Asked in: AMAZON
Image of the Question
All Testcases Passed ✔
Solution
def optimizeCalorieBurn(height):
# Write your code here
height.sort()
arr = []
// ... rest of solution available after purchaseExplanation
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This problem asks us to find the maximum total calories burned by jumping exactly once on each block, starting from the ground (height 0) and never returning to the ground afterwards. The calorie burn for a jump from block i to block j is the square of the difference in heights: (height[i] - height[j])². The goal is to choose the order of jumps to maximize the sum of these calorie burns.
To solve this problem effectively, consider the following thought process and approach directions:
1. **Understanding the problem context and constraints:**
- We have n blocks, each with a specific height.
- The user starts on the ground (height 0).
- The user must visit each block exactly once.
- The user jumps from the ground to some block initially, then from block to block, never returning to the ground.
- The calorie burn is related to the square of the height differences.
- We want to find the order of jumps maximizing total calories burned.
2. **Key observations:**
- Since the calorie burn is based on the square of height differences, larger differences produce significantly more calories burned.
- The jump from the ground to the first block also contributes calories: (0 - height[first])².
- After the first jump, the user must visit all remaining blocks in some order.
- The order can be any permutation of the blocks.
- The main challenge is to determine an order that maximizes the sum of squared height differences between consecutive jumps.
3. **Considering the structure of the problem:**
- This is essentially a permutation optimization problem over n blocks.
- Trying all permutations (n!) is impossible for large n.
- The problem involves maximizing the sum of squared differences between consecutive elements in a sequence.
- This resembles the problem of finding a path that maximizes “adjacent difference squares” in a permutation.
4. **Insights about maximizing squared differences:**
- Squared difference amplifies large gaps, so jumps between very different heights are favorable.
- To maximize sum of squared differences, the order should ideally alternate between the smallest and largest remaining blocks to create large height differences.
- Intuitively, the jumps should "zig-zag" between blocks with very different heights rather than jumping between blocks of similar heights.
5. **Potential strategy - arranging blocks in a zig-zag order:**
- Sort the blocks by height.
- Pick blocks alternately from the ends of the sorted list to maximize height differences between consecutive jumps.
For example, if sorted heights are h1 ≤ h2 ≤ ... ≤ hn, pick h1, then hn, then h2, then hn-1, and so on.
- This arrangement can create large jumps between low and high blocks, maximizing squared differences.
6. **Initial jump from ground:**
- The first jump is from height 0 to the first chosen block.
- Picking a block with a larger height as the first jump is favorable since (0 - height[first])² is higher.
- So consider starting the sequence with either the largest or the smallest block and evaluate which ordering yields the higher total calories.
7. **Dynamic programming approach to maximize calories:**
- Since you must visit each block exactly once in some order, and you want the maximum sum of squared differences, this is similar to a path-finding problem on a fully connected weighted graph.
- The nodes are blocks (plus ground at height 0).
- The weight of an edge between two blocks i and j is (height[i] - height[j])².
- The starting node is ground (height 0), connected to all blocks.
- Find a Hamiltonian path (visiting all nodes exactly once) starting from ground that maximizes sum of edge weights.
- This is a variation of the Traveling Salesman Problem (TSP) with maximization instead of minimization.
8. **Handling the complexity of TSP:**
- The TSP is NP-hard, and with n up to 10^5, exact DP-based TSP solutions are impossible.
- We need a heuristic or greedy strategy guided by the insight about zig-zag ordering.
- Sorting and alternately selecting from the ends is a heuristic that works well to maximize differences.
- Try both possibilities for the first block (smallest or largest) and choose the better total calories.
9. **Outline of heuristic solution:**
- Sort heights ascending.
- Create two candidate sequences:
- Sequence A: Start with smallest block, then largest, then second smallest, then second largest, etc.
- Sequence B: Start with largest block, then smallest, then second largest, then second smallest, etc.
- Calculate total calories burned for both sequences including initial jump from ground.
- Choose the sequence with maximum total calories.
10. **Why this approach works well:**
- Alternating extremes maximize jumps between blocks with large height differences.
- Squared difference ensures larger height gaps yield larger calorie burns.
- Considering starting from either smallest or largest block ensures optimal initial jump from ground.
11. **Additional considerations:**
- If there are repeated heights, this strategy still works as it tries to maximize difference overall.
- Edge cases such as all blocks having the same height result in zero calories except initial jump from ground.
- This approach is efficient, requires only sorting and linear iteration, making it feasible for large input sizes.
12. **Summary:**
- The problem reduces to maximizing sum of squared differences in a permutation of heights.
- Exact combinatorial optimization is infeasible due to input size.
- Use intuition about maximizing jumps between high and low heights.
- Sort the array and create two zig-zag orderings starting from smallest and largest blocks.
- Calculate total calories for both and return the maximum.
- This heuristic is efficient and effective for maximizing calorie burn in this problem.
By thinking in this direction—leveraging the squared difference structure, the importance of alternating between extremes, and starting jump from ground height 0—you can design an efficient and robust solution that provides the maximum possible calorie burn during the exercise.
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