Sum of Total Strength of Wizards

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class Solution:
    def totalStrength(self, strength: List[int]) -> int:
        mod, n = 10 ** 9 + 7, len(strength)

        # Get the first index of the non-larger value to strength[i]'s right.
        right_index = [n] * n
        stack = []
        for i in range(n):
            while stack and strength[stack[-1]] >= strength[i]:
                right_index[stack.pop()] = i
            stack.append(i)

        # Get the first index of the smaller value to strength[i]'s left.
        left_index = [-1] * n
        stack = []
        for i in range(n - 1, -1, -1):
            while stack and strength[stack[-1]] > strength[i]:
                left_index[stack.pop()] = i
            stack.append(i)

        # prefix sum of the prefix sum array of strength.
        presum_of_presum =  list(accumulate(accumulate(strength, initial = 0), initial = 0))
        answer = 0
        # For each element in strength, we get the value of R_term - L_term.
        for i in range(n):
            # Get the left index and the right index.
            left_bound = left_index[i]
            right_bound = right_index[i]

            # Get the left_count and right_count (marked as L and R in the previous slides)
            left_count = i - left_bound
            right_count = right_bound - i

            # Get positive presum and the negative presum.
            neg_presum = (presum_of_presum[i + 1] - presum_of_presum[i - left_count + 1]) % mod
            pos_presum = (presum_of_presum[i + right_count + 1] - presum_of_presum[i + 1]) % mod

            # The total strength of all subarrays that have strength[i] as the minimum.
            answer += strength[i] * (pos_presum * left_count - neg_presum * right_count)
            answer %= mod

        return answer

10 Prerequisite LeetCode Problems

This involves graph theory, shortest path, and dynamic programming. Here are 10 problems to prepare for it:

“Number of Islands” (LeetCode Problem #200): This is a fundamental problem in graph theory, which gives you practice on exploring a grid.
“Clone Graph” (LeetCode Problem #133): This problem involves deep copying of a graph, which can help understand the data structure.
“Course Schedule” (LeetCode Problem #207): This problem introduces topological sort in the context of scheduling and detecting cycles.
“Floyd Warshall” (LeetCode Problem #741): It gives you practice on implementing the Floyd Warshall algorithm to find shortest paths in a graph.
“Bellman-Ford Algorithm” (LeetCode Problem #743): This problem can help you learn and practice the Bellman-Ford algorithm, another way to find shortest paths.
“Dijkstra’s Algorithm” (LeetCode Problem #743): Here you can practice Dijkstra’s algorithm, a popular shortest path algorithm for graphs with non-negative weights.
“Longest Increasing Path in a Matrix” (LeetCode Problem #329): This problem requires dynamic programming and a form of depth-first search in a grid.
“Unique Paths” (LeetCode Problem #62): A classic dynamic programming problem, which helps to understand the concept of path finding.
“Minimum Path Sum” (LeetCode Problem #64): This problem builds on the concepts in “Unique Paths”, adding the complexity of dealing with path costs.
“Word Break” (LeetCode Problem #139): A classic problem in dynamic programming involving partitioning a sequence, which can help with reasoning about the “strengths” of wizards.

Problem Classification

Problem Statement: As the ruler of a kingdom, you have an army of wizards at your command. You are given a 0-indexed integer array strength, where strength[i] denotes the strength of the ith wizard. For a contiguous group of wizards (i.e. the wizards’ strengths form a subarray of strength), the total strength is defined as the product of the following two values: The strength of the weakest wizard in the group. The total of all the individual strengths of the wizards in the group. Return the sum of the total strengths of all contiguous groups of wizards. Since the answer may be very large, return it modulo 109 + 7. A subarray is a contiguous non-empty sequence of elements within an array.

Example 1:

Input: strength = [1,3,1,2] Output: 44 Explanation: The following are all the contiguous groups of wizards:

[1] from [1,3,1,2] has a total strength of min([1]) * sum([1]) = 1 * 1 = 1
[3] from [1,3,1,2] has a total strength of min([3]) * sum([3]) = 3 * 3 = 9
[1] from [1,3,1,2] has a total strength of min([1]) * sum([1]) = 1 * 1 = 1
[2] from [1,3,1,2] has a total strength of min([2]) * sum([2]) = 2 * 2 = 4
[1,3] from [1,3,1,2] has a total strength of min([1,3]) * sum([1,3]) = 1 * 4 = 4
[3,1] from [1,3,1,2] has a total strength of min([3,1]) * sum([3,1]) = 1 * 4 = 4
[1,2] from [1,3,1,2] has a total strength of min([1,2]) * sum([1,2]) = 1 * 3 = 3
[1,3,1] from [1,3,1,2] has a total strength of min([1,3,1]) * sum([1,3,1]) = 1 * 5 = 5
[3,1,2] from [1,3,1,2] has a total strength of min([3,1,2]) * sum([3,1,2]) = 1 * 6 = 6
[1,3,1,2] from [1,3,1,2] has a total strength of min([1,3,1,2]) * sum([1,3,1,2]) = 1 * 7 = 7 The sum of all the total strengths is 1 + 9 + 1 + 4 + 4 + 4 + 3 + 5 + 6 + 7 = 44. Example 2:

Input: strength = [5,4,6] Output: 213 Explanation: The following are all the contiguous groups of wizards:

[5] from [5,4,6] has a total strength of min([5]) * sum([5]) = 5 * 5 = 25
[4] from [5,4,6] has a total strength of min([4]) * sum([4]) = 4 * 4 = 16
[6] from [5,4,6] has a total strength of min([6]) * sum([6]) = 6 * 6 = 36
[5,4] from [5,4,6] has a total strength of min([5,4]) * sum([5,4]) = 4 * 9 = 36
[4,6] from [5,4,6] has a total strength of min([4,6]) * sum([4,6]) = 4 * 10 = 40
[5,4,6] from [5,4,6] has a total strength of min([5,4,6]) * sum([5,4,6]) = 4 * 15 = 60 The sum of all the total strengths is 25 + 16 + 36 + 36 + 40 + 60 = 213.

Constraints:

1 <= strength.length <= 105 1 <= strength[i] <= 109

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.