Maximum Sum Increasing Subsequence
Detailed guide and Python implementation for the 'Maximum Sum Increasing Subsequence' problem.
1. Concept Overview
The 'Maximum Sum Increasing Subsequence' problem is a key challenge in the Dynamic Programming section.
This implementation focuses on hard-level logic in Python.
We prioritize technical accuracy and code readability in our provided solutions.
2. Real-World Applications
3. Visual Intuition
Visualizing the logic flow for Maximum Sum Increasing Subsequence.
4. Prerequisites
5. Step-by-Step Thinking
1. Understand the problem
Read the problem statement for Maximum Sum Increasing Subsequence carefully.
2. Formulate brute force
Draft a simple iterative solution.
3. Identify inefficiency
Look for redundant calculations.
4. Optimize search path
Use hashing or sorting to speed up the process.
5. Final Implementation
Clean up the code for production standards.
Problem Statement
Write a function max_sum_is(arr) that returns the maximum sum of any increasing subsequence in an array of integers arr.
- •1 <= len(arr) <= 1000
- •-10^4 <= arr[i] <= 10^4
Examples
max_sum_is([1, 101, 2, 3, 100, 4, 5])
106
The increasing subsequence with the maximum sum is [1, 2, 3, 100] which sums to 106.
max_sum_is([3, 4, 5, 10])
22
The increasing subsequence is the array itself, summing to 22.
Need a Hint?
Edge Cases to Watch
- Empty input structures
- Single element inputs
- Large numerical bounds
Ready to Solve?
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Interview Insights & Variations
Complexity Analysis Breakdown
Why Time: Directly evaluates all possibilities.
Why Space: Uses standard local memory.
Why Time: Optimized paths reduce total operations.
Why Space: May trade memory for speed.
Optimized Solution Python Code
Optimized Solution Python Code
def msis_opt(arr):
n = len(arr); msis = arr[:]
for i in range(1, n):
for j in range(i):
if arr[i] > arr[j] and msis[i] < msis[j] + arr[i]:
msis[i] = msis[j] + arr[i]
return max(msis)Brute Force Code (Spoiler Guarded)
Brute Force Code (Spoiler Guarded)
def msis_brute(arr, n, i, prev):
if i == n: return 0
excl = msis_brute(arr, n, i + 1, prev)
incl = 0
if arr[i] > prev: incl = arr[i] + msis_brute(arr, n, i + 1, arr[i])
return max(incl, excl)Algorithm Pattern Checklist
When dealing with Dynamic Programming data patterns.
Core Prerequisites
Revision Key Notes
Common Mistakes & Pitfalls
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