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Dynamic Programming Visualization
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Understand how dynamic programming solves complex problems by breaking them into simpler subproblems
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<!-- Algorithm Selection Panel -->
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<h2 class="text-xl font-bold text-white mb-4">DP Problems</h2>
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<h3 class="font-medium text-white">Fibonacci Sequence</h3>
<p class="text-sm text-gray-300 mt-1">Classic recursive problem with overlapping subproblems</p>
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<h3 class="font-medium text-white">0/1 Knapsack</h3>
<p class="text-sm text-gray-300 mt-1">Optimization problem with capacity constraints</p>
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<h3 class="font-medium text-white">Longest Common Subsequence</h3>
<p class="text-sm text-gray-300 mt-1">Finding longest sequence common to both strings</p>
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Time: O(n)
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Space: O(n)
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<h3 class="font-bold text-white mb-4">Computing Fibonacci(6)</h3>
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<th class="dp-cell font-bold">n</th>
<th class="dp-cell font-bold">0</th>
<th class="dp-cell font-bold">1</th>
<th class="dp-cell font-bold">2</th>
<th class="dp-cell font-bold">3</th>
<th class="dp-cell font-bold">4</th>
<th class="dp-cell font-bold">5</th>
<th class="dp-cell font-bold">6</th>
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<td class="dp-cell font-bold">fib(n)</td>
<td class="dp-cell calculated">0</td>
<td class="dp-cell calculated">1</td>
<td class="dp-cell calculated">1</td>
<td class="dp-cell calculated">2</td>
<td class="dp-cell calculated">3</td>
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<span class="text-purple-300">fib(6) = fib(5) + fib(4) = 5 + 3 = </span>
<span class="text-white font-bold text-xl">8</span>
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<h3 class="font-bold text-white mb-4">Recursion Tree Visualization</h3>
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<div class="text-xs text-gray-400 mt-1">Calculated</div>
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Using memoization to avoid redundant calculations
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<p class="text-gray-300">Calculating fib(6) by adding previously computed values fib(5)=5 and fib(4)=3.</p>
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<h3 class="font-bold text-white mb-2">Memoized Fibonacci</h3>
<pre class="text-green-400 text-sm">
def fibonacci(n, memo={}):
if n in memo:
return memo[n]
if n <= 1:
return n
memo[n] = fibonacci(n-1, memo) + fibonacci(n-2, memo)
return memo[n]</pre>
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<h3 class="font-bold text-white mb-3">Mathematical Analysis</h3>
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\( F(n) = F(n-1) + F(n-2) \text{ for } n > 1 \)
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The Fibonacci sequence exhibits optimal substructure and overlapping subproblems,
making it ideal for dynamic programming optimization.
</p>
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\( \text{Time Complexity: } O(n) \text{ with memoization} \)
</div>
<p class="text-gray-300 mt-3">
Without memoization, the naive recursive approach has exponential time complexity
\( O(\phi^n) \) where \( \phi \) is the golden ratio. Memoization reduces this to
linear time by storing previously computed values.
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