What is the time complexity of Maximum Total Damage With Spell Casting?

The optimal solution has a time complexity of O(n).

What data structures are used to solve Maximum Total Damage With Spell Casting?

This problem uses Array, Hash Table, Two Pointers, Binary Search, Dynamic Programming, Sorting, Counting. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Maximum Total Damage With Spell Casting?

Insight 1: The problem can be modeled as finding the maximum weight independent set on a line graph where nodes are spell damages and edges connect incompatible damages. Insight 2: Dynamic programming is a suitable technique because the problem exhibits overlapping subproblems (calculating the maximum damage for sub-arrays of spells).

Which companies ask LeetCode 3186. Maximum Total Damage With Spell Casting in interviews?

This problem is frequently asked at Citadel, PhonePe.

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Maximum Total Damage With Spell Casting - LeetCode 3186 Solution

Maximum Total Damage With Spell Casting - Complete Solution Guide

Maximum Total Damage With Spell Casting is LeetCode problem 3186, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

A magician has various spells. You are given an array power , where each element represents the damage of a spell. Multiple spells can have the same damage value. It is a known fact that if a magician decides to cast a spell with a damage of power[i] , they cannot cast any spell with a damage of power[i] - 2 , power[i] - 1 , power[i] + 1 , or power[i] + 2 . Each spell can be cast only once . Return the maximum possible total damage that a magician can cast. Example 1: Input: power = [1,1,3,4] Ou

Detailed Explanation

The problem asks us to find the maximum total damage a magician can cast, given an array 'power' representing the damage of each spell. There's a constraint: if a spell with damage 'power[i]' is cast, spells with damages 'power[i] - 2', 'power[i] - 1', 'power[i] + 1', and 'power[i] + 2' cannot be cast. Each spell can only be cast once. In essence, we need to select a subset of spells maximizing the sum of their damage, subject to the given adjacency constraint.

Solution Approach

The solution uses dynamic programming to determine the maximum damage. It first counts the occurrences of each spell damage and then sorts the unique spell damage values. A DP array 'dp' is created, where dp[i] represents the maximum damage achievable considering spells up to the i-th unique damage value. For each unique damage value, we have two choices: either include it in the solution or skip it. If we include it, we add its total damage (damage * count) to the maximum damage achievable using compatible spells (those with damage less than current damage - 2). If we skip it, the maximum damage remains the same as the previous damage. The maximum of these two choices is stored in dp[i].

Step-by-Step Algorithm

Step 1: Count the occurrences of each spell damage value using a HashMap/Counter.
Step 2: Extract the unique spell damage values from the HashMap/Counter and sort them in ascending order.
Step 3: Initialize a dynamic programming array 'dp' of size equal to the number of unique damage values.
Step 4: Iterate through the sorted unique damage values. For each damage value 'p', determine the index 'j' of the largest damage value that is strictly less than 'p - 2'. This represents the last 'compatible' power.
Step 5: Calculate the potential damage if 'p' is included: (p * count[p]) + dp[j - 1] (if j > 0, otherwise 0).
Step 6: Calculate the potential damage if 'p' is excluded: dp[i - 1] (if i > 0, otherwise 0).
Step 7: Store the maximum of the two potential damages in dp[i].
Step 8: After iterating through all unique damage values, the last element of 'dp' contains the maximum total damage.

Key Insights

Insight 1: The problem can be modeled as finding the maximum weight independent set on a line graph where nodes are spell damages and edges connect incompatible damages.
Insight 2: Dynamic programming is a suitable technique because the problem exhibits overlapping subproblems (calculating the maximum damage for sub-arrays of spells).
Insight 3: Sorting the spell damages simplifies the dynamic programming process as we can iterate through them in order and consider whether or not to include each spell in our optimal solution.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Array, Hash Table, Two Pointers, Binary Search, Dynamic Programming, Sorting, Counting.

Companies

Asked at: Citadel, PhonePe.

Maximum Total Damage With Spell Casting | LeetCode 3186

Medium

Array Hash Table Two Pointers Binary Search Dynamic Programming Sorting Counting

Time: O(n)Space: O(n)

Solution Code

Available in:

import java.util.*;

class Solution {
    public long maximumTotalDamage(int[] power) {
        Map<Integer, Integer> counts = new HashMap<>();
        for (int p : power) {
            counts.put(p, counts.getOrDefault(p, 0) + 1);
        }

        List<Integer> uniquePowers = new ArrayList<>(counts.keySet());
        Collections.sort(uniquePowers);
        int m = uniquePowers.size();

        if (m == 0) {
            return 0;
        }

        long[] dp = new long[m];
        int j = 0;

        for (int i = 0; i < m; i++) {
            int p = uniquePowers.get(i);

            // Move pointer j to the first element that is not compatible with p
            // A power q is compatible if q < p - 2
            while (j < m && uniquePowers.get(j) < p - 2) {
                j++;
            }

            // Option 1: Take the current power p.
            // The damage would be p * counts[p] plus the max damage from compatible previous powers.
            // The last compatible power is at index j-1, and its max damage is dp[j-1].
            long damageFromP = (long) p * counts.get(p);
            long prevCompatibleDamage = (j > 0) ? dp[j - 1] : 0;
            long damageIfTake = damageFromP + prevCompatibleDamage;

            // Option 2: Skip the current power p.
            // The max damage is the same as the max damage up to the previous power.
            long damageIfSkip = (i > 0) ? dp[i - 1] : 0;

            dp[i] = Math.max(damageIfTake, damageIfSkip);
        }

        return dp[m - 1];
    }
}

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Problem Info

Description

Problem 3186: Maximum Total Damage With Spell Casting - Detailed explanation and solution approaches.

Topics

Array Hash Table Two Pointers Binary Search Dynamic Programming Sorting Counting

Companies

CitadelPhonePe

Stats

Difficulty:Medium

Languages:4

Companies:2

Problem #:3186

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

8 steps

1
Step 1: Count the occurrences of each spell damage value using a HashMap/Counter.
2
Step 2: Extract the unique spell damage values from the HashMap/Counter and sort them in ascending order.
3
Step 3: Initialize a dynamic programming array 'dp' of size equal to the number of unique damage values.
4
Step 4: Iterate through the sorted unique damage values. For each damage value 'p', determine the index 'j' of the largest damage value that is strictly less than 'p - 2'. This represents the last 'compatible' power.
5
Step 5: Calculate the potential damage if 'p' is included: (p * count[p]) + dp[j - 1] (if j > 0, otherwise 0).
6
Step 6: Calculate the potential damage if 'p' is excluded: dp[i - 1] (if i > 0, otherwise 0).
7
Step 7: Store the maximum of the two potential damages in dp[i].
8
Step 8: After iterating through all unique damage values, the last element of 'dp' contains the maximum total damage.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is dominated by the sorting operation. The space complexity arises from storing the counts of each power, the unique powers, and the dynamic programming table, all of which can be at most proportional to the input size.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Recursion with Memoization

A recursive approach with memoization can also be used. It explores all possible combinations of spells, memoizing the results to avoid redundant calculations.

Time: O(n log n)Space: O(n)

Trade-offs: Recursion with memoization can be more intuitive for some, but it might incur a higher overhead due to function calls. The iterative DP approach is generally more efficient.

✨

Key Insights

1Insight 1: The problem can be modeled as finding the maximum weight independent set on a line graph where nodes are spell damages and edges connect incompatible damages.
2Insight 2: Dynamic programming is a suitable technique because the problem exhibits overlapping subproblems (calculating the maximum damage for sub-arrays of spells).
3Insight 3: Sorting the spell damages simplifies the dynamic programming process as we can iterate through them in order and consider whether or not to include each spell in our optimal solution.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Two Pointers

Using two indices to traverse data from different positions simultaneously.

Binary Search

Halving the search space each step. Like finding a word in a dictionary!

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Sorting

Arranging elements in a specific order (ascending/descending).

Edge Cases

Edge case 1: Empty input array. The solution correctly returns 0 as there are no spells to cast.
Edge case 2: Input array with only one element. The solution correctly returns the value of that element.
Edge case 3: Input array with multiple identical elements. The counting using hashmap ensures the damage from those spells are all counted.
Edge case 4: Input with consecutive values, where none of them can be picked together. Example: [1,2,3,4,5]. DP ensures the right subset with max damage is returned.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to sort the unique powers. Without sorting, the DP approach would not work correctly.

✗Common mistake 2: Incorrectly calculating the damage if a power is included, i.e. not considering the count of that damage.

✗Common mistake 3: Using the wrong condition for skipping spells. The constraint is that a spell with damage 'power[i]' prevents casting spells with damages 'power[i] - 2', 'power[i] - 1', 'power[i] + 1', and 'power[i] + 2'.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Resource allocation under constraints. Imagine scheduling tasks on a processor where some tasks cannot be run concurrently due to resource conflicts. The 'power' array could represent the value of each task, and the constraint could represent the conflicts. The goal is to maximize the total value of scheduled tasks.

🏦Real-world scenario 2: Selecting compatible projects. A company needs to select a set of projects from a list, but some projects are incompatible (e.g., they require the same resources or target the same market). The 'power' array could represent the profitability of each project, and the constraint could represent the incompatibility. The goal is to maximize the total profit from the selected projects.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Medium problems are the bread and butter of technical interviews. Practice explaining your approach before coding.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: October 5, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 6

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What are the trade-offs of using a hash table?

Hash tables provide O(1) average-case lookup, insertion, and deletion, making them ideal for frequency counting and duplicate detection. However, they use O(n) extra space and may degrade to O(n) time in worst-case scenarios with hash collisions. Always consider if the space trade-off is acceptable.

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Maximum Total Damage With Spell Casting"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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