in our problem, dp[r][c] is like the "state of Athens"—it summarizes everything that came before?
SOCRATES: Precisely! The value dp[r][c] already encodes all optimal choices from row 0 to row r. You need not trace back through the entire history.
Why This Makes DP Possible
SOCRATES: Consider: without this property, what would your state be?
SEEKER: I would need to remember... every cell I picked? Every path I took?
SOCRATES: And how many such paths are there?
SEEKER: (calculating) If there are n columns and m rows... n^m paths!
SOCRATES: For 100,000 rows and 100,000 columns?
SEEKER: (horrified) 100,000^100,000... the stars in the heavens are fewer!
SOCRATES: But WITH the Markov property, how many states?
SEEKER: Just m × n... just the positions in the grid!
SOCRATES: From impossible to possible. This is the gift of the Markov property.
The present, wisely chosen, contains all necessary past.
Act VI: The Synthesis
The Complete Understanding
SOCRATES: Let us now bring all threads together. When we write:
dp[r][c] = max over all prev_c of:
(dp[r-1][prev_c] + points[r][c] - abs(prev_c - c))
Tell me what each part represents, using all you have learned.
SEEKER: dp[r-1][prev_c] is the chronicle—the accumulated wisdom of all choices from the beginning to row r-1, column prev_c.
SOCRATES: Good. Continue.
SEEKER: points[r][c] is the gain at the current moment.
SOCRATES: And?
SEEKER: abs(prev_c - c) is the price paid for movement—the penalty for distance.
SOCRATES: So the whole formula?
SEEKER: It says: "The best chronicle to reach THIS moment equals the best chronicle to reach the PREVIOUS moment, plus current gain, minus current cost." We try all possible previous moments and choose the best!
SOCRATES: And why do we not need to look back further than row r-1?
SEEKER: Because dp[r-1][prev_c] already contains the optimal history! The Markov property lets us forget the distant past.
SOCRATES: And why is this Dynamic Programming, not greedy?
SEEKER: Because my choice NOW changes what is optimal LATER. Forward causation links all decisions. I cannot be greedy at each step.
SOCRATES: And yet we can solve it efficiently?
SEEKER: Because backward information is LIMITED. I only need the immediate previous state, not the entire history.
The Philosopher's Summary
SOCRATES: Then you have grasped the essence. Let me state it thus:
- Interdependence: Present choices shape future landscapes (forward causation)
- Sequential Dependency: Mathematics, not rules, demand order (computational necessity)
- Accumulated State: DP values are chronicles, not snapshots (running totals)
- Markov Property: The present contains sufficient past (limited backward dependency)
- Efficient Synthesis: We build the future from the immediate past, one stage at a time
SEEKER: Master, I feel I have glimpsed something profound.
SOCRATES: You have learned not just how to solve one problem, but how to THINK about a class of problems. This is the method of Dynamic Programming—to recognize when the present depends on the past, yet can forget most of it; when the future depends on the present, yet can be built stage by stage.
Epilogue: The Code Revealed
SEEKER: And the implementation, Master?
SOCRATES: Ah, implementation is merely the shadow of understanding. But since you have earned understanding, here is its shadow:
def maxPoints(points):
m = len(points)
n = len(points[0])
# dp[r][c] = chronicle of accumulated points to row r, column c
dp = [[0] * n for _ in range(m)]
# Base case: the first row has no history
for c in range(n):
dp[0][c] = points[0][c]
# Build each subsequent chronicle from the previous
for r in range(1, m):
prev_r = r - 1
for c in range(n):
# Consider all possible previous positions
for prev_c in range(n):
# Chronicle from before + current gain - movement cost
score = dp[prev_r][prev_c] + points[r][c] - abs(prev_c - c)
dp[r][c] = max(dp[r][c], score)
# The answer is the best chronicle in the final row
return max(dp[m-1])
SEEKER: It is beautiful in its simplicity, now that I understand.
SOCRATES: That is because you have learned not just WHAT to do, but WHY. Understanding transforms complexity into clarity.
Final Reflection
As the sun sets over Athens, the seeker departs with new wisdom...
The Meta-Lesson
The Socratic method revealed that apparent contradictions often illuminate different facets of truth:
- Forward causation (why DP, not greedy)
- Backward information (why DP is tractable)
The dialogue showed that asking "why?" at each step—especially when something seems contradictory—leads to deeper understanding than memorizing formulas ever could.
Thus ends the dialogue on Dynamic Programming.